I'm looking at this IAU WGCCRE Report: http://aa.usno.navy.mil/publications/reports/Archinaletal2011a.pdf

It gives expressions for Earth orientation relative to the International Celestial Reference Frame in terms of right ascension and declination of the north pole.

The numbers say that the declination of the Earth rotation axis starts from 90 degrees and decreases by 0.557 degrees every century:

δ0 = 90.00 − 0.557T

Which means that in about 16000 years (and 16000 years ago) Earth rotation axis will lay in ecliptic plane. It doesn't make any sense to me.

Is it because this model should be applied on short time scales only? I didn't find any information on the time range for which the model is adequate.

Update: my question is not about how Earth actually rotates, but about the model and how it should be used.

The values

$$alpha_0 = 0.00 - 0.641 T$$ $$delta_0 = 90.00 - 0.557 T$$

provide a *first-order estimate* of the movement of the right ascension ($alpha_0$) and declination ($delta_0$) of the direction of the Earth's north pole for short periods of time after the epoch, expressed to three digits of precision only. $T$ represents the time after epoch expressed in units of Julian centuries (36525 days). The centuries unit was chosen to make the values of order unity, *not because* the values are meant to be used to extrapolate for centuries. You should be thinking of using this for a period of years, *until the next update is published*.

This is because the motion sets off in that direction, but will move in approximately a circle about the pole of the ecliptic, and in about 26,000 years return to roughly the same direction. Watch Steven Senders' snappy video.

This movie was created with Blender and is used in the Spitz Fulldome Curriculum for the SciDome planetariums around the world.

Precession is the wobble of the Earth which makes the poles shift position over ~26,000 years as well as the position of the Vernal and Autumnal equinoxes in the sky. This movie illustrates the movement.

Hopefully this movie gives a little more meaning to the Dawning of the Age of Aquarius.

https://www.spitzinc.com/

The Earth's axis precesses around the ecliptic pole due to gravitational interaction between the Earth's tilted equatorial bulge and (mostly) the Sun. There is also a faster and much smaller amplitude nutation.

**below:** Earth's Rotation, Precession and Nutation. From here

**below:** Screenshots from Steven Senders' video Precession of the earth.

## Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2015

This report continues the practice where the IAU Working Group on Cartographic Coordinates and Rotational Elements revises recommendations regarding those topics for the planets, satellites, minor planets, and comets approximately every 3 years. The Working Group has now become a “functional working group” of the IAU, and its membership is open to anyone interested in participating. We describe the procedure for submitting questions about the recommendations given here or the application of these recommendations for creating a new or updated coordinate system for a given body. Regarding body orientation, the following bodies have been updated: Mercury, based on MESSENGER results Mars, along with a refined longitude definition Phobos Deimos (1) Ceres (52) Europa (243) Ida (2867) Šteins Neptune (134340) Pluto and its satellite Charon comets 9P/Tempel 1, 19P/Borrelly, 67P/Churyumov–Gerasimenko, and 103P/Hartley 2, noting that such information is valid only between specific epochs. The special challenges related to mapping 67P/Churyumov–Gerasimenko are also discussed. Approximate expressions for the Earth have been removed in order to avoid confusion, and the low precision series expression for the Moon’s orientation has been removed. The previously online only recommended orientation model for (4) Vesta is repeated with an explanation of how it was updated. Regarding body shape, text has been included to explain the expected uses of such information, and the relevance of the cited uncertainty information. The size of the Sun has been updated, and notation added that the size and the ellipsoidal axes for the Earth and Jupiter have been recommended by an IAU Resolution. The distinction of a reference radius for a body (here, the Moon and Titan) is made between cartographic uses, and for orthoprojection and geophysical uses. The recommended radius for Mercury has been updated based on MESSENGER results. The recommended radius for Titan is returned to its previous value. Size information has been updated for 13 other Saturnian satellites and added for Aegaeon. The sizes of Pluto and Charon have been updated. Size information has been updated for (1) Ceres and given for (16) Psyche and (52) Europa. The size of (25143) Itokawa has been corrected. In addition, the discussion of terminology for the poles (hemispheres) of small bodies has been modified and a discussion on cardinal directions added. Although they continue to be used for planets and their satellites, it is assumed that the planetographic and planetocentric coordinate system definitions do not apply to small bodies. However, planetocentric and planetodetic latitudes and longitudes may be used on such bodies, following the right-hand rule. We repeat our previous recommendations that planning and efforts be made to make controlled cartographic products newly recommend that common formulations should be used for orientation and size continue to recommend that a community consensus be developed for the orientation models of Jupiter and Saturn newly recommend that historical summaries of the coordinate systems for given bodies should be developed, and point out that for planets and satellites planetographic systems have generally been historically preferred over planetocentric systems, and that in cases when planetographic coordinates have been widely used in the past, there is no obvious advantage to switching to the use of planetocentric coordinates. The Working Group also requests community input on the question submitting process, posting of updates to the Working Group website, and on whether recommendations should be made regarding exoplanet coordinate systems.

This is a preview of subscription content, access via your institution.

## IAU (1976) System of Astronomical Constants

The International Astronomical Union at its XVIth General Assembly in Grenoble in 1976, accepted (Resolution No. 1 [1] ) a whole new consistent set of astronomical constants [2] recommended for reduction of astronomical observations, and for computation of ephemerides. It superseded the IAU's previous recommendations of 1964 (see IAU (1964) System of Astronomical Constants), became in effect in the Astronomical Almanac from 1984 onward, and remained in use until the introduction of the IAU (2009) System of Astronomical Constants. In 1994 [3] the IAU recognized that the parameters became outdated, but retained the 1976 set for sake of continuity, but also recommended to start maintaining a set of "current best estimates". [4]

this "sub group for numerical standards" had published a list, which included new constants (like those for relativistic time scales). [5]

The system of constants was prepared [6] by Commission 4 on ephemerides led by P. Kenneth Seidelmann (after whom asteroid 3217 Seidelmann is named).

At the time, a new standard epoch (J2000.0) was accepted followed later [7] [8] by a new reference system with fundamental catalogue (FK5), and expressions for precession of the equinoxes, and in 1979 by new expressions for the relation between Universal Time and sidereal time, [9] [10] [11] and in 1979 and 1980 by a theory of nutation. [12] [13] There were no reliable rotation elements for most planets, [2] [6] but a joint working group on Cartographic Coordinates and Rotational Elements was installed to compile recommended values. [14] [15]

## IAU Rotation Models

The IAU Working Group on Cartographic Coordinates and Rotational Elements (WGCCRE) is the keeper of official models that describe the cartographic coordinates and rotational elements of planetary bodies (such as the Earth, satellites, minor planets, and comets). Periodically, they release a report containing the coefficients to compute body orientations, based on the latest data available. These coefficients allow one to compute the rotation matrix from the ICRF frame to a body-fixed frame (for example, IAU Earth) by giving the direction of the pole vector and the prime meridian location as functions of time. An example Fortran module illustrating this for the IAU Earth frame is given below. The coefficients are taken from the 2009 IAU report [Reference 1]. Note that the IAU models are also available in SPICE (as “IAU_EARTH”, “IAU_MOON”, etc.). For Earth, the IAU model is not suitable for use in applications that require the highest possible accuracy (for that a more complex model would be necessary), but is quite acceptable for many applications.

### References

#### 1 thought on &ldquo IAU Rotation Models &rdquo

This is interesting. I am from SolarReserve We use the JPL ephemeris amd the Naval Observatory NOVAS software for calculating the sun position in our Solar Plants. We may be able to take advantage of some of these posts and contribute some of our own.

### Leave a Reply Cancel reply

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## IAU rotation model - Astronomy

*Written by Sahar Mohy-Ud-Din, OAD fellow*

**What is a Logic Model?**

A logic model can be defined as:

“A systematic and visual way to present and share your understanding of the relationships among the resources you have to operate your programme, the activities you plan to do and the changes/results you hope to achieve,” (Kellogg Foundation, 2004: 1)

OR

“The model describes logical linkages among program resources, activities, outputs, audiences, and short-, intermediate-, and long-term outcomes related to a specific problem or situation.” (McCawley, n.d: 1)

The purpose of a logic model is to

▫ Clarify underlying beliefs

▫ Challenge assumptions and strengthen logic

▫ Communicate the project’s core purpose to the larger audience

▫ Make a case for financial investment

Logic models are actionable plans, strategies or maps with clear outcomes and explicit steps for solving programme/project problems.

A solid logic model rests on well-reasoned approaches around how and why the stated method of the model will produce the stated outcomes. The best time to develop a logic model is at the planning stage of the project or intervention.

Logic models will vary depending on the project and its complexity. Developing your program’s logic model components will take practice and collaboration. Some programs will involve only a few resources, activities, outputs, and outcomes whilst others will involve many.

**Who benefits from a logic model ?**

i) Evaluators

Who use it to organise data

Who use it to understand how the program works and the logic that guides it

To guide data collection plans (if it is in the logic model)

ii) Stakeholders

By starting with an understanding of program logic, stakeholders are prepared to understand results while the logic underpinning the programme becomes clear and easy to relate to

iii) Evaluator / Stakeholder relationships

Knowledge transfer between parties

Transparency between parties

Assistance with evaluation implementation

Sustainability of the project once the funding has come to an end through the continued implementation of the knowledge/skills gained from the project through the stakeholders

iv) Promote understanding through either

Causal links

o Is X the reason Y happens?

o What are the causes (or at least antecedents) of the problem targeted for intervention?

o Assures that the program addresses important factors involved in the targeted problem thereby providing leanness to the logic underpinning the model making it crisp and concise

Explanatory methods

o Emphasis on investigating reality, through connecting ideas to find meaning and increasing knowledge

o What can help to better understand relationships or describe nature?

**Strengths of the Logic Model**

i) Description

▫ Enables stakeholders to identify their processes, activities and results and to visualise the journey the programme will embark on

ii) Explanation

▫ The Logic Model provides the context specific set of relationships that demonstrates a method of understanding an event for example: How does one understand a Tsunami? A number of variables could be at play such as:

o Weather

o Human error in recognising early warning signs– training, knowledge or individual judgment

o Technology – warning systems, automatic error compensation

o Some combination of all three?

– None of these is “correct” or “incorrect”

– Each provides a different framework for understanding and results in different pathways to policy decisions. One looks for the framework that provides each stakeholder group with the most choice for effective change

iii) Operational

▫ Displays links between resources, activities, and objectives

**Limitations of the Logic Model**

The most common limitations include:

▫ A logic model denotes intention, it is not reality therefore can easily go awry if the underlying logic and assumptions remain inconsistent

▫ It emphasises expected outcomes so unintended outcomes maybe overlooked (positive and negative)

▫ It focuses on positive change – change isn’t always positive

▫ It may simplify the complex nature of causal attribution whereas many factors influence processes and outcomes

▫ It may stifle creativity and spontaneity

**Illustration of a logic model**

A logic model flows from the left to the right moving from the process side to the impact side. It identifies:

**Inputs**

These are the resources available for a program, such as funding, staff, and leadership, expertise, program infrastructure, scientific knowledge and evidence-based strategies, and partnerships.

**Activities**

These are what a program actually does to bring about the intended change,

Such as formation of partnerships for capacity development, referral to services, and the dissemination of prevention messages for healthy families.

**Outputs**

These are the products or direct services resulting from the program activities. Outputs are the direct evidence of implemented activities. Some examples of the outputs of state birth defects surveillance programs might include: improvement in surveillance methodology, dissemination of surveillance information, the number of families linked to services, the number of partnerships channels for referral linkages, and the number of implemented prevention activities. It is important to distinguish early the difference between outputs and outcomes. Outputs relate to what has been done whereas outcomes refer to what difference has been made as a result of the intervention.

**Outcomes**

Identifies the sequence of changes, that is, the results expected to be achieved by the program.

▫ Short-term outcomes

Represent the most immediate effects attributable to a program, such as changes in learning, knowledge, and attitudes. Examples include: knowledge and awareness of malnutrition within the community, and referral and prevention messages through improved dissemination

▫ Intermediate Outcomes

Reflect the changes in actions, such as in behaviours and practices, that are a result of increased knowledge and awareness for example, an increased number of families making more informed decisions around eating and purchasing of food to ensure nutrition within the household

▫ Impact (long term outcomes)

Refer to the conditions that change as a result of actions. Long-term outcomes are what the program is expected to affect, i.e. how have the lives of beneficiaries of the intervention changed (positive and negative change) as a result of the activities of the intervention/project. Examples include a decrease in the number of families reporting malnutrition. These

outcomes are more distant in time, less attributable to the program, and harder to measure.

**Assumptions**

These are facts or conditions you assume to be true. The assumptions that underlie a program’s theory are conditions that are necessary for success, and you believe are true. Your program needs these conditions in order to succeed, but you believe these conditions already exist – they are not something you need to bring about with your program activities. In fact, they are not within your control.

