Astronomy

Orbital parameters of the Sun

Orbital parameters of the Sun



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What are the orbital parameters of the Sun such as orbit velocity etc in it's orbit around the Solar System's center of mass? Consider the Sun pointlike or alternatively when talkin about the Sun's movement I mean it's center of mass.

Do not tell me that the Sun is stationary because the planets' masses can be neglected. I do no want such oversimplifications.


Since the Solar system is a multi-body system (with $N>2$ bodies of significant mass), the orbits of its constituents are not exact Keplerian orbits.

To lowest order, each planet orbits the Sun (or rather the centre of mass of all interior planets) on a Keplerian orbit, but the interactions with the planets as well as the fact that the centre of mass of the interior is not fixed lead to deviations of the true orbit from this simplification. These deviations can be treated either numerically or via perturbation theory, but are non-trivial functions of time.

The same holds for the Sun: to lowest order one can neglect all planets but Jupiter (which is more then twice as massive as all the remaining planets combined), when the Sun follows an elliptic orbit with semi-major axis of about 0.005AU (smaller than that of Jupiter by their mass ratio). This is of the same order as the radius of the Sun, i.e. the barycentre of the Solar system hardly leaves the Sun. However, as above, the pull by all other planets lead to a deviation from this simple model. Again, these deviations are non-trivial.


If we ignore the role of the atmosphere, the insolation at a particular time and location at the Earth's surface is a function of the Sun-Earth distance and the cosine of the solar zenith distance (Eq. 2.20). These two variables can be computed from the time of day, the latitude, and the characteristics of the Earth's orbit. In climatology, the Earth's orbit is determined by three orbital parameters (Fig. 5.16 and 5.17): the obliquity ( ε o b l ) measuring the tilt of the ecliptic compared to the celestial equator (Fig. 2.7), the eccentricity ( ecc ) of the Earth's orbit around the sun and the climatic precession (ecc sin ⁡ ω ˜ ) which is related to the Earth-Sun distance at the summer solstice. In this definition of the climatic precession, ⁡ ω ˜ is the true longitude of the perihelion measured from the moving vernal equinox ( ⁡ ω ˜ = π + PERH on Fig. 2.8).

Figure 5.16: Schematic representation of the changes in the eccentricity ecc and of the obliquity ε o b l of the Earth's orbit. Source: Latsis foundation (2001)

Because of the influence of the Sun, the other planets in the solar system and the Moon, the orbital parameters vary with time. In particular, the torque applied to the Earth by the Sun and the Moon because our planet is not a perfect sphere (the distance from the surface to the Earth's centre is larger at the Equator than at the poles) is largely responsible for the variations of the obliquity and plays an important role in the changes in ⁡ ω ˜ . The eccentricity is particularly influenced by the largest planets of the solar system (Jupiter and Saturn), which also have an impact on ⁡ ω ˜ .

The way those parameters have developed over time has been calculated from the equations representing the perturbations of the Earth-Sun system due to the presence of the other celestial bodies and to the fact that the Earth is not a perfect sphere. The solution can then be expressed as the sum of various terms:

e c c = e c c 0 + ∑ i E i cos ⁡ λ i t + φ i ε o b l = ε obl , 0 + ∑ i A i cos ⁡ γ i t + ξ i ecc sin ⁡ ω ˜ = ∑ i P i cos ⁡ α i t + η i ( 5 . 6 )

The values of the independent parameters ecc0, ε obl,0, of the amplitudes Ei, Ai, Pi, of the frequencies λ t , γ i, α i, and of the phases φ i, ξ i, η i are provided in Berger (1978), updated in Berger and Loutre (1991). The equations (5.6) clearly show that the orbital parameters vary with characteristic periods (Fig. 5.18). The dominant ones for the eccentricity are 413, 95, 123 and 100 ka. For the climatic precession, the dominant periods are 24, 22, and 19 ka and for the obliquity 41 and 54 ka. To completely determine the Earth's orbit, it is also necessary to specify the length of the major axis of the ellipse. However, taking it as a constant is a very good approximation at least for the last 250 million years.

