# What is the magnitude of variation of Earth's orbital inclination?

Please excuse if this has been asked and I just didn't find it.

I have found several references that indicate that the inclination of Earth's orbit varies over a period of ~70,000 years, but I can't seem to find anything that tells what the range of variation is. I know that it's currently at 1.57° to the invariable plane, but I'm looking for the value of the maximum tilt.

In a solar system model by Varadi Runnegar and Ghil Earth's inclination varies between 0 and about 0.05 radians (or about 3 degrees)

The variation is rather chaotic, but you may note that there seems to have been a qualitative change in the pattern about 60-70 million years ago

## Earth: Goldilocks Planet

Earth is the third planet of our solar system. It has one natural moon, 5 natural quasi-satellites, one space station (ISS), a couple dozen space telescopes, and over 3,000 operational artificial satellites (Starlink, OneWeb, Iridium, GPS, etc.) that orbit around it. Earth is the densest planet (and object) in the solar system.

1. Mass (kg)
2. 5.976e+24
2. 6,378.14
1. Density (gm/cm 3 )
2. 5.515
1. Distance from sun (km)
2. 149,600,000
1. Rotational period (days)
2. 23.9345
1. Orbital period (days)
2. 365.256
1. Orbital velocity (km/sec)
2. 29.79
1. Eccentricity of orbit
2. 0.0167
1. Tilt of axis (°)
2. 23.45
1. Inclination of orbit (°)
2. 0.000
1. Surface gravity (m/sec2)
2. 9.78
1. Escape velocity (km/sec)
2. 11.18
1. Albedo
2. 0.37
1. Magnitude
2. -1.9
1. Surface temperature
2. 482°C
1. Atmosphere
2. N2=77%, O2=21%
1. Moons

## Definition of inclination in English:

1 A person's natural tendency or urge to act or feel in a particular way a disposition or propensity.

• &lsquohe was free to follow his inclinations&rsquo
• &lsquoFrom foxhounds to sheep dogs, none can be successful in their natural inclinations without proper training.&rsquo
• &lsquoPrevious conflicts between their natural inclinations and their fears would be resolved firmly in favour of the left.&rsquo
• &lsquoAll of my natural inclinations registered heavily on the Watchtower sin-o-meter.&rsquo
• &lsquoIn that context the audience would include beings of varying capacities, dispositions, and inclinations.&rsquo
• &lsquoThe problem is that many of us are out of touch with our natural inclinations.&rsquo
• &lsquoEvery living being is under the plan of his natural inclinations in terms of the modes of material nature.&rsquo
• &lsquoIntrospection and a compulsion to fleet-footed unexpectedness mean that I sometimes cannot trust my inclinations.&rsquo
• &lsquoFreedom for him is something that belongs to a person when he is not hindered from following his preferences and inclinations.&rsquo
• &lsquoThe way of avoiding such tragedies is for everyone to follow his own inclinations, more or less as they arise.&rsquo
• &lsquoThe other items in each particular circumstance might be different mental events, including desires, inclinations, and so on.&rsquo
• &lsquoAre you ready to finally have your most deafening inclinations and desires voiced for you?&rsquo
• &lsquoThe powers remain, but they now follow the inclinations of man's perverted and self-centred heart.&rsquo
• &lsquoThey are people with special tastes, inclinations and resources.&rsquo
• &lsquoBut they are all still Leftists with the same dictatorial inclinations.&rsquo
• &lsquoThe outcome is clearly a compromise of his own egalitarian inclinations.&rsquo
• &lsquoUnfortunately, this education breeds and dignifies some dangerous inclinations.&rsquo
• &lsquoThey have different approaches, origins, orientations and inclinations.&rsquo
• &lsquoBeing ruled by Venus, planet of love and beauty, you've always had the inclinations of a new romantic, even when grunge dominated.&rsquo
• &lsquoMany of us can think of a dream job… that perfect position matching our aptitudes and our inclinations.&rsquo
• &lsquoThe problem is that my inclinations are in the opposite direction.&rsquo
• &lsquoThe various publics, having other interests or no inclination toward foreign matters short of war, tended toward apathy.&rsquo
• &lsquoThrough her I have satisfied many inclinations to revenge.&rsquo
• &lsquoAnd the image of ordinary, decent boys who showed no inclinations towards extremism and violence began to crumble.&rsquo
• &lsquoSome have inclinations towards activism without ever having really been politicized.&rsquo
• &lsquoWhat I certainly don't feel is guilty about the fact that I have no inclination to watch.&rsquo
• &lsquoMuch of this, I gladly confide, derives from my lifelong inclination for historical geography.&rsquo
• &lsquoHer first inclination was to decline, but before she knew what she was doing she decided that she would accept.&rsquo
• &lsquoStill, he is a bit raw and immature, and he showed no inclination to complete college.&rsquo
• &lsquoMost people don't have the time or inclination to evaluate everything they are told.&rsquo
• &lsquoEither way, if I look at them at all, my inclination to read more than a few lines is heavily influenced by their grasp of, say, punctuation.&rsquo
• &lsquoOne of his more obvious characteristics is his inclination towards exaggeration.&rsquo
• &lsquoAn inclination toward classical art and, most likely, the residual Protestantism of her Canadian-Scottish heritage were also evident.&rsquo
• &lsquo‘The refugees and asylum seekers are generally law-abiding and educated and have no inclination towards crime,’ he said.&rsquo
• &lsquoThe second inconsistency is found in Calvin's insistence that the fallen will retains neither power to choose between good and evil nor any inclination for goodness.&rsquo
• &lsquoBut an inclination for music was not his only love, he also had a passion for film.&rsquo
• &lsquoVicinal faces are typically only hundredths of one degree in inclination from the main crystal face on which they form.&rsquo
• &lsquoAn inclination of 0 degrees would mean the orbit is perfectly aligned with Earth's orbital plane.&rsquo
• &lsquoSlope inclination and aspect were recorded at several locations within each stand.&rsquo
• &lsquoThe great diversity of plants in the formation is due to local variation in soil conditions, topography, slope inclination and resultant microclimates.&rsquo
• &lsquoThe plot was located on a north-west facing slope with an inclination of about 20° and an elevation range of 130 m from the lowest to the highest point.&rsquo
• &lsquoThat most people walk in an ungraceful, ungainly and awkward manner with a forward inclination of the body does not mean that it is the normal way of walking.&rsquo
• &lsquoA slight inclination of Roxy's head indicated to Helen that she knew about her estrangement from Tim.&rsquo
• &lsquoA slight inclination of Alvito's head was all the acknowledgement this pledge received.&rsquo
• &lsquoThe higher coercivity component has a inclination that is steeper than expected and a NW declination.&rsquo
• &lsquoThis component has both reversed and normal polarity, with an average declination of 320 and an inclination of -13 deg.&rsquo