**External Factors**

The environment in which the program exists includes a variety of external factors that can influence the program’s success. External factors include the cultural milieu, the climate, economic structure, housing patterns, demographic patterns, political environment, background and experiences of program participants, media influence, changing policies and priorities. These external factors may have a major influence on the achievement of outcomes. They may affect a variety of things including the following:

• Program implementation

• Participants and recipients

• The speed and degree to which change occurs

• Staffing patterns and resources available

These factors interact with the program. They not only influence the initiative but are influenced by the initiative. A program does not sit in isolation – somehow “outside” or “apart” from its surrounding environment. A program is affected by and affects these external factors.

These are challenges that you foresee that will hinder the implementation of your project. They can come from individuals, organisations or groups.

**Examples of Logic models: OAD Funded Projects**

1. Astronomy for Literacy (AFL)

Astronomy for Literacy (AFL) will work to enhance literacy, numeracy and other foundational skills among up to 1250 struggling junior secondary school students in Sierra Leone. It will do this by adapting and developing high quality curriculum resources across literacy, math and science. The goal will be to use astronomy-related content to teach foundational reading and numeracy skills, while at the same time igniting an interest in astronomy and teaching in an engaging way the core concepts in astronomy that are part of Sierra Leone’s national curriculum for science.

The primary deliverables will be detailed, highly structured lesson plans that enable relatively low-skilled teachers with limited subject knowledge to still deliver high quality learning opportunities. Alongside this, the project will also invest in finding appropriate supplementary digital resources, including videos, presentations and other offline content, and in low-cost tablet devices that will allow groups of students to access this content.

2. Astronomy camp for girls in Abuja, Nigeria

It is believed that gender inequality in Northern Nigeria is promoted by religious and communal customs, which has grave consequences for both the individual and the society, making a girl-child dysfunctional member of the society. This innovative astronomy camp would give the children the first taste of space science and technology. Our target is to select fifty girls from different primary schools in Abuja, Nigeria for a camping exercise. The girls will learn about astronauts and space missions to add to their knowledge of spaceflights and a little bit of mathematics. This is more likely to increase children’s interest in space research and space-based technology. We will run through presentations and exercises between 16:00 and 19:00 to prepare the pupils for night observation session. The night observation session will run from 19:30 to 20:15 and camp closes by 20:30.

## Know Your Planets

During this activity, students play a game and learn the properties of different planets and their relative position in the Solar System.

- Students will be able to describe what the Solar System is.
- Students will be able to describe the properties of different planets and classify them into rocky and gassy.
- Students will be able to name the planets in order of distance from the Sun.

- The teacher discusses with the students about what colour they used for the planets.
- Students have to explain how the colour chosen relates to the planets’ properties that students have learnt over the activity. For example, Venus is very hot so it could be coloured red.
- Ask the students to name the planets in the solar system in order from the Sun.

- Card game PDF (one set per group)
- Planets and Sun model – Universe in a Box or other material
- Coloured pencils (one set per group)
- Scissors (one per student)
- Photos of the Solar System PDF (one set per group)

**Introduction**

The Solar System, in which we live, consists of the Sun as its central star, eight planets with their moons and several dwarf planets. Together with hundreds of thousands of asteroids (boulders from the size of small pebbles to the size of a dwarf planet) and comets, these celestial bodies orbit the Sun.

The Earth is a very special planet among these celestial bodies. It is our home! In order to understand its uniqueness, students need to compare the Earth to the other planets in the Solar System. The Earth is located about 150 million kilometres from the Sun, giving a temperature that is exactly right for liquid water to be present on the surface, unlike on other planets. This proved crucial for the development of life!

The Solar System as a whole is part of the Milky Way galaxy, a collection of about 200 billion stars that are arranged in a spiral, along with gas and dust. Billions of these stars have planets and these, in turn, have moons. This suggests that we are probably not alone in the Milky Way, but the distances between the stars are so big that a visit to another world would be very difficult.

Credit: Wikimedia Common / Nick Risinger / NASA / JPL

Even the star nearest to us, Proxima Centauri, is 4.22 light years (i.e., over 40 trillion kilometres) away from us. This is so distant that a journey there would take generations of human lives.

Credit: UNAWE / C.Provot

Planets that orbit stars other than our Sun are called extrasolar planets or exoplanets for short. Astronomers have already discovered more than 2000 of these exoplanets and regularly discover more. We can categorise the planets of our Solar System into two types: rocky planets, which are nearest to the Sun and have a solid surface, and gas giants, which are farther from the Sun and are more massive and mainly composed of gas. Mercury, Venus, Earth and Mars appear in the former category, and Jupiter, Saturn, Uranus and Neptune make up the latter. Pluto, our formerly outermost planet, has been considered a dwarf planets since 2006. Between Mars and Jupiter is a so-called asteroid belt, which circles the Sun like a ring. It consists of thousands of smaller and larger boulders. The largest of these have their own names, just like the planets. One of them, Ceres, is so large that it is considered a dwarf planet.

**The Planets**

Planets are spherical bodies orbiting a star. They have sufficient mass to have purged their orbits of all larger and smaller boulders thanks to their gravitational pull. Dwarf planets are also spherical and orbit a star, but they have small masses and therefore such weak gravity that they are not capable of attracting smaller boulders in their vicinity. Currently (in 2016) five dwarf planets have been identified: Ceres, Pluto, Haumea, Makemake and Eris. Moons are often spherical as well, depending on their size, but they orbit planets.

Each of the planets in our Solar System has very specific features. We have summarised them in the fact files below. The following rule of thumb is valid in the Solar System: small planets lie close to the Sun and are made of solid material, while large planets are farther away from the Sun and are mainly composed of gas. This is not necessarily true for planets around other stars, some of which have planets like Jupiter much closer to their star than Mercury is to the Sun in our Solar System.

**Rocky Planets**

The four rocky planets (Mercury, Venus, Earth and Mars) are very dense (solid) and comparatively small. Their atmospheres are very thin or non-existent (Mercury), with the exception of that of Venus.

Mercury is the planet nearest to the Sun. It has no atmosphere and its solid surface, like that of our Moon, is covered with many craters. Mercury orbits the Sun once in just 88 days and has no moons. There are severe temperature differences on its surface: 380° C on the side facing the Sun, and -180° C on the night side! This is because day and night shift very slowly on Mercury, because of its slow spin. Also, there is no atmosphere to trap the heat at night.

Credit: NASA

Venus is about as large as the Earth. Carbon dioxide (a greenhouse gas) makes up 99% of its atmosphere, which causes sunlight to get trapped in this mega greenhouse. Whether it is day or night, it is always very hot on Venus: almost 500° C! While the other Solar System planets rotate in the same direction, anticlockwise, Venus rotates backwards, clockwise.

Credit: NASA

Earth is the only planet in the Solar System that has liquid water on its surface, significant amounts of oxygen in the air and moderate temperatures. It orbits the Sun once a year. Its stable axis (inclined 23 degrees) results in seasons. Furthermore, it is the only celestial body on which we have found life so far.

Credit: NASA

Mars is half the size of the Earth. Its reddish colour is caused by iron oxide (rust). It has a very thin atmosphere, which mainly consists of carbon dioxide. One of its special features is its many extinct volcanoes, which reach heights of up to 22,000 metres! Mars has two very small moons and needs about twice as much time as the Earth to orbit the Sun. Like Earth, it also has seasons, as its rotation axis is inclined.

Credit: NASA

The ‘gas giants’ are so-called because they are large compared to other planets and they are composed mostly of gas. They consist of a mighty atmosphere and a relatively small solid core.

Jupiter is the largest planet in our Solar System. Like all giant planets, it mainly consists of gas and has a small solid core and a thin ring system. It has a remarkable red spot on its surface that is twice the size of the Earth! This spot is a huge storm (a cyclone) that has been raging for more than 400 years. It has at least 67 moons (in 2016) and is composed mainly of hydrogen and helium.

Credit: NASA

Saturn is surrounded by large rings and therefore earns its nickname ‘Lord of the rings’. These rings consist of numerous small ice grains. Saturn’s atmosphere has a fairly low density: Saturn is the only planet in the Solar System that could float on water. It has many moons: more than 60 (62 in 2016) and other unnamed ‘moonlets’.

Credit: NASA

Uranus has a few thin rings. Its surface looks very smooth and barely shows any structure. It has 27 known moons (2016).

Credit: NASA

Neptune’s surface has a blue colour, like that of Uranus. White clouds fly over its surface at speeds of over 1000 km/hr. Neptune’s path sometimes crosses the orbit of dwarf planet Pluto. The planet has a thin ring system and 14 known moons (2016).

Credit: NASA

**Dwarf Planet: Pluto**

Pluto is composed of ice and rock. In 2006, astronomers decided that Pluto should no longer be classed as a planet but only a dwarf planet, although it is spherical. Due to its low mass, it cannot attract smaller boulders in its vicinity, as the ‘real’ planets do. Pluto has one larger and two smaller moons. Currently, in 2016, there are 4 other planets considered as dwarf planets that could be added to the game on an equal standing with Pluto: Ceres, Haumea, Makemake, and Eris.

**Our knowledge of the Solar System**

Astronomy research and tools evolve and improve very quickly. Our knowledge about space including our Solar System is extensive but not complete and it is constantly growing. Therefore, data given in the present resource might become incomplete or inaccurate over time. This might be illustrated with Pluto, considered for many years as the ninth planet of the Solar System and today identified as a dwarf planet.

## The Faintest Dwarf Galaxies

Joshua D. Simon

Vol. 57, 2019

### Abstract

The lowest luminosity ( L) Milky Way satellite galaxies represent the extreme lower limit of the galaxy luminosity function. These ultra-faint dwarfs are the oldest, most dark matter–dominated, most metal-poor, and least chemically evolved stellar systems . Read More

### Supplemental Materials

Figure 1: Census of Milky Way satellite galaxies as a function of time. The objects shown here include all spectroscopically confirmed dwarf galaxies as well as those suspected to be dwarfs based on l.

Figure 2: Distribution of Milky Way satellites in absolute magnitude () and half-light radius. Confirmed dwarf galaxies are displayed as dark blue filled circles, and objects suspected to be dwarf gal.

Figure 3: Line-of-sight velocity dispersions of ultra-faint Milky Way satellites as a function of absolute magnitude. Measurements and uncertainties are shown as blue points with error bars, and 90% c.

Figure 4: (a) Dynamical masses of ultra-faint Milky Way satellites as a function of luminosity. (b) Mass-to-light ratios within the half-light radius for ultra-faint Milky Way satellites as a function.

Figure 5: Mean stellar metallicities of Milky Way satellites as a function of absolute magnitude. Confirmed dwarf galaxies are displayed as dark blue filled circles, and objects suspected to be dwarf .

Figure 6: Metallicity distribution function of stars in ultra-faint dwarfs. References for the metallicities shown here are listed in Supplemental Table 1. We note that these data are quite heterogene.

Figure 7: Chemical abundance patterns of stars in UFDs. Shown here are (a) [C/Fe], (b) [Mg/Fe], and (c) [Ba/Fe] ratios as functions of metallicity, respectively. UFD stars are plotted as colored diamo.

Figure 8: Detectability of faint stellar systems as functions of distance, absolute magnitude, and survey depth. The red curve shows the brightness of the 20th brightest star in an object as a functi.

Figure 9: (a) Color–magnitude diagram of Segue 1 (photometry from Muñoz et al. 2018). The shaded blue and pink magnitude regions indicate the approximate depth that can be reached with existing medium.

## 5 Functions

### 5.1 Astronomy functions

- : Calendars. : Ephemerides. : Fundamental args. : Prec nut polar. : Rotation and time. : Space motion. : Star catalogs. : Geodetic geocentric. : Timescales.

#### 5.1.1 Calendars

- The algorithm used is valid from -4800 March 1, but this implementation rejects dates before -4799 January 1.
- The Julian Date is returned in two pieces, in the usual SOFA manner, which is designed to preserve time resolution. The Julian Date is available as a single number by adding djm0 and djm .
- In early eras the conversion is from the &ldquoProleptic Gregorian Calendar&rdquo no account is taken of the date(s) of adoption of the Gregorian Calendar, nor is the AD/BC numbering convention observed.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 12.92 (p604).

- The Julian Date is returned in two pieces, in the usual SOFA manner, which is designed to preserve time resolution. The Julian Date is available as a single number by adding djm0 and djm .

- Lieske, J.H., 1979, Astron.Astrophys. 73, 282.

- The earliest valid date is -68569.5 (-4900 March 1). The largest value accepted is 10^9.
- The Julian Date is apportioned in any convenient way between the arguments dj1 and dj2 . For example, JD = 2450123.7 could be expressed in any of these ways, among others:
- In early eras the conversion is from the &ldquoproleptic Gregorian calendar&rdquo no account is taken of the date(s) of adoption of the Gregorian calendar, nor is the AD/BC numbering convention observed.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 12.92 (p604).

