<p>I am by no means an authority on this area but prompted by user /u/called2voyage I will refer the 1994 Astronomical Journal paper by Shahid-Saless (Colorado) and Yeomans (JPL) <a href="https://www.google.co.uk/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=6&amp;cad=rja&amp;uact=8&amp;ved=0ahUKEwic2LfchcXZAhXL-qQKHeJxAusQFghZMAU&amp;url=http%3A%2F%2Fadsabs.harvard.edu%2Ffull%2F1994AJ....107.1885S&amp;usg=AOvVaw1sOMUCKcBLnUaI427q9N7S" rel="nofollow noreferrer">Relativistic Effects on the Motion of Asteroids and Comets</a>.</p>

<p>To paraphrase their abstract: They study the <strong>predicted</strong> effects arising from relativistic perturbations on the motions of asteroids and comets and show that for a number of such objects, inclusion of relativistic contributions in the equations of motion give rise to significant improvements in the orbital solutions.  They go on to argue that ignoring such corrections to the equations of motion gives incorrect solutions.  They point out how the use of masses derived from relativistic ephemerides together with purely Newtonian equations of motion results in an inconsistent hybridized non-Newtonian, non-relativistic model.</p>

<p>With respect to the posted question:-
<strong>(a) are the existing observational data on any Asteroids or Comets sufficient (in terms of accuracy and extent of observations) to determine/estimate rates of non-Newtonian perihelion advance?</strong></p>

<p>...the authors short-list 15 asteroids with the largest <strong>predicted relativistic</strong> perihelion precession rates (from a long-list of 156 commonly-studied objects).  For example Icarus has the greatest rate (0.101 arcseconds/year) and had (in 1993) been observed over 43 years thereby producing a predicted cumulative relativistic precession over this period of 4.34 arcseconds. Observations referred to are a mix of optical and radar (no spacecraft telemetry) and are those used to produce the ephemerides model from which orbital elements are extracted.</p>

<p>The authors note that, assuming a (then) current observational accuracy of a few tenths of a second of arc, the relativistic contribution to precession should be detectable for Icarus and other asteroids near the top of the (ranked by precession-rate) list. As with the planets the majority of asteroid perihelion precession will be caused by the perturbing influences of other planets, but the authors do not present calculated values for these contributions).</p>

<p><strong>(b) are any such determinations publically available?</strong></p>

<p>...the authors selected six asteroids whose motions are significantly affected by general relativistic effects. They then calculated a set of orbital elements for each orbit using JPL's development ephemeris DE200 (with Earth and Moon perturbations treated separately).  They did this in two ways, firstly with Newtonian equations of motion alone and secondly with the addition of non-Newtonian General Relativisitic equations of motion.</p>

<p>The authors indicate that the residuals (differences between observed and predicted positions at various times) are improved (i.e. reduced) by application of non-Newtonian corrections to the equations of motion. For example the RMS residual for Icarus is improved by 30%. Thus it is technically possible to calculate differences in the orbital behavior (such as perihelion advance) predicted by (i) the pure-Newtonian model and  (ii) the Newtonian plus non-Newtonian model.  </p>

<p>However the paper does not actually present explicit accounts of total observed perihelion precession or a breakdown into Newtonian and non-Newtonian components.  In any case, due to complexity, such a breakdown could only apply over the period of observations which is limited (maximum 61 years in 1993) compared to the timescale (a few hundreds of years) over which (predominantly Newtonian) planetary-induced perturbations might be expected to average out.  </p>

<p>However it would be possible (assuming there are no major perturbing encounters with planets or the asteroid belt) to use the provided asteroid orbital element data in a numerical orbit simulator program to roughly model the perihelion precession rate of an asteroid orbiting the Sun in the presence of the other planets using Newtonian equations of motion (i) without and (ii) with relativistic modifications calculated from $\frac{F}{m} = \frac{GM}{r^2} \left(1+\frac{3V_t^2}{c^2}\right) = \frac{GM}{r^2} \left(1+\frac{3GM.SLR}{c^2.r^2}\right)$ (see <a href="https://astronomy.stackexchange.com/questions/7900/can-general-relativity-indicate-phase-dependent-variations-in-planetary-orbital?rq=1">Walter-Goldstein-Schwartzchild Formulation</a> ).  That way the amount of asteroid precession resulting from each of the two different sources (Newtonian, non-Newtonian) could be modelled.</p>

