<p>First I will specifically address the question referring to <a href="https://ui.adsabs.harvard.edu/abs/1994A%26A...282..663S/abstract" rel="nofollow noreferrer">Numerical expressions for precession formulae and mean elements for the Moon and the planets</a> (<a href="https://articles.adsabs.harvard.edu/pdf/1994A%26A...282..663S" rel="nofollow noreferrer">PDF</a>).  This paper, when discussing the mean elements of the planets states:</p>
<blockquote>
<p>As explained in <a href="https://ui.adsabs.harvard.edu/abs/1988A%26A...202..309B/abstract" rel="nofollow noreferrer">Bretagnon &amp; Francou (1988)</a>, for the variables
<em>k</em>, <em>h</em>, <em>q</em>, <em>p</em> of Mercury, Venus, the Earth and May, <a href="https://en.wikipedia.org/wiki/VSOP_(planets)#VSOP87" rel="nofollow noreferrer">VSOP87</a> contains the mean elements of <a href="https://en.wikipedia.org/wiki/VSOP_(planets)#VSOP82" rel="nofollow noreferrer">VSPO82</a> improved for the terms of
high degree with respect to time by the polynomials taken out of the
general theory of <a href="https://ui.adsabs.harvard.edu/abs/1986A%26A...157...59L/abstract" rel="nofollow noreferrer">Laskar (1986)</a>.</p>
</blockquote>
<p>Following up in Laskar (1086) (<em>Secular terms of classical planetary theories using the results of general theory</em>):</p>
<blockquote>
<p>In VSOP82, some secular terms appear in the semi-major axis, and thus
in the variable <em>p</em>, but the contribution of the term <em>Np</em> in (12) is
negligible for Mercury, Venus, and the Earth; it remains small in the
case of Mars unlike the outer planets... As we deal only with the
inner planets, we shall only compute the contribution of <em>Np</em> for
Mars.</p>
</blockquote>
<p>Based on this, it sounds like for the inner planets, for span of time considered, the accuracy of the semi-major axis was sufficient with just a constant term.  For the outer planets it was necessary to include a <a href="https://en.wikipedia.org/w/index.php?title=Secular_variation&amp;oldid=1044911999" rel="nofollow noreferrer">secular variation</a> (time depancance) as a correction to match the numerically computed evolution of the semi-major axis to the desired degree of accuracy for the time interval considered.</p>
<p>As <a href="https://astronomy.stackexchange.com/users/16685/pm-2ring">PM 2Ring</a> noted in his comment, JPL provides simplified linear approximations to the orbital elements at <a href="https://astronomy.stackexchange.com/questions/48567/constant-semi-major-axes-of-the-planets">Approximate Positions of the Planets</a>.</p>

CC BY-SA 4.0
2022-02-19T16:36:31.877Z
2022-02-19T16:36:31.877Z
GrapefruitIsAwesome
<p>The question can be answered from a historical point of view. It concerns the stability of the solar system, and the question whether it does have long-term stability at all. This question has exercised the minds of theoretical astronomers for a long time.</p>
<p>The first convincing theoretical calculations on the question were made by  Laplace and Lagrange, who &quot;demonstrated that the planets' semi-major axes undergo only small oscillations, and do not have secular terms&quot;. This was &quot;the first major result&quot; on the question of stability for the Solar System&quot;. See (<a href="http://www.scholarpedia.org/article/Stability_of_the_solar_system" rel="nofollow noreferrer">http://www.scholarpedia.org/article/Stability_of_the_solar_system</a>).</p>
<p>On that basis, the mean axes of all the planets should indeed be constant.</p>
<p>But later studies of the 1890s by Poincaré qualified the results of Laplace and Lagrange (see citation above, again): he showed they could not be valid over indefinitely long periods, and the stability question therefore remained open. More recent results and simulations have tended to confirm that there are possibilities for instability, though at low probability.</p>
<p>In the meantime, the small time-dependent component in the case of Mars now detected might be due to either of both of two possible causes.</p>
<p>One such cause would be long-period oscillations of the Laplace-Lagrange type which have not yet been incorporated into the theory of Mars' motion (perhaps due to asteroid perturbations, which certainly do affect Mars and have not yet all been accounted for, because there are so many of them).</p>
<p>The second possible cause, perhaps for part of the time-variation, might be a small non-periodic drift resulting from the effects identified by Poincaré and the simulations. (I don't know whether these latter seeming possibilities have been eliminated from consideration.)</p>

CC BY-SA 4.0
2022-02-19T23:19:34.617Z
2022-02-19T23:34:00.857Z
2022-02-19T23:34:00.857Z
terry-s
<p>in <a href="https://articles.adsabs.harvard.edu/pdf/1994A%26A...282..663S" rel="nofollow noreferrer">Numerical Expressions for Precession Formulae And Mean Elements for the Moon And the Planets (Simon et al., 1994)</a>, the orbital elements of the planets are given for long time durations.</p>
<p>The semi-major axes of Mercury, Venus, and Earth are given as constant, whereas the ones of Mars, Jupiter, Saturn, Uranus, and Neptune are given with time-dependent terms. Why is that so?</p>
<p>In detail: for Mercury is given <span class="math-container">$a = 0.387 098 309 8$</span> AU, yet for Mars <span class="math-container">$a = (1.523 679 341 9 + 3\cdot 10^{-10}t)$</span> AU. <span class="math-container">$t$</span> is measured in millenia of Julian years.</p>