- The Julian Date is apportioned in any convenient way between the arguments dj1 and dj2 . For example, JD = 2450123.7 could be expressed in any of these ways, among others:
- In early eras the conversion is from the &ldquoProleptic Gregorian Calendar&rdquo no account is taken of the date(s) of adoption of the Gregorian Calendar, nor is the AD/BC numbering convention observed.
- Refer to the function iauJd2cal.
- NDP should be 4 or less if internal overflows are to be avoided on machines which use 16-bit integers.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 12.92 (p604).

#### 5.1.2 Ephemerides

- The TDB date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB) = 2450123.7 could be expressed in any of these ways, among others:

The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. However, the accuracy of the result is more likely to be limited by the algorithm itself than the way the date has been expressed. n.b. TT can be used instead of TDB in most applications.

- The date date1 + date2 is in the TDB time scale (in practice TT can be used) and is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. The limited accuracy of the present algorithm is such that any of the methods is satisfactory.

- If an np value outside the range 1-8 is supplied, an error status (function value -1) is returned and the pv vector set to zeroes.
- For np = 3 the result is for the Earth&ndashMoon Barycenter. To obtain the heliocentric position and velocity of the Earth, use instead the SOFA function iauEpv00 .
- On successful return, the array pv contains the following:
The reference frame is equatorial and is with respect to the mean equator and equinox of epoch J2000.0.

- The algorithm is due to J.L. Simon, P. Bretagnon, J. Chapront, M. Chapront-Touze, G. Francou and J. Laskar (Bureau des Longitudes, Paris, France). From comparisons with JPL ephemeris DE102, they quote the following maximum errors over the interval 1800-2050:
Over the interval 1000-3000, they report that the accuracy is no worse than 1.5 times that over 1800-2050. Outside 1000-3000 the accuracy declines.

Comparisons of the present function with the JPL DE200 ephemeris give the following RMS errors over the interval 1960-2025:

Comparisons against DE200 over the interval 1800-2100 gave the following maximum absolute differences. (The results using DE406 were essentially the same.)

- The present SOFA re-implementation of the original Simon et al. Fortran code differs from the original in the following respects:
- C instead of Fortran.
- The date is supplied in two parts.
- The result is returned only in equatorial Cartesian form the ecliptic longitude, latitude and radius vector are not returned.
- The result is in the J2000.0 equatorial frame, not ecliptic.
- More is done in&ndashline: there are fewer calls to subroutines.
- Different error/warning status values are used.
- A different Kepler's&ndashequation&ndashsolver is used (avoiding use of double precision complex).
- Polynomials in t are nested to minimize rounding errors.
- Explicit double constants are used to avoid mixed&ndashmode expressions.

None of the above changes affects the result significantly.

- The returned status indicates the most serious condition encountered during execution of the function. Illegal np is considered the most serious, overriding failure to converge, which in turn takes precedence over the remote date warning.

- Simon, J.L, Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., and Laskar, J., Astron. Astrophys. 282, 663 (1994).

#### 5.1.3 Fundamental args

- Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference.
- The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683.

- Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference.
- The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683.
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.

- Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference.
- The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683.

- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.
- The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.

- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.

- The expression used is as adopted in IERS Conventions (2003) and is adapted from Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- The expression used is as adopted in IERS Conventions (2003). It is taken from Kinoshita & Souchay (1990) and comes originally from Lieske et al. (1977).

- Kinoshita, H. and Souchay J. 1990, Celest.Mech. and Dyn.Astron. 48, 187.
- Lieske, J.H., Lederle, T., Fricke, W. & Morando, B. 1977, Astron.Astrophys. 58, 1-16.
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.

- The expression used is as adopted in IERS Conventions (2003) and is adapted from Simon et al. (1994).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111.

#### 5.1.4 Prec nut polar

- The frame bias corrections in longitude and obliquity (radians) are required in order to correct for the offset between the GCRS pole and the mean J2000.0 pole. They define, with respect to the GCRS frame, a J2000.0 mean pole that is consistent with the rest of the IAU 2000A precession&ndashnutation model.
- In addition to the displacement of the pole, the complete description of the frame bias requires also an offset in right ascension. This is not part of the IAU 2000A model, and is from Chapront et al. (2002). It is returned in radians.
- This is a supplemented implementation of one aspect of the IAU 2000A nutation model, formally adopted by the IAU General Assembly in 2000, namely MHB2000 (Mathews et al. 2002).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience.

- The matrix rb transforms vectors from GCRS to mean J2000.0 by applying frame bias.
- The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession.
- The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp * rb .
- It is permissible to reuse the same array in the returned arguments. The arrays are filled in the order given.

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience.

- The matrix rb transforms vectors from GCRS to mean J2000.0 by applying frame bias.
- The matrix rp transforms vectors from mean J2000.0 to mean of date by applying precession.
- The matrix rbp transforms vectors from GCRS to mean of date by applying frame bias then precession. It is the product rp * rb .

- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855.
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.

- The matrix rbpn transforms vectors from GCRS to true equator (and CIO or equinox) of date, and therefore the Celestial Intermediate Pole unit vector is the bottom row of the matrix.
- The arguments x , y are components of the Celestial Intermediate Pole unit vector in the Geocentric Celestial Reference System.

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience.

- The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix.

- A faster, but slightly less accurate result (about 1 mas), can be obtained by using instead the iauC2i00b function.

- The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix.

- The present function is faster, but slightly less accurate (about 1 mas), than the iauC2i00a function.

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix.

- McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix rbpn transforms vectors from GCRS to true equator (and CIO or equinox) of date. Only the CIP (bottom row) is used.
- The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates:
- Although its name does not include "00", This function is in fact specific to the IAU 2000 models.

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The Celestial Intermediate Pole coordinates are the x , y components of the unit vector in the Geocentric Celestial Reference System.
- The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates:
- Although its name does not include 00 , this function is in fact specific to the IAU 2000 models.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- The Celestial Intermediate Pole coordinates are the x , y components of the unit vector in the Geocentric Celestial Reference System.
- The CIO locator s (in radians) positions the Celestial Intermediate Origin on the equator of the CIP.
- The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates:

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004)

- The TT and UT1 dates tta + ttb and uta + utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb . For example, JD(UT1) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta , utb , the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians to 0 and 90 deg west respectively.
- The matrix rc2t transforms from celestial to terrestrial coordinates:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RC2I is the celestial&ndashto&ndashintermediate matrix, ERA is the Earth rotation angle and RPOM is the polar motion matrix.

- A faster, but slightly less accurate result (about 1 mas), can be obtained by using instead the iauC2t00b function.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004)

- The TT and UT1 dates tta + ttb and uta + utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb . For example, JD(UT1) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta , utb , the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians to 0 and 90 deg west respectively.
- The matrix rc2t transforms from celestial to terrestrial coordinates:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RC2I is the celestial&ndashto&ndashintermediate matrix, ERA is the Earth rotation angle and RPOM is the polar motion matrix.

- The present function is faster, but slightly less accurate (about 1 mas), than the iauC2t00a function.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004)

- The TT and UT1 dates tta + ttb and uta + utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb . For example, JD(UT1) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta , utb , the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians to 0 and 90 deg west respectively.
- The matrix rc2t transforms from celestial to terrestrial coordinates:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RC2I is the celestial&ndashto&ndashintermediate matrix, ERA is the Earth rotation angle and RPOM is the polar motion matrix.

- McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG

- This function constructs the rotation matrix that transforms vectors in the celestial system into vectors in the terrestrial system. It does so starting from precomputed components, namely the matrix which rotates from celestial coordinates to the intermediate frame, the Earth rotation angle and the polar motion matrix. One use of the present function is when generating a series of celestial&ndashto&ndashterrestrial matrices where only the Earth Rotation Angle changes, avoiding the considerable overhead of recomputing the precession&ndashnutation more often than necessary to achieve given accuracy objectives.
- The relationship between the arguments is as follows:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003).

- McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG

- This function constructs the rotation matrix that transforms vectors in the celestial system into vectors in the terrestrial system. It does so starting from precomputed components, namely the matrix which rotates from celestial coordinates to the true equator and equinox of date, the Greenwich Apparent Sidereal Time and the polar motion matrix. One use of the present function is when generating a series of celestial&ndashto&ndashterrestrial matrices where only the Sidereal Time changes, avoiding the considerable overhead of recomputing the precession-nutation more often than necessary to achieve given accuracy objectives.
- The relationship between the arguments is as follows:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003).

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004)

- The TT and UT1 dates tta + ttb and uta + utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb . For example, JD(UT1) = 2450123.7 could be expressed in any of these ways, among others:
- The caller is responsible for providing the nutation components they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date. For high&ndashaccuracy applications, free core nutation should be included as well as any other relevant corrections to the position of the CIP.
- The matrix rc2t transforms from celestial to terrestrial coordinates:
where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RBPN is the bias&ndashprecession&ndashnutation matrix, GST is the Greenwich (apparent) Sidereal Time and RPOM is the polar motion matrix.

- Although its name does not include 00 , this function is in fact specific to the IAU 2000 models.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- The TT and UT1 dates tta + ttb and uta + utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb . For example, JD(UT1) = 2450123.7 could be expressed in any o these ways, among others:
- The Celestial Intermediate Pole coordinates are the x, y components of the unit vector in the Geocentric Celestial Reference System.
- The matrix rc2t transforms from celestial to terrestrial coordinates:
- Although its name does not include 00 , this function is in fact specific to the IAU 2000 models.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The equation of the origins is the distance between the true equinox and the celestial intermediate origin and, equivalently, the difference between Earth rotation angle and Greenwich apparent sidereal time (ERA-GST). It comprises the precession (since J2000.0) in right ascension plus the equation of the equinoxes (including the small correction terms).

- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.

- The equation of the origins is the distance between the true equinox and the celestial intermediate origin and, equivalently, the difference between Earth rotation angle and Greenwich apparent sidereal time (ERA-GST). It comprises the precession (since J2000.0) in right ascension plus the equation of the equinoxes (including the small correction terms).
- The algorithm is from Wallace & Capitaine (2006).

- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855
- Wallace, P. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.

- Naming the following points:
the four Fukushima&ndashWilliams angles are as follows:

- The matrix representing the combined effects of frame bias, precession and nutation is:
- Three different matrices can be constructed, depending on the supplied angles:
- To obtain the nutation x precession x frame bias matrix, generate the four precession angles, generate the nutation components and add them to the psi_bar and epsilon_A angles, and call the present function.
- To obtain the precession x frame bias matrix, generate the four precession angles and call the present function.
- To obtain the frame bias matrix, generate the four precession angles for date J2000.0 and call the present function.

The nutation&ndashonly and precession&ndashonly matrices can if necessary be obtained by combining these three appropriately.

- Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351.

- Naming the following points:
- the four Fukushima&ndashWilliams angles are as follows:
- The matrix representing the combined effects of frame bias, precession and nutation is:
X,Y are elements (3,1) and (3,2) of the matrix.

- Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351.

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(true) = rmatn * V(mean) , where the p -vector V(true) is with respect to the true equatorial triad of date and the p -vector V(mean) is with respect to the mean equatorial triad of date.
- A faster, but slightly less accurate result (about 1 mas), can be obtained by using instead the iauNum00b function.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(true) = rmatn * V(mean) , where the p -vector V(true) is with respect to the true equatorial triad of date and the p -vector V(mean) is with respect to the mean equatorial triad of date.
- The present function is faster, but slightly less accurate (about 1 mas), than the iauNum00a function.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(true) = rmatn * V(mean) , where the p -vector V(true) is with respect to the true equatorial triad of date and the p -vector V(mean) is with respect to the mean equatorial triad of date.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114).

- The supplied mean obliquity epsa , must be consistent with the precession&ndashnutation models from which dpsi and deps were obtained.
- The caller is responsible for providing the nutation components they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The nutation components in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. The obliquity at J2000.0 is assumed to be the Lieske et al. (1977) value of 84381.448 arcsec.
Both the luni&ndashsolar and planetary nutations are included. The latter are due to direct planetary nutations and the perturbations of the lunar and terrestrial orbits.