<p>Note: As /u/UhOh/ commented, an alternative, possibly superior, equation for the extra, relativistic acceleration is given in the Shahid-Sales &amp; Yeomans paper cited above as Eqtn 3.11, on page 1886.</p>

CC BY-SA 3.0
2018-02-28T11:18:28.407Z
`+1` this is a great answer! I'm glad you were persuaded to take the time to post it. I've discussed how to propagate the orbits of solar system bodies including at least some of the GR effects using the equations cited by JPL in their ephemerides in [this answer](https://space.stackexchange.com/a/23409/12102) which turns out to be Eq. 3.11 in the paper you've cited, so I've added a link there as well, thanks!
uhoh
CC BY-SA 3.0
2018-02-28T13:45:26.980Z
@uhoh Thanks for linking your answer which is very interesting to me. I had skimmed over the math in the Shahid-Saless/Yeomans paper but now I see their Eq. 3.11 (and your equivalent eqtn).  It seems that in terms of content they differ a little from the eqtns I presented (although possibly not much difference in effect) e.g. the terms $SLR$ and $4(R.V)V$ are not common to both.
steveOw
CC BY-SA 3.0
2018-03-01T00:32:18.150Z
Thanks for the info! I didn't look too closely because all of the other papers I'd read seemed to end up showing *mathematically equivalent*  equations, though it takes a few minutes to verify. Okay I'll check this one closer.
uhoh
CC BY-SA 3.0
2018-03-01T03:17:40.850Z
@uhoh I am interested to see what you get. It seems to me that Shahid-Saless/Yeomans eqtn 3.11, RHS, 2nd term should have sign '-' not '+'.  Also the $4(r.v)v$ term seems to generate a transverse acceleration term in $4VrVt/c^2$ which leads to an additional 133% of apsidal precession over and above the (correct) amount produced by the other terms.  But there might be a fault in my vector algebra.
steveOw
CC BY-SA 3.0
2018-03-01T05:48:48.817Z
Okay now you've got me curious! I'll take a look, though I'm sure I won't have any additional insight beyond what you've found already
uhoh
CC BY-SA 3.0
2018-03-02T02:19:49.830Z
@uhoh. I went through my sums again and found that Eq.3.11 (and your equation) produce the "correct" relativistic precession.  Sorry for wild goose chase.  But it interests me that this formulation (unlike the Walter-Goldstein-Schwarzchild formulation which I linked) involves transverse acceleration as well as radial acceleration.
steveOw
CC BY-SA 3.0
2018-03-07T20:54:49.507Z
@uhoh I have now added reference to Shahid-Saless & Yeomans eqtn 3.11 to the answer.  That equation would make a useful extra answer to [this old question of mine](https://astronomy.stackexchange.com/questions/7900/can-general-relativity-indicate-phase-dependent-variations-in-planetary-orbital?noredirect=1&lq=1).  Would you maybe like to provide such an answer? If not I would quite like to add such myself sometime.
steveOw
CC BY-SA 3.0
2018-03-07T22:08:51.323Z
I've been "remiss" in my attention to this, and due to constraints that will continue another day or two. Translation: oh! I forgot about this! Please post an answer if you can. If not and I can think of something, it would happen this weekend. Thanks for the ping and reminder!
uhoh
CC BY-SA 3.0
2018-03-08T00:08:49.513Z
@uhoh No worries, I'll have a go at posting an answer :)
steveOw
CC BY-SA 4.0
2023-01-17T18:13:55.067Z
@uhoh If you are still interested in this area you may be interested in my new SE Astronomy question:[https://astronomy.stackexchange.com/questions/51649/problem-with-the-orbital-precession-predicted-by-the-relativistic-acceleration-e](https://astronomy.stackexchange.com/questions/51649/problem-with-the-orbital-precession-predicted-by-the-relativistic-acceleration-e)
steveOw
CC BY-SA 4.0
2023-01-17T23:24:51.040Z
I found it quite interesting! I'm your first and so far only upvoter, but I think it will catch on eventually. In a perfect world Physics SE would be a great place for it, but it's kind of a sad place sometimes and sometimes good questions go there to die. There is also [Physics Overflow](https://physicsoverflow.org/) to consider.
uhoh
CC BY-SA 4.0
2023-01-18T03:32:10.017Z
@uhoh Thanks for the sentiments and suggestions!
steveOw
CC BY-SA 3.0
2018-02-27T04:08:11.973Z
2018-03-22T03:05:38.003Z
2018-03-22T03:05:38.003Z
steveOw
<p>Anomalous (i.e. not predicted from Newtonian theory) advances of the perihelion direction have been observed for many solar system planet orbits and have been accounted for by Einstein's General Relativity.  </p>