3 The angle at which a straight line or plane is inclined to another.

• &lsquoFor example, it is likely that the angle of inclination of the pectoral fin base constrains the range of directions in which force may be applied on the fluid during swimming.&rsquo
• &lsquoFor example, at each location on the globe, the geomagnetic field lines intersect the Earth's surface at a specific angle of inclination.&rsquo
• &lsquoPrevious workers have examined the functional significance of variation in the angle of inclination of the fin base relative to the longitudinal axis of the body.&rsquo
• &lsquoThe first design trend we examine here is in the orientation of the pectoral fin base, defined externally as the angle of inclination of the insertion of the pectoral fin on the body.&rsquo
• &lsquoThe transmembrane helix of subunit VIIc changes its angle of inclination midway through the helix.&rsquo
• &lsquoWe also recorded terrain inclination angle, observer distance, time of day, date and year.&rsquo
1. 3.1 Astronomy The angle between the orbital plane of a planet, comet, etc. and the ecliptic, or between the orbital plane of a satellite and the equatorial plane of its primary.
• &lsquoBecause of the Mercury's high orbital inclination, it can be seen crossing the disk of the sun only rarely.&rsquo
• &lsquoFrom that ellipse one can, in principle, determine the inclination of the planet's orbital plane.&rsquo
• &lsquoHe's based this idea on a study of the angle, or inclination, of asteroid orbits.&rsquo
• &lsquoFirst, the relative inclination of the two orbits means their paths do not intersect.&rsquo
• &lsquoThe orbit plane inclination is from 55 to 60 degrees, which gives good coverage of latitudes up to 75 degrees north.&rsquo
• &lsquoHe also hypothesized that the Mid-Paleogene cooling resulted from a sudden shift in the angle of inclination of the Earth's axis of rotation.&rsquo
• &lsquoThe most plausible conclusion is that the inclination of Jupiter's axis is automatically changing, as we know the Earth's has often done.&rsquo

### Origin

Late Middle English from Latin inclinatio(n-), from inclinare ‘bend towards’ (see incline).

## 1. Introduction

[2] Since the sunspot cycle was discovered, the Sun has been considered as a possible agent of climate change [e.g., Eddy, 1976 ]. However, due to difficulties in distinguishing between solar, volcanic, and anthropogenic influences [ Ammann et al., 2003 Cubasch et al., 2001 Schmidt et al., 2012 ], as well as complex responses related to cloud cover and ocean temperatures [ Hansen, 2000 ], its precise role is still subject to controversy. A common measure of the Sun's energy output is Total Solar Irradiance (TSI), defined as the wavelength-integrated flux of radiation received at the top of Earth's atmosphere. The TSI has a baseline value of approximately 1361 Wm −2 during minima in solar activity [ Kopp and Lean, 2011 ].

[3] Since 1978, TSI variability has been measured with a high accuracy by instruments onboard several space-based platforms. Daily variations up to ∼0.3% are caused by the presence of dark (sunspots) and bright (faculae and network) features on the solar surface [ Willson et al., 1981 ]. Intense magnetic fields within sunspots [e.g., Borrero and Ichimoto, 2011 ] suppress convection and reduce the transport of thermal energy from the solar interior to the photosphere. Such a reduction of the surface temperature within sunspots leads to lower surface opacity. Coupled with the fact that sunspots are partially evacuated, relative to the quiet Sun, surfaces of constant optical depth within sunspot umbra are located at deeper geometric depths, the so-called Wilson depression. The energy blocked by sunspots seems to diffuse in the convection zone on short time scales [ Foukal et al., 2006 ] and is stored and released on longer time scales. On the other hand, the structure of the solar magnetic field governs also the leakage of energy that leads to a positive variation of the TSI during the solar cycle. Most magnetic features on the solar surface other than sunspots appear as faculae and network. These are small, bright structures that also block the convection. However, because the flux tubes are narrow, the inflow of radiation through the hot walls exceeds the energy blocked. The geometry of the small-scale fields causes a non-isotropic radiation field [ Spruit, 1977 Steiner, 2005 ]. The combination of these effects leads to variations in the TSI on time-scales from days, to years (the 11-year sunspot cycle), to millennia [ Shapiro et al., 2011 Steinhilber et al., 2009 Vieira et al., 2011 ].

[4] Previous investigations [ Muller and MacDonald, 1995 ] have indicated that the Earth's orbital inclination and climate records present a strong 100 kyr periodicity signal although no physical mechanism linking the phenomena has been successfully established. Surprisingly, the potential impact of anisotropy on the distribution of active regions on the irradiance variability due to changes in Earth's orbital inclination, on timescales of kyrs, has been overlooked in the literature. Although observational [e.g., Rast et al., 2008 ] and modeling [e.g., Knaack et al., 2001 Schatten, 1993 ] efforts have been performed in the past, since its observation requires an out-of-ecliptic vantage point, the latitudinal dependence of irradiance is not known. To date, only instruments on spacecraft with near-Earth orbits or at the L1 Lagrangian point have measured the TSI. Unfortunately, missions such as Voyager 2 and Ulysses, which reached high solar latitudes, had no measurements of solar irradiance. The Solar Orbiter Mission has an orbit out of the ecliptic plane and is scheduled to be launched in 2017. However, TSI observations will not be a part of this mission. Therefore, any variability outside the terrestrial vantage point (i.e., outside of the ecliptic plane) has not been sampled and will not be measured in the near future. Since the distribution of solar active regions is limited from mid to low solar latitudes, here we investigate the geometric component of solar irradiance variability that has gone undetected. Specifically, we search for an anisotropy in TSI (flux density) from the solar equator to the poles that could be sampled if the Earth had a highly inclined orbit.