- The function computes the MHB2000 nutation series with the associated corrections for planetary nutations. It is an implementation of the nutation part of the IAU 2000A precession&ndashnutation model, formally adopted by the IAU General Assembly in 2000, namely MHB2000 (Mathews et al. 2002), but with the free core nutation (FCN &mdash see Note 4) omitted.
- The full MHB2000 model also contains contributions to the nutations in longitude and obliquity due to the free-excitation of the free&ndashcore&ndashnutation during the period 1979-2000. These FCN terms, which are time&ndashdependent and unpredictable, are NOT included in the present function and, if required, must be independently computed. With the FCN corrections included, the present function delivers a pole which is at current epochs accurate to a few hundred microarcseconds. The omission of FCN introduces further errors of about that size.
- The present function provides classical nutation. The MHB2000 algorithm, from which it is adapted, deals also with (i) the offsets between the GCRS and mean poles and (ii) the adjustments in longitude and obliquity due to the changed precession rates. These additional functions, namely frame bias and precession adjustments, are supported by the SOFA functions iauBi00 and iauPr00.
- The MHB2000 algorithm also provides &ldquototal&rdquo nutations, comprising the arithmetic sum of the frame bias, precession adjustments, luni-solar nutation and planetary nutation. These total nutations can be used in combination with an existing IAU 1976 precession implementation, such as iauPmat76, to deliver GCRS- to-true predictions of sub-mas accuracy at current dates. However, there are three shortcomings in the MHB2000 model that must be taken into account if more accurate or definitive results are required (see Wallace 2002):
- The MHB2000 total nutations are simply arithmetic sums, yet in reality the various components are successive Euler rotations. This slight lack of rigor leads to cross terms that exceed 1 mas after a century. The rigorous procedure is to form the GCRS-to-true rotation matrix by applying the bias, precession and nutation in that order.
- Although the precession adjustments are stated to be with respect to Lieske et al. (1977), the MHB2000 model does not specify which set of Euler angles are to be used and how the adjustments are to be applied. The most literal and straightforward procedure is to adopt the 4-rotation epsilon_0, psi_A, omega_A, xi_A option, and to add DPSIPR to psi_A and DEPSPR to both omega_A and eps_A.
- The MHB2000 model predates the determination by Chapront et al. (2002) of a 14.6 mas displacement between the J2000.0 mean equinox and the origin of the ICRS frame. It should, however, be noted that neglecting this displacement when calculating star coordinates does not lead to a 14.6 mas change in right ascension, only a small second- order distortion in the pattern of the precession-nutation effect.

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The nutation components in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. The obliquity at J2000.0 is assumed to be the Lieske et al. (1977) value of 84381.448 arcsec. (The errors that result from using this function with the IAU 2006 value of 84381.406 arcsec can be neglected.)
The nutation model consists only of luni&ndashsolar terms, but includes also a fixed offset which compensates for certain long&ndashperiod planetary terms (Note 7).

- This function is an implementation of the IAU 2000B abridged nutation model formally adopted by the IAU General Assembly in 2000. The function computes the MHB_2000_SHORT luni&ndashsolar nutation series (Luzum 2001), but without the associated corrections for the precession rate adjustments and the offset between the GCRS and J2000.0 mean poles.
- The full IAU 2000A (MHB2000) nutation model contains nearly 1400 terms. The IAU 2000B model (McCarthy & Luzum 2003) contains only 77 terms, plus additional simplifications, yet still delivers results of 1 mas accuracy at present epochs. This combination of accuracy and size makes the IAU 2000B abridged nutation model suitable for most practical applications.
The function delivers a pole accurate to 1 mas from 1900 to 2100 (usually better than 1 mas, very occasionally just outside 1 mas). The full IAU 2000A model, which is implemented in the function iauNut00a (q.v.), delivers considerably greater accuracy at current dates however, to realize this improved accuracy, corrections for the essentially unpredictable free-core-nutation (FCN) must also be included.

- The present function provides classical nutation. The MHB_2000_SHORT algorithm, from which it is adapted, deals also with (i) the offsets between the GCRS and mean poles and (ii) the adjustments in longitude and obliquity due to the changed precession rates. These additional functions, namely frame bias and precession adjustments, are supported by the SOFA functions iauBi00 and iauPr00.
- The MHB_2000_SHORT algorithm also provides &ldquototal&rdquo nutations, comprising the arithmetic sum of the frame bias, precession adjustments, and nutation (luni-solar + planetary). These total nutations can be used in combination with an existing IAU 1976 precession implementation, such as iauPmat76 , to deliver GCRS&ndashto&ndashtrue predictions of mas accuracy at current epochs. However, for symmetry with the iauNut00a function (q.v. for the reasons), the SOFA functions do not generate the &ldquototal nutations&rdquo directly. Should they be required, they could of course easily be generated by calling iauBi00 , iauPr00 and the present function and adding the results.
- The IAU 2000B model includes &ldquoplanetary bias&rdquo terms that are fixed in size but compensate for long-period nutations. The amplitudes quoted in McCarthy & Luzum (2003), namely Dpsi = -1.5835 mas and Depsilon = +1.6339 mas , are optimized for the &ldquototal nutations&rdquo method described in Note 6. The Luzum (2001) values used in this SOFA implementation, namely -0.135 mas and +0.388 mas, are optimized for the &ldquorigorous&rdquo method, where frame bias, precession and nutation are applied separately and in that order. During the interval 1995-2050, the SOFA implementation delivers a maximum error of 1.001 mas (not including FCN).

- Lieske, J.H., Lederle, T., Fricke, W., Morando, B., &ldquoExpressions for the precession quantities based upon the IAU /1976/ system of astronomical constants&rdquo, Astron.Astrophys. 58, 1-2, 1-16. (1977).
- Luzum, B., private communication, 2001 (Fortran code MHB_2000_SHORT).
- McCarthy, D.D. & Luzum, B.J., &ldquoAn abridged model of the precession&ndashnutation of the celestial pole&rdquo, Cel.Mech.Dyn.Astron. 85, 37-49 (2003).
- Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J., Astron.Astrophys. 282, 663-683 (1994).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The nutation components in longitude and obliquity are in radians and with respect to the mean equinox and ecliptic of date, IAU 2006 precession model (Hilton et al. 2006, Capitaine et al. 2005).
- The function first computes the IAU 2000A nutation, then applies adjustments for (i) the consequences of the change in obliquity from the IAU 1980 ecliptic to the IAU 2006 ecliptic and (ii) the secular variation in the Earth's dynamical form factor J2.
- The present function provides classical nutation, complementing the IAU 2000 frame bias and IAU 2006 precession. It delivers a pole which is at current epochs accurate to a few tens of microarcseconds, apart from the free core nutation.

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The nutation components are with respect to the ecliptic of date.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222 (p111).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The result is the angle between the ecliptic and mean equator of date date1 + date2 .
- Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The result is the angle between the ecliptic and mean equator of date date1 + date2 .
- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Expression 3.222-1 (p114).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- This function returns the set of equinox based angles for the Capitaine et al. &ldquoP03&rdquo precession theory, adopted by the IAU in 2006. The angles are set out in Table 1 of Hilton et al. (2006):
The returned values are all radians.

- Hilton et al. (2006) Table 1 also contains angles that depend on models distinct from the P03 precession theory itself, namely the IAU 2000A frame bias and nutation. The quoted polynomials are used in other SOFA functions: iauXy06 Contains the polynomial parts of the X and Y series.

iauS06 Contains the polynomial part of the s+XY/2 series.

iauPfw06 Implements the series for the Fukushima&ndashWilliams angles that are with respect to the GCRS pole (i.e. the variants that include frame bias). - The IAU resolution stipulated that the choice of parameterization was left to the user, and so an IAU compliant precession implementation can be constructed using various combinations of the angles returned by the present function.
- The parameterization used by SOFA is the version of the Fukushima- Williams angles that refers directly to the GCRS pole. These angles may be calculated by calling the function iauPfw06. SOFA also supports the direct computation of the CIP GCRS X,Y by series, available by calling iauXy06.
- The agreement between the different parameterizations is at the 1 microarcsecond level in the present era.
- When constructing a precession formulation that refers to the GCRS pole rather than the dynamical pole, it may (depending on the choice of angles) be necessary to introduce the frame bias explicitly.
- It is permissible to re-use the same variable in the returned arguments. The quantities are stored in the stated order.
- Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The traditional accumulated precession angles zeta_A, z_A, theta_A cannot be obtained in the usual way, namely through polynomial expressions, because of the frame bias. The latter means that two of the angles undergo rapid changes near this date. They are instead the results of decomposing the precession&ndashbias matrix obtained by using the Fukushima&ndashWilliams method, which does not suffer from the problem. The decomposition returns values which can be used in the conventional formulation and which include frame bias.
- The three angles are returned in the conventional order, which is not the same as the order of the corresponding Euler rotations. The precession&ndashbias matrix is:
- Should zeta_A, z_A, theta_A angles be required that do not contain frame bias, they are available by calling the SOFA function iauP06e.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- Naming the following points:
the four Fukushima&ndashWilliams angles are as follows:

- The matrix representing the combined effects of frame bias and precession is:
- The matrix representing the combined effects of frame bias, precession and nutation is simply:
where dP and dE are the nutation components with respect to the ecliptic of date.

- Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(date) = rbp * V(GCRS) , where the p -vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000) and the p -vector V(date) is with respect to the mean equatorial triad of the given date.
- IAU: Trans. International Astronomical Union, Vol. XXIVB Proc. 24th General Assembly, Manchester, UK. Resolutions B1.3, B1.6. (2000).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(date) = rbp * V(GCRS) , where the p -vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000) and the p -vector V(date) is with respect to the mean equatorial triad of the given date.
- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(date) = RMATP * V(J2000) , where the p -vector V(J2000) is with respect to the mean equatorial triad of epoch J2000.0 and the p -vector V(date) is with respect to the mean equatorial triad of the given date.
- Though the matrix method itself is rigorous, the precession angles are expressed through canonical polynomials which are valid only for a limited time span. In addition, the IAU 1976 precession rate is known to be imperfect. The absolute accuracy of the present formulation is better than 0.1 arcsec from 1960AD to 2040AD, better than 1 arcsec from 1640AD to 2360AD, and remains below 3 arcsec for the whole of the period 500BC to 3000AD. The errors exceed 10 arcsec outside the range 1200BC to 3900AD, exceed 100 arcsec outside 4200BC to 5600AD and exceed 1000 arcsec outside 6800BC to 8200AD.
- Lieske, J.H., 1979, Astron.Astrophys. 73, 282.
- Equations (6) & (7), p283.
- Kaplan,G.H., 1981. USNO circular no. 163, pA2.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The caller is responsible for providing the nutation components they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date. For high&ndashaccuracy applications, free core nutation should be included as well as any other relevant corrections to the position of the CIP.
- The returned mean obliquity is consistent with the IAU 2000 precession&ndashnutation models.
- The matrix rb transforms vectors from GCRS to J2000.0 mean equator and equinox by applying frame bias.
- The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession.
- The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp * rb .
- The matrix rn transforms vectors from mean equator and equinox of date to true equator and equinox of date by applying the nutation (luni&ndashsolar + planetary).
- The matrix rbpn transforms vectors from GCRS to true equator and equinox of date. It is the product rn * rbp , applying frame bias, precession and nutation in that order.
- It is permissible to re-use the same array in the returned arguments. The arrays are filled in the order given.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The nutation components (luni&ndashsolar + planetary, IAU 2000A) in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. Free core nutation is omitted for the utmost accuracy, use the iauPn00 function, where the nutation components are caller&ndashspecified. For faster but slightly less accurate results, use the iauPn00b function.
- The mean obliquity is consistent with the IAU 2000 precession.
- The matrix rb transforms vectors from GCRS to J2000.0 mean equator and equinox by applying frame bias.
- The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession.
- The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp * rb .
- The matrix rn transforms vectors from mean equator and equinox of date to true equator and equinox of date by applying the nutation (luni&ndashsolar + planetary).
- The matrix rbpn transforms vectors from GCRS to true equator and equinox of date. It is the product rn * rbp , applying frame bias, precession and nutation in that order.
- The X, Y, Z coordinates of the IAU 2000B Celestial Intermediate Pole are elements (3,1-3) of the matrix rbpn .
- It is permissible to re-use the same array in the returned arguments. The arrays are filled in the order given.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The nutation components (luni&ndashsolar + planetary, IAU 2000B) in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. For more accurate results, but at the cost of increased computation, use the iauPn00a function. For the utmost accuracy, use the iauPn00 function, where the nutation components are caller&ndashspecified.
- The mean obliquity is consistent with the IAU 2000 precession.
- The matrix rb transforms vectors from GCRS to J2000.0 mean equator and equinox by applying frame bias.
- The matrix rn transforms vectors from mean equator and equinox of date to true equator and equinox of date by applying the nutation (luni&ndashsolar + planetary).
- The matrix rbpn transforms vectors from GCRS to true equator and equinox of date. It is the product rn * rbp , applying frame bias, precession and nutation in that order.
- The X, Y, Z coordinates of the IAU 2000B Celestial Intermediate Pole are elements (3,1-3) of the matrix rbpn.
- It is permissible to re-use the same array in the returned arguments. The arrays are filled in the stated order.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The caller is responsible for providing the nutation components they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date. For high&ndashaccuracy applications, free core nutation should be included as well as any other relevant corrections to the position of the CIP.
- The returned mean obliquity is consistent with the IAU 2006 precession.
- The X, Y, Z coordinates of the IAU 2000B Celestial Intermediate Pole are elements (3,1-3) of the matrix rbpn .
- It is permissible to re-use the same array in the returned arguments. The arrays are filled in the stated order.
- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The nutation components (luni&ndashsolar + planetary, IAU 2000A) in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. Free core nutation is omitted for the utmost accuracy, use the iauPn06 function, where the nutation components are caller&ndashspecified.
- The mean obliquity is consistent with the IAU 2006 precession.
- The matrix rb transforms vectors from GCRS to mean J2000.0 by applying frame bias.
- The matrix rp transforms vectors from mean J2000.0 to mean of date by applying precession.
- The matrix rbp transforms vectors from GCRS to mean of date by applying frame bias then precession. It is the product rp * rb .
- The matrix rn transforms vectors from mean of date to true of date by applying the nutation (luni&ndashsolar + planetary).
- The matrix rbpn transforms vectors from GCRS to true of date (CIP/equinox). It is the product rn * rbp , applying frame bias, precession and nutation in that order.
- The X, Y, Z coordinates of the IAU 2006/2000A Celestial Intermediate Pole are elements (1,1-3) of the matrix rbpn .
- It is permissible to re-use the same array in the returned arguments. The arrays are filled in the stated order.
- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(date) = rbpn * V(GCRS) , where the p -vector V(date) is with respect to the true equatorial triad of date date1 + date2 and the p -vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000).
- A faster, but slightly less accurate result (about 1 mas), can be obtained by using instead the iauPnm00b function.
- IAU: Trans. International Astronomical Union, Vol. XXIVB Proc. 24th General Assembly, Manchester, UK. Resolutions B1.3, B1.6. (2000).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(date) = rbpn * V(GCRS) , where the p -vector V(date) is with respect to the true equatorial triad of date date1 + date2 and the p -vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000).
- The present function is faster, but slightly less accurate (about 1 mas), than the iauPnm00a function.
- IAU: Trans. International Astronomical Union, Vol. XXIVB Proc. 24th General Assembly, Manchester, UK. Resolutions B1.3, B1.6. (2000).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(date) = rnpb * V(GCRS) , where the p -vector V(date) is with respect to the true equatorial triad of date date1 + date2 and the p -vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000).
- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855.
- The TDB date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB) = 2450123.7 could be expressed in any of these ways, among others:
- The matrix operates in the sense V(date) = rmatpn * V(J2000) , where the p -vector V(date) is with respect to the true equatorial triad of date date1 + date2 and the p -vector V(J2000) is with respect to the mean equatorial triad of epoch J2000.0.
- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.3 (p145).
- The argument sp is the TIO locator s', in radians, which positions the Terrestrial Intermediate Origin on the equator. It is obtained from polar motion observations by numerical integration, and so is in essence unpredictable. However, it is dominated by a secular drift of about 47 microarcseconds per century, and so can be taken into account by using s' = -47*t, where t is centuries since J2000.0. The function iauSp00 implements this approximation.
- The matrix operates in the sense V(TRS) = rpom * V(CIP) , meaning that it is the final rotation when computing the pointing direction to a celestial source.
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The precession adjustments are expressed as &ldquonutation components&rdquo, corrections in longitude and obliquity with respect to the J2000.0 equinox and ecliptic.
- Although the precession adjustments are stated to be with respect to Lieske et al. (1977), the MHB2000 model does not specify which set of Euler angles are to be used and how the adjustments are to be applied. The most literal and straightforward procedure is to adopt the 4-rotation epsilon_0, psi_A, omega_A, xi_A option, and to add dpsipr to psi_A and depspr to both omega_A and eps_A.
- This is an implementation of one aspect of the IAU 2000A nutation model, formally adopted by the IAU General Assembly in 2000, namely MHB2000 (Mathews et al. 2002).
- The epochs ep01 + ep02 and ep11 + ep12 are Julian Dates, apportioned in any convenient way between the arguments epn1 and epn2 . For example, JD(TDB) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. The two epochs may be expressed using different methods, but at the risk of losing some resolution.