CC BY-SA 4.0
2022-02-19T13:30:34.260Z
For reference, the units of time are thousands of Julian years from J2000
GrapefruitIsAwesome
CC BY-SA 4.0
2022-02-19T15:07:19.657Z
For what it's worth, NASA uses constant RADII for calculations in CSPICE +-15000 years from now. Of course, that's a pretty small time interval compared to the numbers you're using. If you visit ftp://naif.jpl.nasa.gov/pub/naif/generic_kernels/pck/ and click on "pck00010.tpc" you'll see the (constant) values NASA uses
Barry Carter
CC BY-SA 4.0
2022-02-19T15:30:31.630Z
Also, JPL gives a couple of linear approximations for orbital elements of all the planets [here](https://ssd.jpl.nasa.gov/planets/approx_pos.html), derived from E.M. Standish (1992).
PM 2Ring
<orbit><solar-system><celestial-mechanics>
constant semi-major axes of the planets
theorist
# constant semi-major axes of the planets

(Asked by: username_0 on 2022-02-19)

in [Numerical Expressions for Precession Formulae And Mean Elements for the Moon And the Planets (Simon et al., 1994)](https://articles.adsabs.harvard.edu/pdf/1994A%26A...282..663S), the orbital elements of the planets are given for long time durations.

The semi-major axes of Mercury, Venus, and Earth are given as constant, whereas the ones of Mars, Jupiter, Saturn, Uranus, and Neptune are given with time-dependent terms. Why is that so?

In detail: for Mercury is given $a = 0.387 098 309 8$ AU, yet for Mars $a = (1.523 679 341 9 + 3\cdot 10^{-10}t)$ AU. $t$ is measured in millenia of Julian years.

**Tags:** orbit, solar-system, celestial-mechanics

> **Comments:**
> **username_1**: For reference, the units of time are thousands of Julian years from J2000  
> **username_2**: For what it's worth, NASA uses constant RADII for calculations in CSPICE +-15000 years from now. Of course, that's a pretty small time interval compared to the numbers you're using. If you visit ftp://naif.jpl.nasa.gov/pub/naif/generic_kernels/pck/ and click on "pck00010.tpc" you'll see the (constant) values NASA uses  
> **username_3**: Also, JPL gives a couple of linear approximations for orbital elements of all the planets [here](https://ssd.jpl.nasa.gov/planets/approx_pos.html), derived from E.M. Standish (1992).  

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## 2 answers

### Answer by username_1 on 2022-02-19. Score: 2 | Accepted answer

First I will specifically address the question referring to [Numerical expressions for precession formulae and mean elements for the Moon and the planets](https://ui.adsabs.harvard.edu/abs/1994A%26A...282..663S/abstract) ([PDF](https://articles.adsabs.harvard.edu/pdf/1994A%26A...282..663S)). This paper, when discussing the mean elements of the planets states:

> As explained in [Bretagnon & Francou (1988)](https://ui.adsabs.harvard.edu/abs/1988A%26A...202..309B/abstract), for the variables
> *k*, *h*, *q*, *p* of Mercury, Venus, the Earth and May, [VSOP87](https://en.wikipedia.org/wiki/VSOP_(planets)#VSOP87) contains the mean elements of [VSPO82](https://en.wikipedia.org/wiki/VSOP_(planets)#VSOP82) improved for the terms of
> high degree with respect to time by the polynomials taken out of the
> general theory of [Laskar (1986)](https://ui.adsabs.harvard.edu/abs/1986A%26A...157...59L/abstract).

Following up in Laskar (1086) (*Secular terms of classical planetary theories using the results of general theory*):

> In VSOP82, some secular terms appear in the semi-major axis, and thus
> in the variable *p*, but the contribution of the term *Np* in (12) is
> negligible for Mercury, Venus, and the Earth; it remains small in the
> case of Mars unlike the outer planets... As we deal only with the
> inner planets, we shall only compute the contribution of *Np* for
> Mars.

Based on this, it sounds like for the inner planets, for span of time considered, the accuracy of the semi-major axis was sufficient with just a constant term. For the outer planets it was necessary to include a [secular variation](https://en.wikipedia.org/w/index.php?title=Secular_variation&oldid=1044911999) (time depancance) as a correction to match the numerically computed evolution of the semi-major axis to the desired degree of accuracy for the time interval considered.

As [username_3](https://astronomy.stackexchange.com/users/16685/pm-2ring) noted in his comment, JPL provides simplified linear approximations to the orbital elements at [Approximate Positions of the Planets](https://astronomy.stackexchange.com/questions/48567/constant-semi-major-axes-of-the-planets).


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### Answer by username_4 on 2022-02-19. Score: 2

The question can be answered from a historical point of view. It concerns the stability of the solar system, and the question whether it does have long-term stability at all. This question has exercised the minds of theoretical astronomers for a long time.

The first convincing theoretical calculations on the question were made by Laplace and Lagrange, who "demonstrated that the planets' semi-major axes undergo only small oscillations, and do not have secular terms". This was "the first major result" on the question of stability for the Solar System". See (<http://www.scholarpedia.org/article/Stability_of_the_solar_system>).

On that basis, the mean axes of all the planets should indeed be constant.

But later studies of the 1890s by Poincaré qualified the results of Laplace and Lagrange (see citation above, again): he showed they could not be valid over indefinitely long periods, and the stability question therefore remained open. More recent results and simulations have tended to confirm that there are possibilities for instability, though at low probability.

In the meantime, the small time-dependent component in the case of Mars now detected might be due to either of both of two possible causes.

One such cause would be long-period oscillations of the Laplace-Lagrange type which have not yet been incorporated into the theory of Mars' motion (perhaps due to asteroid perturbations, which certainly do affect Mars and have not yet all been accounted for, because there are so many of them).

The second possible cause, perhaps for part of the time-variation, might be a small non-periodic drift resulting from the effects identified by Poincaré and the simulations. (I don't know whether these latter seeming possibilities have been eliminated from consideration.)


---