Figure 5.17: Because of the climatic precession, the Earth was closest to the Sun during the boreal summer 11 ka ago while it is closest to the Sun during the present boreal winter. Source: Latsis foundation (2001)

The eccentricity of the Earth's orbit (Fig. 5.16) has varied over the last million years between nearly zero, corresponding nearly to a circular orbit, to 0.054 (Fig. 5.18). Using Eq. 2.26, it can be shown that the annual mean energy received by the Earth is inversely proportional to 1 - e c c 2 . As expected, this value is independent of the obliquity because of the integration over all latitudes, and is independent of ⁡ ω ˜ because of the integration over a whole year. The annual mean energy received by the Earth is thus at its smallest when Earth's orbit is circular and increases with the eccentricity. However, as the variations in eccentricity are relatively small (Fig. 5.18), there are only minor differences in the annual mean radiations received by the Earth. The maximum relative variation is equal to 0.15% (1.510 -3 = 1 - 1/ 1 - 0.05 4 2 ), corresponding to about 0.5 W m -2 (0.5=1.5 10 -3 x 342 W m -2 ).

Figure 5.18: Long-term variations in eccentricity, climatic precession and obliquity (in degrees) for the last million years and the next 100 thousand years (zero corresponds to 1950 AD). The minimum value of the climatic precession corresponds to boreal winter (December) solstice at perihelion. Computed from Berger (1978).

The obliquity is responsible for the existence of seasons on Earth. If ε o b l were equal to zero night and day would be 12 hours long everywhere (Eq. 2.22 and 2.25) and if ecc were also equal to zero, each location on Earth would have the same daily mean insolation throughout the year (Eq. 2.22 and 2.26). With a large obliquity, the insolation is much higher in polar regions in summer, while it is zero in winter during the polar night. Over the last million years, the obliquity has varied from 22 o to 24.5 o (Fig. 5.18). This corresponds to maximum changes in daily mean insolation at the poles of up to 50 W m -2 (Fig. 5.19). Obliquity also has an influence on the annual mean insolation, increasing it by a few W m -2 at high latitudes and decreasing it (but to a lesser extent) at the Equator.


The motion of planets around the Sun can be described by Kepler’s three laws of planetary motion:

  • First Law: All planets move along elliptical orbits with the Sun at one focus.
  • Second Law: A line connecting a planet and the Sun sweeps out equal areas in equal time intervals.
  • Third Law: The square of a planet’s orbital period about the Sun is proportional to the cube of its semi-major axis.

However, these laws are not sufficient to describe exactly where in an orbit a planet is, or how that orbit is oriented. Instead, we have to specify the values of the orbital elements. Note that several of these elements are redundant, as they can be used to derive other orbital elements.

Type of orbital parameter Name Symbol
These orbital elements tell us information about the shape of an orbit: Semi-major axis a
Orbital eccentricity e
Perihelion distance p
These orbital elements tell us the orientation of an orbit: Orbital inclination i
Ascending node Ω
Argument of perihelion ω
These orbital elements tell us the position and speed of a planet in its orbit: Orbital velocity v
Mean anomaly M
Mean daily motion d

Finally, we need to specify the epoch (t0), or reference date of the coordinate system. This is usually given as the time when the planet is at its closest approach to the Sun.

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Résumé

La découverte des périodes glaciaires au xix e siècle suscite les premières interrogations scientifiques sur l’évolution du climat au cours du temps et marque ainsi la naissance de la paléoclimatologie. Dès lors, les scientifiques se sont attachés à reconstruire les changements climatiques passés et à en comprendre les bases physiques. Ainsi, depuis cette époque, deux théories se sont affrontées pour tenter d’expliquer les alternances glaciaire–interglaciaire : les variations des paramètres orbitaux de la Terre, et des changements de la composition atmosphérique en dioxide de carbone. Si la théorie astronomique a pu être largement confirmée depuis une trentaine d’années, la modélisation physique des changements climatiques mis en œuvre reste encore balbutiante. Par ailleurs, les résultats les plus récents de la paléoclimatologie nous démontrent qu’il est maintenant de plus en plus nécessaire de construire une synthèse de ces deux hypothèses historiques.