- The accumulated precession angles zeta , z , theta are expressed through canonical polynomials which are valid only for a limited time span. In addition, the IAU 1976 precession rate is known to be imperfect. The absolute accuracy of the present formulation is better than 0.1 arcsec from 1960AD to 2040AD, better than 1 arcsec from 1640AD to 2360AD, and remains below 3 arcsec for the whole of the period 500BC to 3000AD. The errors exceed 10 arcsec outside the range 1200BC to 3900AD, exceed 100 arcsec outside 4200BC to 5600AD and exceed 1000 arcsec outside 6800BC to 8200AD.
- The three angles are returned in the conventional order, which is not the same as the order of the corresponding Euler rotations. The precession matrix is:
- Lieske, J.H., 1979, Astron.Astrophys. 73, 282, equations (6) & (7), p283.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The CIO locator s is the difference between the right ascensions of the same point in two systems: the two systems are the GCRS and the CIP, CIO, and the point is the ascending node of the CIP equator. The quantity s remains below 0.1 arcsecond throughout 1900-2100.
- The series used to compute s is in fact for s + X Y/2, where X and Y are the x and y components of the CIP unit vector this series is more compact than a direct series for s would be. This function requires X, Y to be supplied by the caller, who is responsible for providing values that are consistent with the supplied date.
- The model is consistent with the IAU 2000A precession&ndashnutation.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The CIO locator s is the difference between the right ascensions of the same point in two systems. The two systems are the GCRS and the CIP, CIO, and the point is the ascending node of the CIP equator. The CIO locator s remains a small fraction of 1 arcsecond throughout 1900-2100.
- The series used to compute s is in fact for s + X Y/2, where X and Y are the x and y components of the CIP unit vector this series is more compact than a direct series for s would be. The present function uses the full IAU 2000A nutation model when predicting the CIP position. Faster results, with no significant loss of accuracy, can be obtained via the function iauS00b , which uses instead the IAU 2000B truncated model.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The CIO locator s is the difference between the right ascensions of the same point in two systems. The two systems are the GCRS and the CIP, CIO, and the point is the ascending node of the CIP equator. The CIO locator s remains a small fraction of 1 arcsecond throughout 1900-2100.
- The series used to compute s is in fact for s + X Y/2, where X and Y are the x and y components of the CIP unit vector this series is more compact than a direct series for s would be. The present function uses the IAU 2000B truncated nutation model when predicting the CIP position. The function iauS00a uses instead the full IAU 2000A model, but with no significant increase in accuracy and at some cost in speed.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The CIO locator s is the difference between the right ascensions of the same point in two systems: the two systems are the GCRS and the CIP, CIO, and the point is the ascending node of the CIP equator. The quantity s remains below 0.1 arcsecond throughout 1900-2100.
- The series used to compute s is in fact for s + X Y/2, where X and Y are the x and y components of the CIP unit vector this series is more compact than a direct series for s would be. This function requires X, Y to be supplied by the caller, who is responsible for providing values that are consistent with the supplied date.
- The model is consistent with the &ldquoP03&rdquo precession (Capitaine et al. 2003), adopted by IAU 2006 Resolution 1, 2006, and the IAU 2000A nutation (with P03 adjustments).
- Capitaine, N., Wallace, P.T. & Chapront, J., 2003, Astron. Astrophys. 432, 355.
- McCarthy, D.D., Petit, G. (eds.) 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The CIO locator s is the difference between the right ascensions of the same point in two systems. The two systems are the GCRS and the CIP, CIO, and the point is the ascending node of the CIP equator. The CIO locator s remains a small fraction of 1 arcsecond throughout 1900-2100.
- The series used to compute s is in fact for s + X Y/2, where X and Y are the x and y components of the CIP unit vector this series is more compact than a direct series for s would be. The present function uses the full IAU 2000A nutation model when predicting the CIP position.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The TIO locator s' is obtained from polar motion observations by numerical integration, and so is in essence unpredictable. However, it is dominated by a secular drift of about 47 microarcseconds per century, which is the approximation evaluated by the present function.
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The X, Y coordinates are those of the unit vector towards the celestial intermediate pole. They represent the combined effects of frame bias, precession and nutation.
- The fundamental arguments used are as adopted in IERS Conventions (2003) and are from Simon et al. (1994) and Souchay et al. (1999).
- This is an alternative to the angles&ndashbased method, via the SOFA function iauFw2xy and as used in iauXys06a for example. The two methods agree at the 1 microarcsecond level (at present), a negligible amount compared with the intrinsic accuracy of the models. However, it would be unwise to mix the two methods (angles&ndashbased and series&ndashbased) in a single application.
- Capitaine, N., Wallace, P.T. & Chapront, J., 2003, Astron.Astrophys., 412, 567.
- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855
- McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG.
- Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G. & Laskar, J., Astron.Astrophys., 1994, 282, 663
- Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M., 1999, Astron.Astrophys.Supp.Ser. 135, 111
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The Celestial Intermediate Pole coordinates are the x, y components of the unit vector in the Geocentric Celestial Reference System.
- The CIO locator s (in radians) positions the Celestial Intermediate Origin on the equator of the CIP.
- A faster, but slightly less accurate result (about 1 mas for X, Y), can be obtained by using instead the iauXys00b function.
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The Celestial Intermediate Pole coordinates are the X, Y components of the unit vector in the Geocentric Celestial Reference System.
- The CIO locator s (in radians) positions the Celestial Intermediate Origin on the equator of the CIP.
- The present function is faster, but slightly less accurate (about 1 mas in X, Y), than the iauXys00a function.
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and

- The Celestial Intermediate Pole coordinates are the x , y components of the unit vector in the Geocentric Celestial Reference System.
- Series&ndashbased solutions for generating X and Y are also available: see Capitaine & Wallace (2006) and iauXy06 .
- Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The obliquity, in radians, is mean of date.
- The result, which is in radians, operates in the following sense:
- The result is compatible with the IAU 2000 resolutions. For further details, see IERS Conventions 2003 and Capitaine et al. (2002).
- Capitaine, N., Wallace, P.T. and McCarthy, D.D., &ldquoExpressions to implement the IAU 2000 definition of UT1&rdquo, Astronomy & Astrophysics, 406, 1135-1149 (2003)
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The result, which is in radians, operates in the following sense:
- The result is compatible with the IAU 2000 resolutions. For further details, see IERS Conventions 2003 and Capitaine et al. (2002).
- Capitaine, N., Wallace, P.T. and McCarthy, D.D., &ldquoExpressions to implement the IAU 2000 definition of UT1&rdquo, Astronomy & Astrophysics, 406, 1135-1149 (2003).
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The result, which is in radians, operates in the following sense:
- The result is compatible with the IAU 2000 resolutions except that accuracy has been compromised for the sake of speed. For further details, see McCarthy & Luzum (2001), IERS Conventions 2003 and Capitaine et al. (2003).
- Capitaine, N., Wallace, P.T. and McCarthy, D.D., &ldquoExpressions to implement the IAU 2000 definition of UT1&rdquo, Astronomy & Astrophysics, 406, 1135-1149 (2003).
- McCarthy, D.D. & Luzum, B.J., &ldquoAn abridged model of the precession&ndashnutation of the celestial pole&rdquo, Celestial Mechanics & Dynamical Astronomy, 85, 37-49 (2003).
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The result, which is in radians, operates in the following sense:
- McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The &ldquocomplementary terms&rdquo are part of the equation of the equinoxes (EE), classically the difference between apparent and mean Sidereal Time:
with:

where dpsi is the nutation in longitude and eps is the obliquity of date. However, if the rotation of the Earth were constant in an inertial frame the classical formulation would lead to apparent irregularities in the UT1 timescale traceable to side&ndasheffects of precession&ndashnutation. In order to eliminate these effects from UT1, &ldquocomplementary terms&rdquo were introduced in 1994 (IAU, 1994) and took effect from 1997 (Capitaine and Gontier, 1993):

By convention, the complementary terms are included as part of the equation of the equinoxes rather than as part of the mean Sidereal Time. This slightly compromises the &ldquogeometrical&rdquo interpretation of mean sidereal time but is otherwise inconsequential.

The present function computes CT in the above expression, compatible with IAU 2000 resolutions (Capitaine et al., 2002, and IERS Conventions 2003).

- Capitaine, N. & Gontier, A.-M., Astron. Astrophys., 275, 645-650 (1993).
- Capitaine, N., Wallace, P.T. and McCarthy, D.D., &ldquoExpressions to implement the IAU 2000 definition of UT1&rdquo, Astronomy & Astrophysics, 406, 1135-1149 (2003).
- IAU Resolution C7, Recommendation 3 (1994).
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The date date1 date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The result, which is in radians, operates in the following sense:
- IAU Resolution C7, Recommendation 3 (1994).
- Capitaine, N. & Gontier, A.-M., 1993, Astron. Astrophys., 275, 645-650.
- The UT1 date dj1 + dj2 is a Julian Date, apportioned in any convenient way between the arguments dj1 and dj2 . For example, JD(UT1) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. The date & time method is best matched to the algorithm used: maximum precision is delivered when the dj1 argument is for 0hrs UT1 on the day in question and the dj2 argument lies in the range 0 to 1, or vice versa.