[5] In Section 2we describe the method employed to compute the out-of-the-ecliptic solar irradiance. Then the modeled evolution of the out-of-the-ecliptic TSI during the ascending phase of cycle 24 is discussed inSection 3. Next, an estimate of TSI variations due to changes in orbital inclination is presented in Section 4. Finally, conclusions are provided in Section 5.

## Eclipsing binaries

An eclipsing binary consists of two close stars moving in an orbit so placed in space in relation to Earth that the light of one can at times be hidden behind the other. Depending on the orientation of the orbit and sizes of the stars, the eclipses can be total or annular (in the latter, a ring of one star shows behind the other at the maximum of the eclipse) or both eclipses can be partial. The best known example of an eclipsing binary is Algol (Beta Persei), which has a period (interval between eclipses) of 2.9 days. The brighter (B8-type) star contributes about 92 percent of the light of the system, and the eclipsed star provides less than 8 percent. The system contains a third star that is not eclipsed. Some 20 eclipsing binaries are visible to the naked eye.

The light curve for an eclipsing binary displays magnitude measurements for the system over a complete light cycle. The light of the variable star is usually compared with that of a nearby (comparison) star thought to be fixed in brightness. Often, a deep, or primary, minimum is produced when the component having the higher surface brightness is eclipsed. It represents the total eclipse and is characterized by a flat bottom. A shallower secondary eclipse occurs when the brighter component passes in front of the other it corresponds to an annular eclipse (or transit). In a partial eclipse neither star is ever completely hidden, and the light changes continuously during an eclipse.

The shape of the light curve during an eclipse gives the ratio of the radii of the two stars and also one radius in terms of the size of the orbit, the ratio of luminosities, and the inclination of the orbital plane to the plane of the sky.

If radial-velocity curves are also available—i.e., if the binary is spectroscopic as well as eclipsing—additional information can be obtained. When both velocity curves are observable, the size of the orbit as well as the sizes, masses, and densities of the stars can be calculated. Furthermore, if the distance of the system is measurable, the brightness temperatures of the individual stars can be estimated from their luminosities and radii. All of these procedures have been carried out for the faint binary Castor C (two red-dwarf components of the six-member Castor multiple star system) and for the bright B-type star Mu Scorpii.

Close stars may reflect each other’s light noticeably. If a small, high-temperature star is paired with a larger object of low surface brightness and if the distance between the stars is small, the part of the cool star facing the hotter one is substantially brightened by it. Just before (and just after) secondary eclipse, this illuminated hemisphere is pointed toward the observer, and the total light of the system is at a maximum.

The properties of stars derived from eclipsing binary systems are not necessarily applicable to isolated single stars. Systems in which a smaller, hotter star is accompanied by a larger, cooler object are easier to detect than are systems that contain, for example, two main-sequence stars (see below Hertzsprung-Russell diagram). In such an unequal system, at least the cooler star has certainly been affected by evolutionary changes, and probably so has the brighter one. The evolutionary development of two stars near one another does not exactly parallel that of two well-separated or isolated ones.

Eclipsing binaries include combinations of a variety of stars ranging from white dwarfs to huge supergiants (e.g., VV Cephei), which would engulf Jupiter and all the inner planets of the solar system if placed at the position of the Sun.

Some members of eclipsing binaries are intrinsic variables, stars whose energy output fluctuates with time (see below Variable stars). In many such systems, large clouds of ionized gas swirl between the stellar members. In others, such as Castor C, at least one of the faint M-type dwarf components might be a flare star, one in which the brightness can unpredictably and suddenly increase to many times its normal value (see below Peculiar variables).

## Astronomy and Climate-Earth System: Can Magma Motion under Sun-Moon Gravitation Contribute to Paleoclimatic Variations and Earth’s Heat?

100 ky. Thermodynamic changes resulting from orbital eccentricity, obliquity, and precession have been ascribed as the cause of the variations although processes within the oceans and atmosphere may have too short memory to explain such variations. In this work, the dynamics of Sun-Moon gravitation (SMG) were explored for a rotating Earth and were determined to have a long memory in magma, a mostly ignored geophysical fluid with a mass

3,400 times that of the atmosphere plus the oceans. Using the basic motion and gravitation (including obliquity) of the Sun and the Moon, we determined that SMG-induced magma motion could produce paleoclimatic variations with multiple periods (e.g.,

100 ky), with considerable power for Earth’s heat. Such “reproducible” power could possibly maintain an energetic Earth against collapse, radioactivity, and cooling.

#### 1. Introduction

Orbital eccentricity, obliquity, and precession, with periodicities of

23 ky, respectively, are three important drivers of paleoclimatic variations [1–4]. Each type of orbital forcing thermodynamically impacts climate, by altering either the total amount or the distribution of insolation received on Earth. For example, latitude-averaged insolation is reduced during Northern Hemisphere summer, leading to the reduction of summer snow-melt at high latitudes and over long time periods to glacial expansion due to lower obliquity [5]. Ice core records obtained from Antarctica indicate that orbitally driven Antarctic climate change lags behind Northern Hemisphere insolation variations by a few millennia [6, 7]. Insolation (i.e., Milankovitch Cycles) summarizes the thermodynamics of active orbital forcing. However, as indicated in Figure 1, we determined that insolation yielded a low correlation (

24.9%) to and a low contribution to temperature on paleoclimatic scale and yielded different anomalies and spectra from those of observed temperature and CO2 concentrations over the past 800 ky [6]. The observed temperature and CO2 concentrations over the past 800 ky highly correlated to each other, with an

88.6% correlation with a 99% confidence. The spectra of both temperature and CO2 displayed periods of

100 (96.97–103.17) ky, while the spectra of insolation produced shorter periods of

41 (39.75–42.30) ky. Period ranges within the parentheses above were from error estimations via a chi-square test [8, 9] for a 95% significance level. Also, temperature and CO2 yielded a much longer total phase during negative-anomaly phases (443 and 412 ky) than during positive-anomaly phases (357 and 389 ky), while insolation had a shorter total phase during negative-anomaly phases (391 ky) than during positive-anomaly phases (409 ky). Finally, temperature and CO2 yielded much smaller amplitudes during negative-anomaly phases (