Orbital parameters of the Sun - Astronomy

2. Jacobson, R.A. (2003) ``Reconstruction of the Voyager Saturn Encounter Orbits in the ICRF System'', Paper No. AAS 03-198, 13th AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico.

3. Jacobson, R.A. (2000) ``The orbits of the outer Jovian satellites'', Astronomical Journal 120 , 2679-2686. Note: the mean motions and periods appearing in the reference are incorrect for the retrograde satellites. The correct values appear here.

4. Jacobson, R.A., Riedel, J.E. and Taylor, A.H. (1991) ``The orbits of Triton and Nereid from spacecraft and Earthbased observations'', Astronomy & Astrophysics 247 , 565.

5. Jacobson, R.A. (2008) ``Ephemerides of the Martian Satellites - MAR080'', JPL IOM 343R-08-006

6. Jacobson, R.A. (1996) ``Orbits of the Saturnian Satellites from Earthbased and Voyager Observations'', Bull. American Astronomical Society 28 (3), 1185.

7. Jacobson, R.A. (1997) JUP120 - JPL satellite ephemeris.

8. Jacobson, R.A. (1998) ``The orbit of Phoebe from Earthbased and Voyager observations'', Astronomy & Astrophysics Supp. 128 , 7.

9. Jacobson, R.A. (1998) ``The orbits of the inner Uranian satellites from Hubble Space Telescope and Voyager 2 observations'', Astronomical Journal 115 , 1195.

10. Laskar, J. and Jacobson, R.A. (1987) ``GUST86. An analytical ephemeris of the Uranian satellites'', Astronomy & Astrophysics 188 , 212.

11. Jacobson, R.A. (2003) JUP230 - JPL satellite ephemeris.

12. Jacobson, R.A. (2007) SAT270, SAT271 - JPL satellite ephemerides.

13. Tholen, D .J. and Buie, M. W. (1990) ``Further Analysis of Pluto-Charon Mutual Event Observations'', Bull. American Astronomical Society 22 (3), 1129.

14. Owen, W.M., Vaughan, R.M, and Synnott, S.P. (1991) ``Orbits of the Six New Satellites of Neptune'', Astronomical Journal 101 , 1511.

15. Showalter, M.R. (1991) ``Visual detection of 1981S13, Saturn's eighteenth satellite, and its role in the Encke gap'', Nature 351 , 709.

16. Brozovic, M. and Jacobson, R. A. (2009) ``The Orbits of the Outer Uranian Satellites'', Astronomical Journal 137 , 3834.

17. Jacobson, R.A. (2000) URA047 - JPL satellite ephemeris.

18. Jacobson, R.A. (2013) JUP300 - JPL satellite ephemeris for the irregular Jovian satellites.

19. Jacobson, R.A. (2004) ``The orbits of the major Saturnian satellites and the gravity field of Saturn from spacecraft and Earthbased observations'', Astronomical Journal 128 , 492.

20. Jacobson, R.A. (2001c) ``The Orbits of Jupiter and its Galilean Satellites and the Gravity Field of the Jovian System'', Jupiter: The Planet, Satellites, and Magnetosphere, Boulder, Colorado.

21. Jacobson, R.A. (2001d) ``The Gravity Field of the Jovian System and the Orbits of the Regular Jovian Satellites'', 33rd Annual Meeting of the Division for Planetary Sciences, New Orleans, Louisiana.