<p>Basic Orbital elements have been obtained and published for many Asteroids and Comets.  But are the existing observational data on any Asteroids or Comets sufficient (in terms of accuracy and extent of observations) to determine/estimate rates of non-Newtonian perihelion advance and are any such determinations publically available?</p>

CC BY-SA 3.0
2014-09-11T19:07:44.013Z
If you want to write a scientific publication, then search the published literature.
Walter
CC BY-SA 3.0
2014-09-12T23:15:11.570Z
@Walter. I wish I could but I don't have easy access to journals or funds to pay, that's why I use public internet including Stack Exchange.
steveOw
CC BY-SA 3.0
2014-09-13T11:10:52.850Z
Use [ADS](http://www.adsabs.harvard.edu/abstract_service.html) to search for publications (via author names, title words, keywords, or words from the abstract) and even read their abstract for free. Almost all astronomy papers are also published on the arXiv (linked from ADS), where you can obtain (a preprint version) for free.
Walter
CC BY-SA 3.0
2014-09-13T17:21:42.667Z
@Walter.  Many thanks. ADS is a potential goldmine!
steveOw
<gravity><observational-astronomy><general-relativity><orbital-migration>
Are there sufficient observational data to measure non-Newtonian perihelion advances of any Asteroid and Comet orbits?
steveOw
# Are there sufficient observational data to measure non-Newtonian perihelion advances of any Asteroid and Comet orbits?

(Asked by: username_0 on 2014-09-10)

Anomalous (i.e. not predicted from Newtonian theory) advances of the perihelion direction have been observed for many solar system planet orbits and have been accounted for by Einstein's General Relativity.

Basic Orbital elements have been obtained and published for many Asteroids and Comets. But are the existing observational data on any Asteroids or Comets sufficient (in terms of accuracy and extent of observations) to determine/estimate rates of non-Newtonian perihelion advance and are any such determinations publically available?

**Tags:** gravity, observational-astronomy, general-relativity, orbital-migration

> **Comments:**
> **username_1**: If you want to write a scientific publication, then search the published literature.  
> **username_0**: @username_1. I wish I could but I don't have easy access to journals or funds to pay, that's why I use public internet including Stack Exchange.  
> **username_1**: Use [ADS](http://www.adsabs.harvard.edu/abstract_service.html) to search for publications (via author names, title words, keywords, or words from the abstract) and even read their abstract for free. Almost all astronomy papers are also published on the arXiv (linked from ADS), where you can obtain (a preprint version) for free.  
> **username_0**: @username_1.  Many thanks. ADS is a potential goldmine!  

---

## 1 answers

### Answer by username_0 on 2018-02-27. Score: 4

I am by no means an authority on this area but prompted by user /u/called2voyage I will refer the 1994 Astronomical Journal paper by Shahid-Saless (Colorado) and Yeomans (JPL) [Relativistic Effects on the Motion of Asteroids and Comets](https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&cad=rja&uact=8&ved=0ahUKEwic2LfchcXZAhXL-qQKHeJxAusQFghZMAU&url=http%3A%2F%2Fadsabs.harvard.edu%2Ffull%2F1994AJ....107.1885S&usg=AOvVaw1sOMUCKcBLnUaI427q9N7S).

To paraphrase their abstract: They study the **predicted** effects arising from relativistic perturbations on the motions of asteroids and comets and show that for a number of such objects, inclusion of relativistic contributions in the equations of motion give rise to significant improvements in the orbital solutions. They go on to argue that ignoring such corrections to the equations of motion gives incorrect solutions. They point out how the use of masses derived from relativistic ephemerides together with purely Newtonian equations of motion results in an inconsistent hybridized non-Newtonian, non-relativistic model.