52 ppm) than during positive-anomaly phases (

75 ppm), while insolation had approximately the same amplitude during negative- and positive-anomaly phases (

66 Wm −2 , resp.). Both temperature and CO2 yielded a “buffer” or a slower accumulation during negative anomaly phases, but a relatively quick release during positive anomaly phases, while insolation did not.

and 4 μm/s for the lower mantle and outer core, resp., see Section 2). Left column: the time series (red-green curve) was 100% correlated to its reconstruction (dashed blue curve) that was produced via the Discrete Fourier Transform (see Methods, Section 2.3), with a confidence of 99.99%, and was evaluated using the following: the positive/negative amplitude

, the total phase length during positive-/negative-phases

, and the correlation to temperature (CT). Right column: the squared amplitude spectrum (see Methods, Section 2.3, black curve) with the

-axis representing the value and the

-axis representing the period (ky). Error bars were plotted with a dashed blue (upper limit) and red (lower limit) line, estimated via a chi-square test [8, 9] for a 95% significance level.

Past studies on slow processes within the atmosphere and oceans have indicated climatic variations of shorter periods from

100 ky [12–16]. For example, the approximate 1.5 ky period in the north Atlantic Ocean climate system is caused by a combination of weak periodic forcing and “noise” from ice sheet related events [12]. The 1 ky Northern Hemisphere temperature and the 100 ky cycle for tropical sea surface temperature are associated with greenhouse gases [13, 14]. Regional climate shifts during the Pliocene epoch (

4.5 to 3.0 million years ago) are related to gradual global cooling [15]. Changes in sea level and temperature are also influenced by freshwater forcing via the phase change [16].

However, the slow processes within the atmosphere and oceans may only explain shorter variation probably because the atmosphere and oceans have memories shorter than those of paleoclimatic variations from the viewpoint of dynamics [17], and processes within the atmosphere and oceans may largely serve as the responsive portion of paleoclimatic variations. Slower driver(s) should exist that can influence CO2 and temperature on paleoclimate scales. Ice core data obtained from Antarctica over the past 800 ky provide detailed insights into the aerosol (dust) load of the atmosphere [10]. We determined that dust is highly correlated with temperature, with a correlation coefficient of −67.5% (confidence > 97%, via a double-sided

-test). Dust lags temperature by approximately 500 years and has almost the same spectra as temperature, with periods of

100 (96.97–103.17) ky (from Figure 1(d), the period ranges within the parentheses above were estimated via a chi-square test [8, 9] for a 95% significance level). During the period of time from 800 ky (BP) to the preindustrial period, the dust load should largely result from natural processes associated with volcanism and, therefore, should be related to the activity of magma associated with variations in temperature and dust.

Using a theoretical model established for SMG-induced magma motion (see Section 2), we examined the Power of SMG-Induced Magma (PSMGIM) motion and determined that PSMGIM results in significant variations on multiple paleoclimatic scales for an 800 ky time window that began from the time period set in the model (instead of 800 ky BP). Our experimental solutions for the PSMGIM yielded spectra similar to those for temperature and CO2 for “buffer” and “quick release” mechanisms during negative and positive anomaly phases, respectively. The spectra of the PSMGIM displayed periods of

100 (96.97–103.17) ky. Period ranges within the parentheses above were estimated via a chi-square test [8, 9] for a 95% significance level. As shown in Figure 1(e), the total phase length was much longer during negative anomaly phases (417 ky) than during positive anomaly phases (383 ky), and the amplitude was much smaller during negative anomaly phases (1,206 TW, 1 TW = 10 12 W) than during positive anomaly phases (2,063 TW). Shorter periods (e.g., of

22.3–23.7 ky) were also determined to appear when smaller diffusion coefficients were employed (Figure 3).

The following questions were further answered in this study. (1) How can orbital drivers (note that orbital obliquity was included while orbital eccentricity and precession were excluded) with limited periodicity produce period-abundant paleoclimatic variations within the PSMGIM? (2) Can the PSMGIM be significant for Earth’s heat budget? (3) How will PSMGIM variations contribute to paleoclimatic variations? The last question was partially explained and needed further studies.

#### 2. Methods

This section contains three major parts, as follows: (1) the dynamic model for SMG-induced magma motion, (2) the periodicity calculations included for SMG-induced magma motion and the probability for a period to occur in SMG-induced magma motion, and (3) a spectrum analysis for data and modeling results.

##### 2.1. The Dynamic Model for SMG-Induced Magma Motion

As far as methodology is concerned, omitting small-magnitude SMG in climate is an extension of the scale-analysis method that has been effectively applied for linear processes and short-term weather systems. However, for climate and paleoclimate studies with a large space and a long time duration, small driving factors such as SMG that accumulatively and actively acts on climate and paleoclimate systems may not be omitted. The size of SMG is even comparable to Coriolis in the “SMG dynamical zone (SMDZ)” [17], especially for slow magma. As compared to magma whose mass is approximately 3,400 times that of the atmosphere plus the oceans, the atmosphere and the oceans are only a thin layer of fluids having much smaller thermal and dynamic inertia or memory [17] to produce weather-climate variations driven by both SMG and solar radiation. However, magma, the third geophysical fluid, has been omitted in climate-paleoclimate studies, although magma has a large mass through which the thermal contribution can be significant even if solar radiation hardly reaches magma. SMG can drive magma to move and the kinetic energy from motion can be transferred into thermal energy via the friction associated with sticky magma.

Obtaining an accurate nonlinear solution for magma motion under SMG in Eulerian system is currently impossible. SMG changes with a relative location between a float and the Sun or the Moon. For a numerical model to determine the changing location of a moving float, grid spacing in Eulerian system must be smaller than the distance the float moves within one time step. The longer the temporal scale is required to study or predict, the smaller the speed dynamically contributes to the corresponding temporal variation [17], indicating that a higher resolution is required for a longer-term climatic model. Pinpointing accurate relative locations between a float and the Sun or the Moon given Earth’s, the fast Earth’s rotation makes the time step much shorter than that used by classic climate models. If, for example, the time step is one hour and the smallest speed that needs to be simulated is 10 −5 m/s (as is typically required for a variation of

## Space Solar

### 24.3.3 Other space solar architectures

Many architectures take distinctly different approaches than those embodied by the perpendicular to orbital plane or sandwich module concepts. One concept is shown in Fig. 24.7 . Created by Japanese researchers in 1994, this scheme would employ an equatorial low earth orbit (LEO) at 1100 km, use 2.45 GHz as the microwave downlink frequency, and transmit 10 MW [43] . Because of its location in LEO rather than GEO, a single satellite could not offer continuous ground coverage, as it would frequently enter earth's shadow and would only be visible to a given ground site for less than about 15 min at a time. However, multiple countries could take advantage of the downlinked power, and a constellation of the proposed satellites could increase the ground coverage time. Much smaller in capacity and size than the DOE/NASA reference concept, it only slightly exceeds 300 m in length along each edge.