22. Jacobson, R.A. (2009) JUP269 - JPL satellite ephemeris.

23. Lieske, J.H. (1998) ``Galilean satellite ephemerides E5'', Astronomy & Astrophysics Supp. 129 , 205.

24. Jacobson, R.A. (2003) URA066 - JPL satellite ephemeris.

25. Jacobson, R.A. (2002) JUP242 - JPL satellite ephemeris.

26. Jacobson, R.A. (2003) JUP219 - JPL satellite ephemeris.

27. Jacobson, R.A. (2003) NEP029 - JPL satellite ephemeris.

28. Jacobson, R.A. (2003) JUP222 - JPL satellite ephemeris.

29. Jacobson, R.A. (2003) JUP224 - JPL satellite ephemeris.

30. Jacobson, R.A. (2003) JUP226 - JPL satellite ephemeris.

31. Jacobson, R.A. (2003) JUP227 - JPL satellite ephemeris.

32. Jacobson, R.A. (2010) SAT339 - JPL satellite ephemeris.

33. Jacobson, R.A. (2010) SAT342 - JPL satellite ephemeris.

34. Jacobson, R.A. (2013) SAT361 - JPL satellite ephemeris.

35. Jacobson, R.A. (2004) JUP252 - JPL satellite ephemeris

36. Jacobson, R.A. (2003) NEP041 - JPL satellite ephemeris

37. Jacobson, R.A. (2003) URA067 - JPL satellite ephemeris

38. Jacobson, R.A. (2007) NEP057 - JPL satellite ephemeris

39. Jacobson, R.A. (2003) URA068 - JPL satellite ephemeris

40. Jacobson, R.A. (2003) URA072 - JPL satellite ephemeris

41. Jacobson, R.A. (2009) SAT317 - JPL satellite ephemeris.

42. Jacobson, R.A. (2006) JUP261 - JPL satellite ephemeris.

43. Jacobson, R.A. (2009) JUP268 - JPL satellite ephemeris.

44. Jacobson, R.A. (2009) JUP270 - JPL satellite ephemeris.

45. Jacobson, R. A. and Owen, Jr., W. M. (2004) ``The orbits of the inner Neptunian satellites from Voyager, Earthbased, and Hubble Space Telescope observations'', Astronomical Journal 128 , 1412.

46. Jacobson, R. A. and French, R. G. (2004) ``Orbits and Masses of Saturn's Coorbital and F-ring Shepherding Satellites'', Icarus 172 , 382.

47. Spitale, J. N., Jacobson, R. A., Porco, C. C., and Owen, Jr., W. M. (2006) ``The Orbits of Saturn's Small Satellites Derived from Combined Historic and Cassini Imaging Observations'', Astronomical Journal 132 , 692.

48. Jacobson, R.A. (2007) PLU017 - JPL satellite ephemeris.

49. Jacobson, R.A. (2008) SAT295 - JPL satellite ephemeris.

50. Jacobson, R.A. (2008) SAT296 - JPL satellite ephemeris.

51. Showalter, M. R. and Lissauer, J. J. (2006) ``The Second Ring-Moon System of Uranus: Discovery and Dynamics'', Science 311 , 973.

52. Jacobson, R.A. (2008) SAT297 - JPL satellite ephemeris.

53. Jacobson, R.A. (2008) SAT298 - JPL satellite ephemeris.

54. Jacobson, R. A. (2009) ``The Orbits of the Neptunian Satellites and the Orientation of the Pole of Neptune'', Astronomical Journal 137 , 4322.

55. Jacobson, R.A. (2008) NEP077 - JPL satellite ephemeris.

56.Brozovic, M., Jacobson, R. A., and Sheppard, S. S. (2011) ``The Orbits of the Outer Neptunian Satellites'', Astronomical Journal 141 , 135.

57. Jacobson, R.A. (2013) NEP087 - JPL satellite ephemeris.

58. Brozovic, M. and Jacobson, R. A. (2013) ``The Orbits, Masses of Pluto's Satellites'', Presented at The Pluto System on the Eve of Exploration by New Horizons: Perspectives, Predictions held at APL, Laurel, MD - PLU042 - JPL satellite ephemeris.