- The algorithm is adapted from Expression 22 of Capitaine et al. 2000. The time argument has been expressed in days directly, and, to retain precision, integer contributions have been eliminated. The same formulation is given in IERS Conventions (2003), Chap. 5, Eq. 14.
- Capitaine N., Guinot B. and McCarthy D.D, 2000, Astron. Astrophys., 355, 398-405.
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The UT1 and TT dates uta + utb and tta + ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession. If UT1 is used for both purposes, errors of order 100 microarcseconds result.
- This GMST is compatible with the IAU 2000 resolutions and must be used only in conjunction with other IAU 2000 compatible components such as precession-nutation and equation of the equinoxes.
- The result is returned in the range 0 to 2pi.
- The algorithm is from Capitaine et al. (2003) and IERS Conventions 2003.
- Capitaine, N., Wallace, P.T. and McCarthy, D.D., &ldquoExpressions to implement the IAU 2000 definition of UT1&rdquo, Astronomy & Astrophysics, 406, 1135-1149 (2003)
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The UT1 and TT dates uta + utb and tta + ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth rotation angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession. If UT1 is used for both purposes, errors of order 100 microarcseconds result.
- This GMST is compatible with the IAU 2006 precession and must not be used with other precession models.
- The result is returned in the range 0 to 2pi.
- Capitaine, N., Wallace, P.T. & Chapront, J., 2005, Astron.Astrophys. 432, 355.
- The UT1 date dj1 + dj2 is a Julian Date, apportioned in any convenient way between the arguments dj1 and dj2 . For example, JD(UT1) = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. The date & time method is best matched to the algorithm used: maximum accuracy (or, at least, minimum noise) is delivered when the dj1 argument is for 0hrs UT1 on the day in question and the dj2 argument lies in the range 0 to 1, or vice versa.

- The algorithm is based on the IAU 1982 expression. This is always described as giving the GMST at 0 hours UT1. In fact, it gives the difference between the GMST and the UT, the steady 4&ndashminutes&ndashper&ndashday drawing&ndashahead of ST with respect to UT. When whole days are ignored, the expression happens to equal the GMST at 0 hours UT1 each day.
- In this function, the entire UT1 (the sum of the two arguments dj1 and dj2 ) is used directly as the argument for the standard formula, the constant term of which is adjusted by 12 hours to take account of the noon phasing of Julian Date. The UT1 is then added, but omitting whole days to conserve accuracy.
- Transactions of the International Astronomical Union, XVIII B, 67 (1983).
- Aoki et al., Astron. Astrophys. 105, 359-361 (1982).
- The UT1 and TT dates uta + utb and tta + ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession&ndashnutation. If UT1 is used for both purposes, errors of order 100 microarcseconds result.
- This GAST is compatible with the IAU 2000 resolutions and must be used only in conjunction with other IAU 2000 compatible components such as precession&ndashnutation.
- The result is returned in the range 0 to 2pi.
- The algorithm is from Capitaine et al. (2003) and IERS Conventions 2003.
- Capitaine, N., Wallace, P.T. and McCarthy, D.D., &ldquoExpressions to implement the IAU 2000 definition of UT1&rdquo, Astronomy & Astrophysics, 406, 1135-1149 (2003)
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The UT1 date uta + utb is a Julian Date, apportioned in any convenient way between the argument pair. For example, JD = 2450123f.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- The result is compatible with the IAU 2000 resolutions, except that accuracy has been compromised for the sake of speed and convenience in two respects:
- UT is used instead of TDB (or TT) to compute the precession component of GMST and the equation of the equinoxes. This results in errors of order 0.1 mas at present.
- The IAU 2000B abridged nutation model (McCarthy & Luzum, 2001) is used, introducing errors of up to 1 mas.

- This GAST is compatible with the IAU 2000 resolutions and must be used only in conjunction with other IAU 2000 compatible components such as precession-nutation.
- The result is returned in the range 0 to 2pi.
- The algorithm is from Capitaine et al. (2003) and IERS Conventions 2003.
- McCarthy, D.D. & Luzum, B.J., &ldquoAn abridged model of the precession&ndashnutation of the celestial pole&rdquo, Celestial Mechanics & Dynamical Astronomy, 85, 37-49 (2003).
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- The UT1 and TT dates uta + utb and tta + ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth rotation angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession&ndashnutation. If UT1 is used for both purposes, errors of order 100 microarcseconds result.
- Although the function uses the IAU 2006 series for s + XY/2 , it is otherwise independent of the precession&ndashnutation model and can in practice be used with any equinox&ndashbased NPB matrix.
- The result is returned in the range 0 to 2pi.
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.
- The UT1 and TT dates uta + utb and tta + ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth rotation angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession-nutation. If UT1 is used for both purposes, errors of order 100 microarcseconds result.
- This GAST is compatible with the IAU 2000/2006 resolutions and must be used only in conjunction with IAU 2006 precession and IAU 2000A nutation.
- The result is returned in the range 0 to 2pi.
- Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981.
- The UT1 date uta + utb is a Julian Date, apportioned in any convenient way between the argument pair. For example, JD = 2450123.7 could be expressed in any of these ways, among others:
The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa.

- The result is compatible with the IAU 1982 and 1994 resolutions, except that accuracy has been compromised for the sake of convenience in that UT is used instead of TDB (or TT) to compute the equation of the equinoxes.
- This GAST must be used only in conjunction with contemporaneous IAU standards such as 1976 precession, 1980 obliquity and 1982 nutation. It is not compatible with the IAU 2000 resolutions.
- The result is returned in the range 0 to 2pi.
- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992).
- IAU Resolution C7, Recommendation 3 (1994).
- The specified pv -vector is the coordinate direction (and its rate of change) for the date at which the light leaving the star reached the solar&ndashsystem barycenter.
- The star data returned by this function are &ldquoobservables&rdquo for an imaginary observer at the solar&ndashsystem barycenter. Proper motion and radial velocity are, strictly, in terms of barycentric coordinate time, TCB. For most practical applications, it is permissible to neglect the distinction between TCB and ordinary &ldquoproper&rdquo time on Earth (TT/TAI). The result will, as a rule, be limited by the intrinsic accuracy of the proper&ndashmotion and radial&ndashvelocity data moreover, the supplied pv -vector is likely to be merely an intermediate result (for example generated by the function iauStarpv), so that a change of time unit will cancel out overall.
In accordance with normal star&ndashcatalog conventions, the object's right ascension and declination are freed from the effects of secular aberration. The frame, which is aligned to the catalog equator and equinox, is Lorentzian and centered on the SSB.

Summarizing, the specified pv -vector is for most stars almost identical to the result of applying the standard geometrical &ldquospace motion&rdquo transformation to the catalog data. The differences, which are the subject of the Stumpff paper cited below, are:

- In stars with significant radial velocity and proper motion, the constantly changing light&ndashtime distorts the apparent proper motion. Note that this is a classical, not a relativistic, effect.
- The transformation complies with special relativity.

- Stumpff, P., 1985, Astron.Astrophys. 144, 232-240.

- The star data accepted by this function are &ldquoobservables&rdquo for an imaginary observer at the solar&ndashsystem barycenter. Proper motion and radial velocity are, strictly, in terms of barycentric coordinate time, TCB. For most practical applications, it is permissible to neglect the distinction between TCB and ordinary &ldquoproper&rdquo time on Earth (TT/TAI). The result will, as a rule, be limited by the intrinsic accuracy of the proper&ndashmotion and radial&ndashvelocity data moreover, the pv -vector is likely to be merely an intermediate result, so that a change of time unit would cancel out overall.
In accordance with normal star&ndashcatalog conventions, the object's right ascension and declination are freed from the effects of secular aberration. The frame, which is aligned to the catalog equator and equinox, is Lorentzian and centered on the SSB.

- The resulting position and velocity pv -vector is with respect to the same frame and, like the catalog coordinates, is freed from the effects of secular aberration. Should the &ldquocoordinate direction&rdquo, where the object was located at the catalog epoch, be required, it may be obtained by calculating the magnitude of the position vector pv [0][0-2] dividing by the speed of light in AU/day to give the light&ndashtime, and then multiplying the space velocity pv [1][0-2] by this light&ndashtime and adding the result to pv [0][0-2] .
Summarizing, the pv -vector returned is for most stars almost identical to the result of applying the standard geometrical &ldquospace motion&rdquo transformation. The differences, which are the subject of the Stumpff paper referenced below, are:

- In stars with significant radial velocity and proper motion, the constantly changing light&ndashtime distorts the apparent proper motion. Note that this is a classical, not a relativistic, effect.
- The transformation complies with special relativity.

- Care is needed with units. The star coordinates are in radians and the proper motions in radians per Julian year, but the parallax is in arcseconds the radial velocity is in km/s, but the pv -vector result is in AU and AU/day.
- The RA proper motion is in terms of coordinate angle, not true angle. If the catalog uses arcseconds for both RA and Dec proper motions, the RA proper motion will need to be divided by cos(Dec) before use.
- Straight&ndashline motion at constant speed, in the inertial frame, is assumed.
- An extremely small (or zero or negative) parallax is interpreted to mean that the object is on the &ldquocelestial sphere&rdquo, the radius of which is an arbitrary (large) value (see the constant PXMIN ). When the distance is overridden in this way, the status, initially zero, has 1 added to it.
- If the space velocity is a significant fraction of c (see the constant VMAX ), it is arbitrarily set to zero. When this action occurs, 2 is added to the status.
- The relativistic adjustment involves an iterative calculation. If the process fails to converge within a set number ( IMAX ) of iterations, 4 is added to the status.
- The inverse transformation is performed by the function iauPvstar .

- Stumpff, P., 1985, Astron.Astrophys. 144, 232-240.

#### 5.1.7 Star catalogs

- This function transforms FK5 star positions and proper motions into the system of the Hipparcos catalog.
- The proper motions in RA are dRA/dt rather than cos(Dec) * dRA/dt , and are per year rather than per century.
- The FK5 to Hipparcos transformation is modeled as a pure rotation and spin zonal errors in the FK5 catalog are not taken into account.
- See also iauH2fk5 , iauFk5hz , iauHfk5z .

- F.Mignard & M.Froeschle, Astron. Astrophys. 354, 732-739 (2000).

- This function models the FK5 to Hipparcos transformation as a pure rotation and spin zonal errors in the FK5 catalogue are not taken into account.
- The r -matrix r5h operates in the sense:
where P_FK5 is a p -vector in the FK5 frame, and P_Hipparcos is the equivalent Hipparcos p -vector.

- The r -vector s5h represents the time derivative of the FK5 to Hipparcos rotation. The units are radians per year (Julian, TDB).

- F.Mignard & M.Froeschle, Astron. Astrophys. 354, 732-739 (2000).

- This function converts a star position from the FK5 system to the Hipparcos system, in such a way that the Hipparcos proper motion is zero. Because such a star has, in general, a non-zero proper motion in the FK5 system, the function requires the date at which the position in the FK5 system was determined.
- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The FK5 to Hipparcos transformation is modeled as a pure rotation and spin zonal errors in the FK5 catalogue are not taken into account.
- The position returned by this function is in the Hipparcos reference system but at date date1+date2.
- See also iauFk52h , iauH2fk5 , iauHfk5z .

- F.Mignard & M.Froeschle, 2000, Astron.Astrophys. 354, 732-739.

- This function transforms Hipparcos star positions and proper motions into FK5 J2000.0.
- The proper motions in RA are dRA/dt rather than cos(Dec) * dRA/dt , and are per year rather than per century.
- The FK5 to Hipparcos transformation is modeled as a pure rotation and spin zonal errors in the FK5 catalog are not taken into account.
- See also iauFk52h , iauFk5hz , iauHfk5z .

- F.Mignard & M.Froeschle, Astron. Astrophys. 354, 732-739 (2000).

- The TT date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
- The proper motion in RA is dRA/dt rather than cos(Dec) * dRA/dt .
- The FK5 to Hipparcos transformation is modeled as a pure rotation and spin zonal errors in the FK5 catalogue are not taken into account.
- It was the intention that Hipparcos should be a close approximation to an inertial frame, so that distant objects have zero proper motion such objects have (in general) non-zero proper motion in FK5, and this function returns those fictitious proper motions.
- The position returned by this function is in the FK5 J2000.0 reference system but at date date1 + date2 .
- See also iauFk52h , iauH2fk5 , iauFk5zhz .

- F.Mignard & M.Froeschle, 2000, Astron.Astrophys. 354, 732-739.

- The starting and ending TDB dates ep1a + ep1b and ep2a + ep2b are Julian Dates, apportioned in any convenient way between the two parts (A and B). For example, JD(TDB) = 2450123.7 could be expressed in any of these ways, among others:
- In accordance with normal star&ndashcatalog conventions, the object's right ascension and declination are freed from the effects of secular aberration. The frame, which is aligned to the catalog equator and equinox, is Lorentzian and centered on the SSB.
The proper motions are the rate of change of the right ascension and declination at the catalog epoch and are in radians per TDB Julian year.

The parallax and radial velocity are in the same frame.