Figure 24.7 . SPS 2000 Japanese concept [43] , circa 1994.

The SunTower concept seen in Fig. 24.8 is one that resulted from the NASA studies of the late 1990s. It employs concentrating PV and could be emplaced in sun-synchronous LEO, middle earth orbits between 6000 km and 12,000 km, or GEO, depending on the implementation [44] . The microwave link frequency proposed is 5.8 GHz and the amount of power delivered to the grid ranges from 50 to 250 MW depending on the architectural options exercised.

Figure 24.8 . SunTower concept [8] , circa 1999.

In 2017 Ian Cash introduced a novel, new space solar concept titled CASSIOPeiA, standing for Constant Aperture, Solid-State, Integrated, Orbital Phased Array. It employs a three-dimensional phased array for power transmission in conjunction with an integrated helix photovoltaic collection structure, allowing for the elimination or minimization of any mechanisms or other moving parts. Several simulations suggest that this approach may be feasible [45] . One variant of the concept is shown in Fig. 24.9 .

Figure 24.9 . The constant aperture, solid-state, integrated, orbital phased array (CASSIOPeiA) solar power satellite concept, circa 2018.

Courtesy of Ian Cash, International Electric, used with permission.

Many other solar power satellite concepts have been developed and studied, some entailing the use of lunar or asteroidal materials to minimize the mass needed to be launched from Earth, such as Peter Schubert's Tin Can SPS [46] . The vast majority employ microwave power transmission, the theory of which has been definitively chronicled, and which enjoys numerous instances of practical demonstration. However, concepts using laser transmission have been investigated as well, in part because the shorter wavelengths result in smaller transmit and receive apertures, and consequently smaller and lighter satellites versus those using microwave power transmission. One such concept, developed by the Aerospace Corporation, is shown in Fig. 24.10 . A 2001 Aerospace Corporation study commissioned by NASA found that laser-based space solar offers many potential advantages over microwave-based systems [47] , results affirmed by a 2009 Lawrence Livermore National Laboratory commissioned by DOE [48] .

Figure 24.10 . A laser-based space solar concept explored by the Aerospace Corporation [47] .

## Obliquity of the Earth

In the solar system, planets have orbits that are all roughly in the same plane. That of the Earth is called the ecliptic. Each planet rotates around its axis of rotation, causing a succession of local days in each planet. The slow change in direction of the axis of rotation of the Earth is called the precession of the equinoxes.
The axial tilt or obliquity is the angle between the axis of rotation of the Earth and its orbital plane, it remains confined between 21.8° and 24.4°. Currently, it is 23°26'14'' but the axis is recovering about 0.46" per year or &asymp1 degree every 7800 years. Moreover, this axis oscillates around a cone, the full cycle (360°) lasts 25,765 years. This angle (&asymp23°26') made the changing seasons. Indeed, in summer, the sun is higher in the northern hemisphere than in the southern hemisphere.
The sun is higher in the sky of the northern part of the globe, in the southern part. Sun rays coming to Earth with more intensity. The sun rises early, goes to bed later, and the days are longer. In the south it is winter. The Sun also appears lower on the horizon and the days are shorter, the sun rises later and sets earlier.
At the equator the length of day and night does not vary (although the Sun's position in the sky varies). At the poles, day and night lasts six months each.
The obliquity characterizes therefore the tilt of the Earth's axis relative to the ecliptic varying between 21.8° and 24.4°. But the Earth is slightly flattened at the poles, gravitational forces exerted by the Sun and the Moon rotate on itself not as a perfectly spherical ball but like a top. This small variation from 21.8° to 24.4° is due to the presence of the moon acts as a stabilizer on the equatorial bulge of the Earth.
Nevertheless, small variations in the obliquity have broad implications for the sunshine at latitude 65°, which is considered the most reliable criterion of melting ice sheets.

The combination of these two effects produces an oscillation of the Earth's obliquity, very limited, about 1.3° around a mean value close to 23.5°.
The combined period of these oscillations is about 41 000 years. The obliquity has a great importance on high latitudes because it is the cause of the seasons, if the obliquity were zero, there would be no seasons, and thus little variation in temperature. It is a parameter or Milanković Milanković cycles corresponding to three astronomical phenomena affecting the Earth's eccentricity, obliquity and precession.
They are used in the context of the astronomical theory of paleoclimatology. They are partly responsible for natural climate changes that have major consequence, the glacial and interglacial periods.

## NUCLEAR STRUCTURE FAR ABOVE THE YRAST LINE

### 4 SUMMARY AND CONCLUSIONS

High-energy gamma radiation produced in the decay of giant resonance modes can be used to probe the structure of highly excited nuclei. The progress of the experimental techniques which has taken place during the last few years, now makes it possible to study gamma ray spectra covering a transition energy range which includes the L=1 and L=2 multipoles of isovector giant resonances.

The success in interpreting such gamma ray spectra in terms of a statistical deexcitation of the observed giant resonances, combined with the high experimental statistics achieved in the GDR region, permits the quantitative analysis of the properties of the GDR in hot rotating heavy nuclei. Nevertheless, the absolute magnitudes of the extracted parameters describing the GDR are still strongly dependent on the assumptions about the parameters entering the statistical model calculations. With this in mind, differential measurements of the GDR properties in a given nucleus or among neighbouring systems, provide a powerful tool to study changes of the nuclear structure as a function of T and I. In this way the influence of possible systematic errors is reduced.