The First Two Laws of Planetary Motion

The path of an object through space is called its orbit. Kepler initially assumed that the orbits of planets were circles, but doing so did not allow him to find orbits that were consistent with Brahe&rsquos observations. Working with the data for Mars, he eventually discovered that the orbit of that planet had the shape of a somewhat flattened circle, or ellipse. Next to the circle, the ellipse is the simplest kind of closed curve, belonging to a family of curves known as conic sections (Figure (PageIndex<2>)).

Figure (PageIndex<2>) Conic Sections.The circle, ellipse, parabola, and hyperbola are all formed by the intersection of a plane with a cone. This is why such curves are called conic sections.

You might recall from math classes that in a circle, the center is a special point. The distance from the center to anywhere on the circle is exactly the same. In an ellipse, the sum of the distance from two special points inside the ellipse to any point on the ellipse is always the same. These two points inside the ellipse are called its foci (singular: focus), a word invented for this purpose by Kepler.

This property suggests a simple way to draw an ellipse (Figure (PageIndex<3>)). We wrap the ends of a loop of string around two tacks pushed through a sheet of paper into a drawing board, so that the string is slack. If we push a pencil against the string, making the string taut, and then slide the pencil against the string all around the tacks, the curve that results is an ellipse. At any point where the pencil may be, the sum of the distances from the pencil to the two tacks is a constant length&mdashthe length of the string. The tacks are at the two foci of the ellipse.

The widest diameter of the ellipse is called its major axis. Half this distance&mdashthat is, the distance from the center of the ellipse to one end&mdashis the semimajor axis, which is usually used to specify the size of the ellipse. For example, the semimajor axis of the orbit of Mars, which is also the planet&rsquos average distance from the Sun, is 228 million kilometers.

Figure (PageIndex<3>) Drawing an Ellipse. (a) We can construct an ellipse by pushing two tacks (the white objects) into a piece of paper on a drawing board, and then looping a string around the tacks. Each tack represents a focus of the ellipse, with one of the tacks being the Sun. Stretch the string tight using a pencil, and then move the pencil around the tacks. The length of the string remains the same, so that the sum of the distances from any point on the ellipse to the foci is always constant. (b) In this illustration, each semimajor axis is denoted by a. The distance 2a is called the major axis of the ellipse.

The shape (roundness) of an ellipse depends on how close together the two foci are, compared with the major axis. The ratio of the distance between the foci to the length of the semimajor axis is called the eccentricity of the ellipse.

If the foci (or tacks) are moved to the same location, then the distance between the foci would be zero. This means that the eccentricity is zero and the ellipse is just a circle thus, a circle can be called an ellipse of zero eccentricity. In a circle, the semimajor axis would be the radius.

Next, we can make ellipses of various elongations (or extended lengths) by varying the spacing of the tacks (as long as they are not farther apart than the length of the string). The greater the eccentricity, the more elongated is the ellipse, up to a maximum eccentricity of 1.0, when the ellipse becomes &ldquoflat,&rdquo the other extreme from a circle.

The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity. Using Brahe&rsquos data, Kepler found that Mars has an elliptical orbit, with the Sun at one focus (the other focus is empty). The eccentricity of the orbit of Mars is only about 0.1 its orbit, drawn to scale, would be practically indistinguishable from a circle, but the difference turned out to be critical for understanding planetary motions.

Kepler generalized this result in his first law and said that the orbits of all the planets are ellipses. Here was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an acceptable cosmos. The universe could be a bit more complex than the Greek philosophers had wanted it to be.

Kepler&rsquos second law deals with the speed with which each planet moves along its ellipse, also known as its orbital speed. Working with Brahe&rsquos observations of Mars, Kepler discovered that the planet speeds up as it comes closer to the Sun and slows down as it pulls away from the Sun. He expressed the precise form of this relationship by imagining that the Sun and Mars are connected by a straight, elastic line. When Mars is closer to the Sun (positions 1 and 2 in Figure (PageIndex<4>)), the elastic line is not stretched as much, and the planet moves rapidly. Farther from the Sun, as in positions 3 and 4, the line is stretched a lot, and the planet does not move so fast. As Mars travels in its elliptical orbit around the Sun, the elastic line sweeps out areas of the ellipse as it moves (the colored regions in our figure). Kepler found that in equal intervals of time (t), the areas swept out in space by this imaginary line are always equal that is, the area of the region B from 1 to 2 is the same as that of region A from 3 to 4.