- Care is needed with units. The star coordinates are in radians and the proper motions in radians per Julian year, but the parallax is in arcseconds.
- The RA proper motion is in terms of coordinate angle, not true angle. If the catalog uses arcseconds for both RA and Dec proper motions, the RA proper motion will need to be divided by cos(Dec) before use.
- Straight&ndashline motion at constant speed, in the inertial frame, is assumed.
- An extremely small (or zero or negative) parallax is interpreted to mean that the object is on the &ldquocelestial sphere&rdquo, the radius of which is an arbitrary (large) value (see the iauStarpv function for the value used). When the distance is overridden in this way, the status, initially zero, has 1 added to it.
- If the space velocity is a significant fraction of c (see the constant VMAX in the function iauStarpv ), it is arbitrarily set to zero. When this action occurs, 2 is added to the status.
- The relativistic adjustment carried out in the iauStarpv function involves an iterative calculation. If the process fails to converge within a set number of iterations, 4 is added to the status.

#### 5.1.8 Geodetic geocentric

- The identifier n is a number that specifies the choice of reference ellipsoid. The following are supported:
The n value has no significance outside the SOFA software. For convenience, symbols WGS84 etc. are defined in sofam.h .

- The ellipsoid parameters are returned in the form of equatorial radius in meters (a) and flattening (f). The latter is a number around 0.00335, i.e. around 1/298.
- For the case where an unsupported n value is supplied, zero a and f are returned, as well as error status.

- Department of Defense World Geodetic System 1984, National Imagery and Mapping Agency Technical Report 8350.2, Third Edition, p3-2.
- Moritz, H., Bull. Geodesique 66-2, 187 (1992).
- The Department of Defense World Geodetic System 1972, World Geodetic System Committee, May 1974.
- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), p220.

- The identifier n is a number that specifies the choice of reference ellipsoid. The following are supported:
The n value has no significance outside the SOFA software. For convenience, symbols WGS84 etc. are defined in sofam.h .

- The geocentric vector (xyz, given) and height (height, returned) are in meters.
- An error status -1 means that the identifier n is illegal. An error status -2 is theoretically impossible. In all error cases, phi and height are both set to -1e9.
- The inverse transformation is performed in the function iauGd2gc .

- This function is based on the GCONV2H Fortran subroutine by Toshio Fukushima (see reference).
- The equatorial radius, a, can be in any units, but meters is the conventional choice.
- The flattening, f, is (for the Earth) a value around 0.00335, i.e. around 1/298.
- The equatorial radius, a, and the geocentric vector, xyz, must be given in the same units, and determine the units of the returned height, height.
- If an error occurs (status < 0), elong, phi and height are unchanged.
- The inverse transformation is performed in the function iauGd2gce .
- The transformation for a standard ellipsoid (such as WGS84) can more conveniently be performed by calling iauGc2gd, which uses a numerical code to identify the required A and F values.

- Fukushima, T., &ldquoTransformation from Cartesian to geodetic coordinates accelerated by Halley's method&rdquo, J.Geodesy (2006) 79: 689-693

- The identifier n is a number that specifies the choice of reference ellipsoid. The following are supported:
The n value has no significance outside the SOFA software. For convenience, symbols WGS84 etc. are defined in sofam.h .

- The height (height, given) and the geocentric vector (xyz, returned) are in meters.
- No validation is performed on the arguments elong, phi and height. An error status -1 means that the identifier n is illegal. An error status -2 protects against cases that would lead to arithmetic exceptions. In all error cases, xyz is set to zeros.
- The inverse transformation is performed in the function iauGc2gd .

- The equatorial radius, a, can be in any units, but meters is the conventional choice.
- The flattening, f, is (for the Earth) a value around 0.00335, i.e. around 1/298.
- The equatorial radius, a, and the height, height, must be given in the same units, and determine the units of the returned geocentric vector, xyz.
- No validation is performed on individual arguments. The error status -1 protects against (unrealistic) cases that would lead to arithmetic exceptions. If an error occurs, xyz is unchanged.
- The inverse transformation is performed in the function iauGc2gde .
- The transformation for a standard ellipsoid (such as WGS84) can more conveniently be performed by calling iauGd2gc , which uses a numerical code to identify the required a and f values.

- Green, R.M., Spherical Astronomy, Cambridge University Press, (1985) Section 4.5, p96.
- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 4.22, p202.

#### 5.1.9 Timescales

- scale identifies the time scale. Only the value UTC (in upper case) is significant, and enables handling of leap seconds (see Note 4).
- ndp is the number of decimal places in the seconds field, and can have negative as well as positive values, such as:
The limits are platform dependent, but a safe range is -5 to +9.

- d1 + d2 is Julian Date, apportioned in any convenient way between the two arguments, for example where d1 is the Julian Day Number and d2 is the fraction of a day. In the case of UTC, where the use of JD is problematical, special conventions apply: see the next note.
- JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The SOFA internal convention is that the quasi-JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds.
- The warning status &ldquodubious year&rdquo flags UTCs that predate the introduction of the time scale and that are too far in the future to be trusted. See iauDat for further details.
- For calendar conventions and limitations, see iauCal2jd .

- A new line must be added to the set of statements that initialize the array &ldquochanges&rdquo.
- The parameter IYV must be set to the current year.
- The &ldquoLatest leap second&rdquo comment below must be set to the new leap second date.
- The &ldquoThis revision&rdquo comment, later, must be set to the current date.

- UTC began at 1960 January 1.0 (JD 2436934.5) and it is improper to call the function with an earlier date. If this is attempted, zero is returned together with a warning status.

Because leap seconds cannot, in principle, be predicted in advance, a reliable check for dates beyond the valid range is impossible. To guard against gross errors, a year five or more after the release year of the present function (see parameter IYV) is considered dubious. In this case a warning status is returned but the result is computed in the normal way.

- For dates from 1961 January 1 onwards, the expressions from the file ftp://maia.usno.navy.mil/ser7/tai-utc.dat are used.
- The 5ms timestep at 1961 January 1 is taken from 2.58.1 (p87) of the 1992 Explanatory Supplement.

Called: iauCal2jd Gregorian calendar to Julian Day number.

- The date date1 + date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT) = 2450123.7 could be expressed in any of these ways, among others:
Although the date is, formally, barycentric dynamical time (TDB), the terrestrial dynamical time (TT) can be used with no practical effect on the accuracy of the prediction.

- TT can be regarded as a coordinate time that is realized as an offset of 32.184s from International Atomic Time, TAI. TT is a specific linear transformation of geocentric coordinate time TCG, which is the time scale for the Geocentric Celestial Reference System, GCRS.
- TDB is a coordinate time, and is a specific linear transformation of barycentric coordinate time TCB, which is the time scale for the Barycentric Celestial Reference System, BCRS.
- The difference TCG-TCB depends on the masses and positions of the bodies of the solar system and the velocity of the Earth. It is dominated by a rate difference, the residual being of a periodic character. The latter, which is modeled by the present function, comprises a main (annual) sinusoidal term of amplitude approximately 0.00166 seconds, plus planetary terms up to about 20 microseconds, and lunar and diurnal terms up to 2 microseconds. These effects come from the changing transverse Doppler effect and gravitational red-shift as the observer (on the Earth's surface) experiences variations in speed (with respect to the BCRS) and gravitational potential.
- TDB can be regarded as the same as TCB but with a rate adjustment to keep it close to TT, which is convenient for many applications. The history of successive attempts to define TDB is set out in Resolution 3 adopted by the IAU General Assembly in 2006, which defines a fixed TDB(TCB) transformation that is consistent with contemporary solar-system ephemerides. Future ephemerides will imply slightly changed transformations between TCG and TCB, which could introduce a linear drift between TDB and TT however, any such drift is unlikely to exceed 1 nanosecond per century.
- The geocentric TDB-TT model used in the present function is that of Fairhead & Bretagnon (1990), in its full form. It was originally supplied by Fairhead (private communications with P.T.Wallace, 1990) as a Fortran subroutine. The present C function contains an adaptation of the Fairhead code. The numerical results are essentially unaffected by the changes, the differences with respect to the Fairhead & Bretagnon original being at the 1e-20 s level.
The topocentric part of the model is from Moyer (1981) and Murray (1983), with fundamental arguments adapted from Simon et al. 1994. It is an approximation to the expression ( v / c ) . ( r / c ) , where v is the barycentric velocity of the Earth, r is the geocentric position of the observer and c is the speed of light.

By supplying zeroes for u and v , the topocentric part of the model can be nullified, and the function will return the Fairhead & Bretagnon result alone.

- During the interval 1950-2050, the absolute accuracy is better than +/- 3 nanoseconds relative to time ephemerides obtained by direct numerical integrations based on the JPL DE405 solar system ephemeris.
- It must be stressed that the present function is merely a model, and that numerical integration of solar-system ephemerides is the definitive method for predicting the relationship between TCG and TCB and hence between TT and TDB.

- Fairhead, L., & Bretagnon, P., Astron.Astrophys., 229, 240-247 (1990).
- IAU 2006 Resolution 3.
- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Moyer, T.D., Cel.Mech., 23, 33 (1981).
- Murray, C.A., Vectorial Astrometry, Adam Hilger (1983).
- Seidelmann, P.K. et al., Explanatory Supplement to the Astronomical Almanac, Chapter 2, University Science Books (1992).
- Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G. & Laskar, J., Astron.Astrophys., 282, 663-683 (1994).

- scale identifies the time scale. Only the value UTC (in upper case) is significant, and enables handling of leap seconds (see Note 4).
- For calendar conventions and limitations, see iauCal2jd.
- The sum of the results, d1 + d2 , is Julian Date, where normally d1 is the Julian Day Number and d2 is the fraction of a day. In the case of UTC, where the use of JD is problematical, special conventions apply: see the next note.
- JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The SOFA internal convention is that the quasi-JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds.
- The warning status &ldquotime is after end of day&rdquo usually means that the sec argument is greater than 60.0. However, in a day ending in a leap second the limit changes to 61.0 (or 59.0 in the case of a negative leap second).
- The warning status &ldquodubious year&rdquo flags UTCs that predate the introduction of the time scale and that are too far in the future to be trusted. See iauDat for further details.
- Only in the case of continuous and regular time scales (TAI, TT, TCG, TCB and TDB) is the result d1+d2 a Julian Date, strictly speaking. In the other cases (UT1 and UTC) the result must be used with circumspection in particular the difference between two such results cannot be interpreted as a precise time interval.

- tai1 + tai2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tai1 is the Julian Day Number and tai2 is the fraction of a day. The returned tt1 , tt2 follow suit.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992).

- tai1 + tai2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tai1 is the Julian Day Number and tai2 is the fraction of a day. The returned ut11 , ut12 follow suit.
- The argument dta, i.e. UT1-TAI, is an observed quantity, and is available from IERS tabulations.

- Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992).

- tai1 + tai2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tai1 is the Julian Day Number and tai2 is the fraction of a day. The returned utc1 and utc2 form an analogous pair, except that a special convention is used, to deal with the problem of leap seconds&mdashsee the next note.
- JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds.
- The function iauD2dtf can be used to transform the UTC quasi-JD into calendar date and clock time, including UTC leap second handling.
- The warning status &ldquodubious year&rdquo flags UTCs that predate the introduction of the time scale and that are too far in the future to be trusted. See iauDat for further details.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- tcb1 + tcb2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tcb1 is the Julian Day Number and tcb2 is the fraction of a day. The returned tdb1 , tdb2 follow suit.
- The 2006 IAU General Assembly introduced a conventional linear transformation between TDB and TCB. This transformation compensates for the drift between TCB and terrestrial time TT, and keeps TDB approximately centered on TT. Because the relationship between TT and TCB depends on the adopted solar system ephemeris, the degree of alignment between TDB and TT over long intervals will vary according to which ephemeris is used. Former definitions of TDB attempted to avoid this problem by stipulating that TDB and TT should differ only by periodic effects. This is a good description of the nature of the relationship but eluded precise mathematical formulation. The conventional linear relationship adopted in 2006 sidestepped these difficulties whilst delivering a TDB that in practice was consistent with values before that date.
- TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides.

- IAU 2006 Resolution B3.

- tcg1 + tcg2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tcg1 is the Julian Day Number and tcg2 is the fraction of a day. The returned tt1 , tt2 follow suit.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003),. IERS Technical Note No. 32, BKG (2004).
- IAU 2000 Resolution B1.9.

- tdb1 + tdb2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tdb1 is the Julian Day Number and tdb2 is the fraction of a day. The returned tcb1 , tcb2 follow suit.
- The 2006 IAU General Assembly introduced a conventional linear transformation between TDB and TCB. This transformation compensates for the drift between TCB and terrestrial time TT, and keeps TDB approximately centered on TT. Because the relationship between TT and TCB depends on the adopted solar system ephemeris, the degree of alignment between TDB and TT over long intervals will vary according to which ephemeris is used. Former definitions of TDB attempted to avoid this problem by stipulating that TDB and TT should differ only by periodic effects. This is a good description of the nature of the relationship but eluded precise mathematical formulation. The conventional linear relationship adopted in 2006 sidestepped these difficulties whilst delivering a TDB that in practice was consistent with values before that date.
- TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides.

- IAU 2006 Resolution B3.