It now seems well established that the structure of the GDR in hot nuclei can be correlated with the nuclear shape. This opens for the possibility of using excited-state giant resonances to investigate the transition from the shell-structure dominated region, characteristic of cold nuclei, to the Fermi gas regime expected at higher E. Recent measurements, indicate a marked angular momentum dependence of the GDR in heavy deformed systems. The observed behaviour is consistent with a change of the nuclear shape from prolate to oblate in the temperature range T=1.1-1.6 MeV.

Studies of the temperature dependence of the damping width of the GDR are potentially interesting, since they may provide new insight into the mechanisms underlying the relaxation of excitations in hot nuclei. The presently available experimental information suffers, however, from the difficulty in separating out the contributions to the GDR width arising from deformation changes.

## What is the magnitude of variation of Earth's orbital inclination? - Astronomy

GS6777 Satellite Geodesy Class Schedule

September 21 (Wednesday) : Quarter begins

September 22 (Thursday) : Class begins.

September 22 (Thursday) : Class begins. Roll call. Course outline. Discuss course prerequisites, requirements, and lab/homework requirements. What is satellite geodesy? Classical definition of geodesy versus the contemporary definition - size and shape of the Earth, which is changing with time, and the measurements are becoming increasingly more accurate, and able to detect these changes: Geodetic Science at Ohio State University , Wikipedia Geodesy .

History of satellite geodesy: NGA/NIMA Geodesy for Layman . The launch of Sputnik in October 4, 1957 started the space age. Interdisciplinary applications and science of geodesy. What is time in geodesy? Reference books for the course (e.g., Satellite Geodesy by Bill Kaula ). Handouts: (1) Mulholland, Measures of time in Astronomy , Publ. Astron. Soc. of the Pacific, June 1972 (Time & Coordinate System or Geometric Geodesy is a prerequisite of this course), (2) Lecture Notes on Time System . Discussion of dynamical time (or uniform time), time measurement devices, and time keeping practices, due in part to the variable rate of Earth rotation. Discussed the occurrence of maximum ice height during the Last Ice Age (LGM or Last Glacial Maximum), and the variability of the Earth's orbit around the Sun, as the main forcing mechanism for the occurrence of Ice Ages (with a time scale of

September 22 (Thursday) : No Lab

September 27 (Tuesday) : Discussions of the last class questions. Handout: (1) discuss a plot of observed Excess Length-of-day (LOD), 1820-1980, showing various signals from seasonal, interannual , decadal to secular periodicities. Earth rotation is slowing down (LOD is longer) at an approximate rate of 2 millisecond/century due to lunar tidal dissipation on Earth. A brief history of the discovery of planets evolves around the Sun (Nicholas Copernicus, Tycho Brahe, Johannes Kepler , 1543 to 1609) and Keplers Three Laws (The law of ellipses, the law of areas, the harmonic law). The 2nd Law was basically ignored for about 80 years. Modified Keplers 3 rd law. Design of 24-hour circular geosynchronous orbit, and design of GPS orbit (12-hour synchronous orbit) using Kepler  s 3 rd Law (problems also on Lab. 1). Most accurate clocks are built, e.g., by US National Institute of Standards and Technology (NIST): the aluminum clock, accurate to 1 second over 3.7 billion years, as compared to the cesium fountain clock (US civilian time standard), which drifts 1 second over 100 million years. Other new technology is the optical clock. ESAs mission on the International Space Station, the Atomic Clock Ensemble in Space (ACES), intends to deplore very accurate clocks in space to study relativity and using clock to measure gravtiy . Discussion on an inertial coordinate system, and Earth-fixed coordinate system, and their conversion. Realization of ITRF (International Terrestrial Reference Frame), and the transformation to or from the J2000 inertial system: current modeling (precession, nutation, tides, polar motion, UT1, tidal loading, plate tectonics), and signals (atmosphere and hydrologic loading, geocenter , vertical motion including the effect of glacial isostatic adjustment or postglacial rebound).

Class Assigned Questions : (1) Oblateness of the Earth: is the oblateness of the Earth (J2) changing? If so, what is the physical cause? (2) What is the mean orbital period ( Keplers 2 nd law) for low Earth orbiters (250 km to 1500 km)? (3) What is the orbital altitude of the satellite LAGEOS (SLR satellites) and what is the science studied by LAGEOS and other SLR ranged satellites (Lageos-1, -II, Starlette , Stella, Ajisa ) (related to #1)?

Study Question Answers : (1) The oblateness of the Earth (J2or -C20) is changing at a rate of -3x10 -11 / yr , indicating that the Earth is getting "rounder" because of melting of ice-sheets since the Last Ice Age [Yoder et al., 1983 ] by analyzing satellite laser ranging (SLR) observations to the satellite LAGEOS. (2) 90 to 120 minutes. (3) 6,000 km orbital altitude, geodynamics (see No. 1).

September 29 (Thursday , 1:00 PM ), Lab, Lei Wang: Tutorials about MATLAB (if needed) and School of Earth Sciences computing lab. Lab. No. 1.

October 4 (Tuesday) : Handout: Yoder et al. [1983] Nature paper. More examples on applications of Keplers Laws and Keplers Equation. Use of "measurements" from Explorer 27 to estimate GM of the Earth. Derivation of Keplers equation . Derivation of the two-body problem in relative coordinate system. Its solution starting from Newtons Law of gravitational and using relative coordinate system. Note all the assumptions in the derivation. Orbit in space (3D). Physical meaning of the six Keplerian orbit elements. Conservation of angular momentum – its physical meaning: invariant of the orbital plane. Energy integral. (1) Check out and learn about the Gravity Recovery and Climate Experiment (GRACE) twin-satellite mission, TOPEX/POSEIDON satellite mission. (2) Derive the integration constant (E) for the energy integral.

October 6 (Thursday) : Handouts: (1) Review of Two-Body Program and Orbits in Space (lecture notes), (2) Orbits in 3D, Relationship between orbit elements and Cartesian coordinates. Derive of Two-Body Program in polar coordinates. Review the assumptions made (inertial coordinates, point mass or bodies with constant densities). Explanation of the so-called Central Body Term. Discussion of the term 'non-spherical geopotential or gravity field of the Earth: related to the applications of GRACE (last time's study question). Orbit in space, review on physical meaning of the six Kelperian orbit elements, prograde and retrograde orbits. Transformation between orbit elements and Cartesian position and velocity vectors (handout, Relationship between orbit elements and Cartesian coordinates , not discussed in detail in class). Illustration of the fuel, Delta-v, needed to change the inclination of an orbit (not discussed in class). The largest term is when the inclination change, Delta- i , is 60 0 , it requires Delta-v=v

=7 km/sec, which is almost impossible. Study question: what are the Lagrangian Points (of the Earth-Sun system)?