If a planet moves in a circular orbit, the elastic line is always stretched the same amount and the planet moves at a constant speed around its orbit. But, as Kepler discovered, in most orbits that speed of a planet orbiting its star (or moon orbiting its planet) tends to vary because the orbit is elliptical.

Figure (PageIndex<4>) Kepler's Second Law: The Law of Equal Areas. The orbital speed of a planet traveling around the Sun (the circular object inside the ellipse) varies in such a way that in equal intervals of time (t), a line between the Sun and a planet sweeps out equal areas (A and B). Note that the eccentricities of the planets&rsquo orbits in our solar system are substantially less than shown here.


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In: Astronomical Journal , Vol. 159, No. 2, ab5c1d, 02.2020.

Research output : Contribution to journal › Article › peer-review

T1 - Mutual Orbital Inclinations between Cold Jupiters and Inner Super-Earths

N1 - Publisher Copyright: © 2020. The American Astronomical Society. All rights reserved.

N2 - Previous analyses of Doppler and Kepler data have found that Sun-like stars hosting "cold Jupiters" (giant planets with a ≈ 1 au) almost always host "inner super-Earths" (1-4 R ⊕, a ≲ 1 au). Here we attempt to determine the degree of alignment between the orbital planes of the cold Jupiters and the inner super-Earths. The key observational input is the fraction of Kepler stars with transiting super-Earths that also have transiting cold Jupiters. This fraction depends on both the probability for cold Jupiters to occur in such systems and the mutual orbital inclinations. Since the probability of occurrence has already been measured in Doppler surveys, we can use the data to constrain the mutual inclination distribution. We find σ = 11.°8-5.°5+12.°7 (68% confidence) and σ > 3.°5 (95% confidence), where σ is the scale parameter of the Rayleigh distribution. This suggests that planetary orbits in systems with cold Jupiters tend to be coplanar - although not quite as coplanar as those in the solar system, which have a mean inclination from the invariable plane of 1.°8. We also find evidence that cold Jupiters have lower mutual inclinations relative to inner systems with higher transit multiplicity. This suggests a link between the dynamical excitation in the inner and outer systems. For example, perturbations from misaligned cold Jupiters may dynamically heat or destabilize systems of inner super-Earths.

AB - Previous analyses of Doppler and Kepler data have found that Sun-like stars hosting "cold Jupiters" (giant planets with a ≈ 1 au) almost always host "inner super-Earths" (1-4 R ⊕, a ≲ 1 au). Here we attempt to determine the degree of alignment between the orbital planes of the cold Jupiters and the inner super-Earths. The key observational input is the fraction of Kepler stars with transiting super-Earths that also have transiting cold Jupiters. This fraction depends on both the probability for cold Jupiters to occur in such systems and the mutual orbital inclinations. Since the probability of occurrence has already been measured in Doppler surveys, we can use the data to constrain the mutual inclination distribution. We find σ = 11.°8-5.°5+12.°7 (68% confidence) and σ > 3.°5 (95% confidence), where σ is the scale parameter of the Rayleigh distribution. This suggests that planetary orbits in systems with cold Jupiters tend to be coplanar - although not quite as coplanar as those in the solar system, which have a mean inclination from the invariable plane of 1.°8. We also find evidence that cold Jupiters have lower mutual inclinations relative to inner systems with higher transit multiplicity. This suggests a link between the dynamical excitation in the inner and outer systems. For example, perturbations from misaligned cold Jupiters may dynamically heat or destabilize systems of inner super-Earths.