- tdb1 + tdb2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tdb1 is the Julian Day Number and tdb2 is the fraction of a day. The returned tt1 , tt2 follow suit.
- The argument dtr represents the quasi&ndashperiodic component of the GR transformation between TT and TCB. It is dependent upon the adopted solar&ndashsystem ephemeris, and can be obtained by numerical integration, by interrogating a precomputed time ephemeris or by evaluating a model such as that implemented in the SOFA function iauDtdb . The quantity is dominated by an annual term of 1.7 ms amplitude.
- TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- IAU 2006 Resolution 3.

- tt1 + tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned tai1 , tai2 follow suit.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- tt1 + tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned tcg1 , tcg2 follow suit.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- IAU 2000 Resolution B1.9.

- tt1 + tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned tdb1 , tdb2 follow suit.
- The argument dtr represents the quasi&ndashperiodic component of the GR transformation between TT and TCB. It is dependent upon the adopted solar&ndashsystem ephemeris, and can be obtained by numerical integration, by interrogating a precomputed time ephemeris or by evaluating a model such as that implemented in the SOFA function iauDtdb . The quantity is dominated by an annual term of 1.7 ms amplitude.
- TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).
- IAU 2006 Resolution 3.

- tt1 + tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned ut11 , ut12 follow suit.
- The argument dt is classical Delta T.

- ut11 + ut12 is Julian Date, apportioned in any convenient way between the two arguments, for example where ut11 is the Julian Day Number and ut12 is the fraction of a day. The returned tai1 , tai2 follow suit.
- The argument dta , i.e. UT1-TAI, is an observed quantity, and is available from IERS tabulations.

- ut11 + ut12 is Julian Date, apportioned in any convenient way between the two arguments, for example where ut11 is the Julian Day Number and ut12 is the fraction of a day. The returned tt1 , tt2 follow suit.
- The argument dt is classical Delta T.

- ut11 + ut12 is Julian Date, apportioned in any convenient way between the two arguments, for example where ut11 is the Julian Day Number and ut12 is the fraction of a day. The returned utc1 and utc2 form an analogous pair, except that a special convention is used, to deal with the problem of leap seconds&mdashsee Note 3.
- Delta UT1 can be obtained from tabulations provided by the International Earth Rotation and Reference Systems Service. The value changes abruptly by 1 second at a leap second however, close to a leap second the algorithm used here is tolerant of the &ldquowrong&rdquo choice of value being made.
- JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the returned quasi JD day UTC1 + utc2 represents UTC days whether the length is 86399, 86400 or 86401 SI seconds.
- The function iauD2dtf can be used to transform the UTC quasi-JD into calendar date and clock time, including UTC leap second handling.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- utc1 + utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day.
- JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds.
- The function iauDtf2d converts from calendar date and time of day into 2-part Julian Date, and in the case of UTC implements the leap&ndashsecond&ndashambiguity convention described above.
- The returned TAI1 , TAI2 are such that their sum is the TAI Julian Date.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

- utc1 + utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day.
- JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds.
- The function iauDtf2d converts from calendar date and time of day into 2-part Julian Date, and in the case of UTC implements the leap&ndashsecond&ndashambiguity convention described above.
- Delta UT1 can be obtained from tabulations provided by the International Earth Rotation and Reference Systems Service. It It is the caller's responsibility to supply a DUT argument containing the UT1-UTC value that matches the given UTC.
- The returned ut11 , ut12 are such that their sum is the UT1 Julian Date.

- McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004).

### 5.2 Vector and matrix functions

- : Angle operations. : Build rotations. : Copy extend extract. : Initialization. : Matrix operations. : Matrix vector products. : Rotation vectors. : Separation and angle. : Spherical cartesian. : Vector operations.

#### 5.2.1 Angle operations

- The argument ndp is interpreted as follows:
- The largest positive useful value for ndp is determined by the size of angle, the format of doubles on the target platform, and the risk of overflowing idmsf[3] . On a typical platform, for angle up to 2pi , the available floating&ndashpoint precision might correspond to ndp=12 . However, the practical limit is typically ndp=9 , set by the capacity of a 32-bit int, or ndp=4 if int is only 16 bits.
- The absolute value of angle may exceed 2pi . In cases where it does not, it is up to the caller to test for and handle the case where angle is very nearly 2pi and rounds up to 360 degrees, by testing for idmsf[0]=360 and setting idmsf[0-3] to zero.

- The argument ndp is interpreted as follows:
- The largest positive useful value for ndp is determined by the size of angle , the format of doubles on the target platform, and the risk of overflowing ihmsf[3] . On a typical platform, for angle up to 2pi , the available floating&ndashpoint precision might correspond to ndp=12 . However, the practical limit is typically ndp=9 , set by the capacity of a 32-bit int, or ndp=4 if int is only 16 bits.
- The absolute value of angle may exceed 2pi . In cases where it does not, it is up to the caller to test for and handle the case where angle is very nearly 2pi and rounds up to 24 hours, by testing for ihmsf[0]=24 and setting ihmsf(0-3) to zero.

- The result is computed even if any of the range checks fail.
- Negative ideg , iamin and/or asec produce a warning status, but the absolute value is used in the conversion.
- If there are multiple errors, the status value reflects only the first, the smallest taking precedence.

Normalize angle into the range 0 <= a < 2pi .

Status: vector/matrix support function.

Given:

Returned (function value):

Normalize angle into the range -pi <= a < +pi .

Status: vector/matrix support function.

Given:

Returned (function value):

- The argument ndp is interpreted as follows:
- The largest positive useful value for ndp is determined by the size of days, the format of double on the target platform, and the risk of overflowing ihmsf[3] . On a typical platform, for days up to 1.0, the available floating&ndashpoint precision might correspond to ndp = 12 . However, the practical limit is typically ndp = 9 , set by the capacity of a 32-bit int, or ndp = 4 if int is only 16 bits.
- The absolute value of days may exceed 1.0. In cases where it does not, it is up to the caller to test for and handle the case where days is very nearly 1.0 and rounds up to 24 hours, by testing for ihmsf [0] = 24 and setting ihmsf [0-3] to zero.

- The result is computed even if any of the range checks fail.
- Negative ihour , imin and/or sec produce a warning status, but the absolute value is used in the conversion.
- If there are multiple errors, the status value reflects only the first, the smallest taking precedence.

- The result is computed even if any of the range checks fail.
- Negative ihour , imin and/or sec produce a warning status, but the absolute value is used in the conversion.
- If there are multiple errors, the status value reflects only the first, the smallest taking precedence.

#### 5.2.2 Build rotations

- Calling this function with positive phi incorporates in the supplied r -matrix r an additional rotation, about the X-axis, anticlockwise as seen looking towards the origin from positive x .
- The additional rotation can be represented by this matrix:

- Calling this function with positive theta incorporates in the supplied r -matrix r an additional rotation, about the Y-axis, anticlockwise as seen looking towards the origin from positive Y.
- The additional rotation can be represented by this matrix:

- Calling this function with positive psi incorporates in the supplied r -matrix r an additional rotation, about the Z-axis, anticlockwise as seen looking towards the origin from positive Z.
- The additional rotation can be represented by this matrix:

#### 5.2.3 Copy extend extract

Copy a p -vector.

Status: vector/matrix support function.

Given:

Returned:

Copy a position/velocity vector.

Status: vector/matrix support function.

Given:

Returned:

Called: iauCp Copy p -vector.

Copy an r -matrix.

Status: vector/matrix support function.

Given:

Returned:

Called: iauCp Copy p -vector.

Extend a p -vector to a pv -vector by appending a zero velocity.

Status: vector/matrix support function.

Given:

Returned:

Called: iauCp Copy p -vector.

iauZp Zero p -vector.Discard velocity component of a pv -vector.

Status: vector/matrix support function.

Given:

Returned:

Called: iauCp Copy p -vector.

#### 5.2.4 Initialization

Initialize an r -matrix to the identity matrix.

Status: vector/matrix support function.

Returned:

Zero a p -vector.

Status: vector/matrix support function.

Returned:

Zero a pv -vector.

Status: vector/matrix support function.

Returned:

Called: iauZp Zero p -vector.

Initialize an r -matrix to the null matrix.

Status: vector/matrix support function.

Returned:

#### 5.2.5 Matrix operations

#### 5.2.6 Matrix vector products

#### 5.2.7 Rotation vectors

- A rotation matrix describes a rotation through some angle about some arbitrary axis called the Euler axis. The &ldquorotation vector&rdquo returned by this function has the same direction as the Euler axis, and its magnitude is the angle in radians. (The magnitude and direction can be separated by means of the function iauPn .)
- If r is null, so is the result. If r is not a rotation matrix the result is undefined r must be proper (i.e. have a positive determinant) and real orthogonal (inverse = transpose).
- The reference frame rotates clockwise as seen looking along the rotation vector from the origin.

- A rotation matrix describes a rotation through some angle about some arbitrary axis called the Euler axis. The &ldquorotation vector&rdquo supplied to This function has the same direction as the Euler axis, and its magnitude is the angle in radians.
- If w is null, the unit matrix is returned.
- The reference frame rotates clockwise as seen looking along the rotation vector from the origin.

#### 5.2.8 Separation and angle

- The result is the position angle, in radians, of direction b with respect to direction a . It is in the range -pi to +pi. The sense is such that if b is a small distance &ldquonorth&rdquo of a the position angle is approximately zero, and if b is a small distance &ldquoeast&rdquo of a the position angle is approximately +pi/2.
- The vectors a and b need not be of unit length.
- Zero is returned if the two directions are the same or if either vector is null.
- If vector a is at a pole, the result is ill&ndashdefined.

- The result is the bearing (position angle), in radians, of point B with respect to point A. It is in the range -pi to +pi. The sense is such that if B is a small distance &ldquoeast&rdquo of point A, the bearing is approximately +pi/2.
- Zero is returned if the two points are coincident.

- If either vector is null, a zero result is returned.
- The angular separation is most simply formulated in terms of scalar product. However, this gives poor accuracy for angles near zero and pi. The present algorithm uses both cross product and dot product, to deliver full accuracy whatever the size of the angle.

Angular separation between two sets of spherical coordinates.

Status: vector/matrix support function.

Given:

Returned (function value):

Called: iauS2c Spherical coordinates to unit vector.

iauSepp Angular separation between two p -vectors.#### 5.2.9 Spherical cartesian

- The vector p can have any magnitude only its direction is used.
- If p is null, zero theta and phi are returned.
- At either pole, zero theta is returned.

- If p is null, zero theta , phi and r are returned.
- At either pole, zero theta is returned.

- If the position part of pv is null, theta , phi , td and pd are indeterminate. This is handled by extrapolating the position through unit time by using the velocity part of pv . This moves the origin without changing the direction of the velocity component. If the position and velocity components of pv are both null, zeroes are returned for all six results.
- If the position is a pole, theta , td and pd are indeterminate. In such cases zeroes are returned for all three.

Convert spherical coordinates to Cartesian.

Status: vector/matrix support function.

Given:

Returned:

Convert spherical polar coordinates to p -vector.

Status: vector/matrix support function.

Given:

Returned:

Called: iauS2c Spherical coordinates to unit vector.

iauSxp Multiply p -vector by scalar.Convert position/velocity from spherical to Cartesian coordinates.

Status: vector/matrix support function.

Given:

Returned:

#### 5.2.10 Vector operations

p -vector inner (=scalar=dot) product.

Status: vector/matrix support function.

Given:

Returned (function value):

Modulus of p -vector.

Status: vector/matrix support function.

Given:

Returned (function value):

- If p is null, the result is null. Otherwise the result is a unit vector.
- It is permissible to re-use the same array for any of the arguments.

- If the position and velocity components of the two pv -vectors are (ap, av) and (bp, bv) , the result, a * b , is the pair of numbers (ap * bp, ap * bv + av bp) . The two numbers are the dot&ndashproduct of the two p -vectors and its derivative.

Modulus of pv -vector.

Status: vector/matrix support function.

Given:

Returned:

Called: iauPm Modulus of p -vector.

- &ldquoUpdate&rdquo means &ldquorefer the position component of the vector to a new date dt time units from the existing date&rdquo.
- The time units of dt must match those of the velocity.
- It is permissible for pv and upv to be the same array.

- &ldquoUpdate&rdquo means &ldquorefer the position component of the vector to a new date dt time units from the existing date&rdquo.
- The time units of dt must match those of the velocity.

- If the position and velocity components of the two pv -vectors are (ap, av) and (bp, bv) , the result, a x b , is the pair of vectors (ap x bp, ap x bv + av x bp) . The two vectors are the cross&ndashproduct of the two p -vectors and its derivative.
- It is permissible to re-use the same array for any of the arguments.

### 5.3 Fortran language constants

These must be used exactly as presented below. Pi

2Pi

Radians to hours

Radians to seconds

Radians to degrees

Radians to arc seconds

Hours to radians

Seconds to radians

Degrees to radians

Arc seconds to radians### 5.4 C language constants

The constants used by the C version of SOFA are defined in the header file sofam.h .

## Watch the video: Model of Earths Rotation u0026 Revolution. School Science Project. 3D Model of Rotation u0026 Revolution (October 2021).

#### 5.1.5 Rotation and time