October 6 (Thursday, 1:00 PM) : Lab Session. Continued discussion on Lab. No. 1.

October 11 (Tuesday): More discussions on Keplers 3 rd Law: T=2*pi* SQRT( a 3 /G(M1+M2)) Is this equation accounted for elliptical orbits? If the GM (or density) of the body is less dense, e.g., water instead of rock. The orbital period, T, increases. Brief discussion on the innovative use of GPS (GPS occultation, GPS water level measurements over ocean, rivers, GPS altimetry or reflectometry ). Relationship between satellite launch site latitude and azimuth angle and orbital inclination for the satellites to be launched. Derivation of N-body equation of motion, energy and angular momentum integrals. N-body problem equation of motion: direct and indirect terms. Express the equation of motion in the form of a Central Body Term, and another term, which could be Perturbing Term. One needs the values of the GM of the heavenly bodies, and their positions and velocities from each other in time, to formulate the equation of motion of the N-body problem (NASA/JPLs Planetary Ephemerides). Three-Body and the Restricted Three-Body Problem (applicable to the Earth-Moon-spacecraft problem). Periodic orbits, Lagrange (Josef Lagrange, an Italian-French mathematician) Points (L1, L2, L3, L4, L5) of the Earth-Sun system: possibility of stable points (actually L1, L2, L3 are less stable L1 & L2 are unstable on a time scale of about 23 days L4 and L5 are more stable) for satellites to "park" in the L1 (Solar & Heliospheric Observatory Satellite, SOHO), and Al Gores envisioned and proposed NASA mission in 1998, the Deep Space Climate Observatory (DSCOVR, formerly known as Triana ), L2 (e.g., next Hubble), L5 Society (space colony), L3 (locations of the Planet X?). Columbus Dispatch (2004) story on Lagrange Points . Class Assigned Questions: (1) the Earth-sun (the ecliptic) & Earth-moon orbital planes, Earth's (and Moon"s ) orbital inclination around the sun (obliquity) and around the Earth, respectively: what are they, and what do they reflect the Keplerian motion that you have learned so far? (2) What are the Milankovitch Cycles and its relationship between the Earth-sun orbit variations and paleoclimate ? Students asked to review or be familiar with Legendre functions.

Study Question Answers : (1) There are variations of the obliquity (inclination of the Earths orbit around the sun, 23.4 0 ), eccentricity, gravitational pull of the sun and the moon on the Earths equatorial bulge ( oblated Earth) causes a slow change in the orientation of the Earths of rotation (precession, with a period of 26,000 years), Earth wobbles (polar motion), rotates

24 hours/day and rotation rate is slowing down because of the Moons tidal dissipation, and nutation (periodic changes in the obliquity or inclination of 23.4 0 . Lunar orbit around the Earth: eccentricity is 0.05, variations of 0.03 to 0.06 due to the Suns perturbation, semi-major axis oscillates about 5,000 to 10,000 km about the mean semi-major axis, inclination varies from about 5 0 to 5.25 0 , or the Earth-Moon orbital plane is almost the same as the ecliptic (Earth-Sun orbital plane) in orientation. Moons inclination angle to Earths equator thus varies between 18 0 to 28 0 . (2) Milankovitch cycles refers to the changes of the Earths orbit around the sun, which resulted in Ice Ages to form and have periodicities of 100,000 years (eccentricity change), 41,000 years (inclination or obliquity change or axial tilt), and 23,000 years (precession). This is the link between natural climate change on Earth and orbital variations ( Adhemar , 1842, Croll , 1875, Berger, 1991, Laskar , 1993, etc ). First modern studies of ocean sediment cores, and link of orbital variations to paleoclimate is by [ Hay et al., 1976] . Ice core studies are among the other tools of studying paleoclimate . Discussions of using the N-Body equations of motion, based on JPLs Planetary Ephemerides, which provides the positions and velocities of planets as a function of time, and the GMs of the planets, to integrate backwards the Earths orbit around the Sun, to obtain an orbit as a function of time, say for 1 million years. Spectral analysis, provided that the modeling is adequate and the JPL ephemeride is accurate enough, can reveal various periodicities of the orbit and identify the Milankovitch Cycle periodicities.

October 13 (Thursday): Practical considerations of the Earth-sun coordinate system. Earth-Moon-Sun system. Motion of the moon. Earth-moon orbit (perturbed by the Sun): precession (20,000 years), nutation, 18 .6 year lunar node regression (the Saros cycle, or the Antiquity). General perturbation theory. The perturbed equation of orbital motion. Discussion of osculating (orbit) elements, assumption of precessing ellipse, and variations of parameter techniques, and the basic assumptions (perturbation is small compared to central body term, and the true velocity equals to the osculating velocity) which yields the Lagrange Brackets, and Lagrange planetary equations (Italian-French mathematician, Joseph Louis Lagrange, 1736-1813). The goal is to derive the Lagranges Equation. Class Assigned Questions: (1) What are Sidereal and Synodic periods (orbital periods)? (2) Consider the Earth-Moon system, where is the center of mass of the system? (3) Consider Laplaces equation, look into its solution, and assumptions. Discuss a possible method to test the geopotential model, U, from Laplaces equation, and determined using data. For example, evaluate U using Laplaces equation globally and as a function of radial distance.

October 13 (Thursday, 1:00 PM): Lab. No. 2 posted. No Lab.

Study Question Answers : (1) The synodic period is the temporal interval that it takes for an object to reappear at the same point in relation to two other objects (linear nodes), i.e. the Moon relative to the Sun as observed from the Earth. Computation of the "moon phase" (lunar synodic period, 29.53087 days)=(1/27.32166 days – 1/365.256363 days) -1 . (2) Center of mass of the Earth-Moon system lies inside the Earth. (3) Solution of Laplaces equation, U, or the geopotential can be expressed in terms of spherical harmonics.