Eccentricity

We have now given a parameter of the size of the orbit, but to completely define the orbit we also need to know it’s shape. This is given by the second orbital parameter, the eccentricity . It can be defined from and , and is quite simple to calculate. For an elliptical orbit it is given as

It does not matter what units the two radii are given in as the eccentricity is unitless. As an example, what is the eccentricity of a circle? Since a circle has constant radius, we must have that making the eccentricity . An ellipse will have an eccentricity from up to (but not including) . What kind of orbits we get for even higher eccentricities, we will come back to shortly. Ellipses with different eccentricities are shown in figure 2.

In figure 1 we can see that the radius r is given from the central body to the center of the satellite. The angle is the angle between the semi-major axis and the line between the central body to the satellite and varies in time when the satellite orbits the central body. When the satellite is at perigee the angle is and when it is at apogee the angle is . The angle is most often called the true anomaly and is the third orbital parameter. It describes the orbital position of the satellite at any specific time.

Now we have found the first three parameters describing the orbit and the satellite position in it. The following expression

describes the orbit in polar coordinates as a function of the true anomaly (see below). We states previously that ellipses has eccentricities from 0 up to 1, and for these cases the radius will be well defined for all angles as can easily be seen.

Figure 3: Definition of the true and eccentric anomalies. Illustration derived from a derived work by the Wikimedia-user CheChe (original by Brews ohare). Licenced under CC BY-SA 4.0.

The true anomaly is one of the three anomalies/parameters describing the position around the orbit. The other two anomalies are called the eccentric anomaly and mean anomaly, and they are used to relate the position of the satellite (from the true anomaly) to the time since passing of the perigee. The eccentric anomaly is an actual angle shown in figure 3. The position of the satellite is at the point P with true anomaly . The blue (largest) circular orbit has a constant radius equal to the semi-major axis of the satellite orbit (shown in red in the figure). The eccentric anomaly is the left angle in the triangle CQP’, where the line segment QP’ is the line segment perpendicular to the periapsis–apoapsis line through the point P (the actual position of the satellite) to the point P’, which is the intersection with the circular blue orbit (the point Q is not marked on the figure). This makes the length of the hypotenuse (the line segment CP’) the satellite semi-major axis. The relation between the true anomaly and the eccentric anomaly is

The mean anomaly is not a true angle. It is defined as where is the time since the last crossing of periapsis and , where is the orbital period of the satellite orbit. The mean anomaly is an angle an imaginary satellite would have if it where in a circular arbit around the point C (figure 3) with a time period equal to the true satellite orbit. The reason of why we define the eccentric and mean anomaly is to find the relationship between the true anomaly and the time since periapsis. The relationship between the mean and eccentric anomaly is

where is the eccentricity of the (real) satellite orbit. This equation is a transcendental equation that cannot be solved analytically for but several resources online can help with solving equations like this, e.g. WolframAlpha.com. Knowing the true anomaly, the time since periapsis can be found by first calculating the eccentric anomaly and then the mean anomaly. Knowing the time since periapsis, however, the mean anomaly is first calculated, but finding the eccentric anomaly need to be found numerically. That solution is then used to find the true anomaly.


Summary:

With only 30 days of observation, and assuming that this 30 days is < 10% of a year , similar to Earth, you can get:

  • Latitude, accurate to 500m. Seriously! It's only limited by your resolution of photo, and measurement of horizon. (and Oblateness of Planet, but day length and gravity will give you an order-of-magnitude estimate of that, and it is a very small error)
  • Day length, Accurate to about 1-2 seconds. Limited by: visual observation of star occultation by horizon.
  • Year length, accurate to well under one day. Accurate to under an hour if the orbit is circular, but it never is, quite.
  • Longitude: perfect, because its arbitrary
  • Moon orbit periods: Accurate to <10 minutes, IF your stay is longer than one orbit. Otherwise To the hour. Limited by: visual observation of occultation of a star(if there long enough), else visual of horizon touch.

Moon size, planet size, sun size/ orbit distance: rough guesswork based on how heavy you feel. Really just guesswork, unless your cellphone includes some form of accurate distance measurement.


Watch the video: ClassicalKeplerian Orbital Elements (August 2022).