October 18 (Tuesday): Reference: Kaulas book (Section 3.2, derivation of Lagrange Planetary Equation, p. 25-29). Derivation of Lagrange Planetary Equation (1) & (2). Hand-out (LPE derivation).

October 20 (Thursday): Discussion one of the largest perturbations on a near-Earth satellite, the non-spherical geopotential , U, which could be the perturbation function in the Lagrange Planetary Equation (LPE). Outline of solution of Laplaces equation (spherical coordinate system, the differences between spherical harmonics and ellipsoidal harmonics, geocentric versus geodetic latitude). Discuss Poissons equation. Solution of Laplaces equation using separation of variables (example derivation). Our focus is on the particular solution of U, which represents the external geopotential . Discussion of the solution, U, expressed in spherical harmonics, and the physical meanings of J2 and J3 (or –C20 and –C30), oblateness / oblongness and pear-shape, respectively. Discussion of l=0 and l=1 ( geocenter ) terms. Zonal, sectorial and tesseral harmonics. Study Questions: (1) look at contemporary geopotential models, e.g., EGM96, EGM08, and other models, (2) what is geocenter motion ? Its magnitude and physical reason for its motion, and (3) compare the magnitudes, for a near-Earth satellite orbit, the central body term, the J2 and the J3 terms.

Study Question Answers : (1) EGM96 ( lmax =360), EGM08 ( lmax =2,160), and GRACE static geopotential models, (2) geocenter motion is associated with l=1 geopotential coefficients (C10, C11, S11) and is due primarily to hydrologic variations with a seasonal amplitude of approximately several mm (up to 1 cm), causing the center of the mass of the Earth to move.

October 20 (Thursday, 1:00 PM): Lab.

October 25 (Tuesday): No Class.

October 27 (Thursday): Seminar by Chris Wright, South Dakota State University, Russia Browning: The 2010 Heat Wave Was Not an Isolated Event, ML291, students are asked to write a concise summary of the talk, due before October 28 (Friday). The summary should include: (1) summary of the talk including what you think is his major findings, (2) interesting aspects of his approach, (3) even though you may not be a hydrologist, if you are asked to work on this problem, your suggestion as how to enhance the study (observations, modeling, or other type of analysis).

October 27 (Thursday, 1:00 PM): Lab.

October 28 (Friday, 3:30𔃂:30 PM): Make-up class (review for mid-term exam).

November 1 (Tuesday): No class (review for exam optional)

November 3 (Thursday): No class (exam).

November 3 (Thursday, 1:00𔃁:00 PM): Mid-term exam ( no books, no computer usage except use as calculators, 2 page notes allowed).

November 8 (Tuesday): Return of examination and discussions of exam answers (Lei Wang)

November 10 (Thursday): Discussion of mid-term exam (continued). Review on Lagrange bracket derivations.

November 10 (Thursday, 1:00 PM) : Lab.

November 15 (Tuesday): Discuss procedures to derive Kaulas theory of linear perturbation due to non-spherical geopotential for an Earth-orbiting or planet-orbiting satellite. Transformation of angular arguments into orbit elements. Inclination function. Eccentricity function. Function of the spherical harmonic coefficient of the geopotential and the angular argument. Kaulas introduction of indexes, p and q, and the generalized expression of the perturbation function due to the perturbation of the non-spherical geopotential . Schematics on the meaning of orbit element variations (secular, and periodic signals about the mean or averaged orbit element) as a function of time. Derive a general expression of the solution of da as a function of time and with Kaulas perturbation function.

November 17 (Thursday): Kaulas theory of linear perturbation (continued). Solution in the general form for the orbit element variations: da, di, de, dW , dw , and dM. Derived dW . Example of the perturbation function (or secular perturbation on the orbit elements due to J2): Ulmpq =U2010. Derive da, di, de, dW , dw , and dM. Discuss sun-synchronous orbit, critical inclination using the generalized perturbation function due to the geopotential on a planet-orbiting satellite. Discuss long-period, short-period, m-daily, secular perturbations and resonance. Example problem to derive first order resonance (m=14). Discussion of special perturbation and numerical integration of ordinary differential equations. Contemporary integrators use the so-called multi-step, multi-order integrators [Krogh , 1994 ]. Test of integrators [ Berry & Healy, 2005 ] especially for short arcs indicate good agreement.

November 17 (Thursday, 1:00 PM): No Lab. Lab. No. 3 posted.

November 22 (Tuesday): More discussion on resonances. The eccentricity function converges rapidly if the orbit is near circular or e = 0. Study Questions: (1) what is the differences between the conditions to have a secular perturbation and in resonance for the orbit element variations? And (2) what are the geopotential resonant periods and the geopotential order, m, responsible for, if any, resonances in the GPS orbit? Methods of special perturbation include Cowell and Encke methods. H. Cowell used his method to compute orbits for the moon of Jupiter and to predict Halleys comet revisiting the Earth in 1920. Various force (N-body, non-spherical geopotential perturbation, general relativity, Earth and ocean tides, atmospheric drag, solar and Earth (albedo) radiation pressure, spacecraft thermal forces) and measurement models for a near-Earth satellite orbit determination and inversion problem. Derivation of the nonlinear statistical orbit determination and parameter recovery estimation formulation [ Tapley , 1973 ]. State transition matrix approach. Discussion of how does one assess accuracy of computed orbits and how Kaula`s perturbation theory could be of help assuming that the orbit errors containing geopotential errors. Example problem: Uniform gravity field estimation. Note additional question in the Problem: solve for station coordinates ( xs , ys ).

November 24󈞇 (Thursday-Sunday): Thanksgiving Holiday, no classes.

November 29 (Tuesday): ( Continued:) Nonlinear statistical orbit determination (OD) and parameter recovery estimation formulation. Application of the nonlinear OD approach to estimate position and velocity and constant parameters. Example: Various perturbations on Earth orbiters: size of various perturbations in terms of acceleration. Study Question: what is the difference between accelerometer and drag-free instruments? Review of final exam topics.

December 1 (Thursday): Last day of class. Types of numerical orbit determination methods: dynamic, kinematic, reduced-dynamics.

December 5 (Monday) : Final Exam, 9:30 AM 󈝷:18 AM, ML 255. No books, no computer usage except use as calculators, 2 page notes allowed.