With respect to the posted question:-
**(a) are the existing observational data on any Asteroids or Comets sufficient (in terms of accuracy and extent of observations) to determine/estimate rates of non-Newtonian perihelion advance?**

...the authors short-list 15 asteroids with the largest **predicted relativistic** perihelion precession rates (from a long-list of 156 commonly-studied objects). For example Icarus has the greatest rate (0.101 arcseconds/year) and had (in 1993) been observed over 43 years thereby producing a predicted cumulative relativistic precession over this period of 4.34 arcseconds. Observations referred to are a mix of optical and radar (no spacecraft telemetry) and are those used to produce the ephemerides model from which orbital elements are extracted.

The authors note that, assuming a (then) current observational accuracy of a few tenths of a second of arc, the relativistic contribution to precession should be detectable for Icarus and other asteroids near the top of the (ranked by precession-rate) list. As with the planets the majority of asteroid perihelion precession will be caused by the perturbing influences of other planets, but the authors do not present calculated values for these contributions).

**(b) are any such determinations publically available?**

...the authors selected six asteroids whose motions are significantly affected by general relativistic effects. They then calculated a set of orbital elements for each orbit using JPL's development ephemeris DE200 (with Earth and Moon perturbations treated separately). They did this in two ways, firstly with Newtonian equations of motion alone and secondly with the addition of non-Newtonian General Relativisitic equations of motion.

The authors indicate that the residuals (differences between observed and predicted positions at various times) are improved (i.e. reduced) by application of non-Newtonian corrections to the equations of motion. For example the RMS residual for Icarus is improved by 30%. Thus it is technically possible to calculate differences in the orbital behavior (such as perihelion advance) predicted by (i) the pure-Newtonian model and (ii) the Newtonian plus non-Newtonian model.

However the paper does not actually present explicit accounts of total observed perihelion precession or a breakdown into Newtonian and non-Newtonian components. In any case, due to complexity, such a breakdown could only apply over the period of observations which is limited (maximum 61 years in 1993) compared to the timescale (a few hundreds of years) over which (predominantly Newtonian) planetary-induced perturbations might be expected to average out.

However it would be possible (assuming there are no major perturbing encounters with planets or the asteroid belt) to use the provided asteroid orbital element data in a numerical orbit simulator program to roughly model the perihelion precession rate of an asteroid orbiting the Sun in the presence of the other planets using Newtonian equations of motion (i) without and (ii) with relativistic modifications calculated from $\frac{F}{m} = \frac{GM}{r^2} \left(1+\frac{3V\_t^2}{c^2}\right) = \frac{GM}{r^2} \left(1+\frac{3GM.SLR}{c^2.r^2}\right)$ (see [username_1-Goldstein-Schwartzchild Formulation](https://astronomy.stackexchange.com/questions/7900/can-general-relativity-indicate-phase-dependent-variations-in-planetary-orbital?rq=1) ). That way the amount of asteroid precession resulting from each of the two different sources (Newtonian, non-Newtonian) could be modelled.

Note: As /u/UhOh/ commented, an alternative, possibly superior, equation for the extra, relativistic acceleration is given in the Shahid-Sales & Yeomans paper cited above as Eqtn 3.11, on page 1886.

> **Comments:**
> **username_2**: `+1` this is a great answer! I'm glad you were persuaded to take the time to post it. I've discussed how to propagate the orbits of solar system bodies including at least some of the GR effects using the equations cited by JPL in their ephemerides in [this answer](https://space.stackexchange.com/a/23409/12102) which turns out to be Eq. 3.11 in the paper you've cited, so I've added a link there as well, thanks!  
> **username_0**: @username_2 Thanks for linking your answer which is very interesting to me. I had skimmed over the math in the Shahid-Saless/Yeomans paper but now I see their Eq. 3.11 (and your equivalent eqtn).  It seems that in terms of content they differ a little from the eqtns I presented (although possibly not much difference in effect) e.g. the terms $SLR$ and $4(R.V)V$ are not common to both.  
> **username_2**: Thanks for the info! I didn't look too closely because all of the other papers I'd read seemed to end up showing *mathematically equivalent*  equations, though it takes a few minutes to verify. Okay I'll check this one closer.  
> **username_0**: @username_2 I am interested to see what you get. It seems to me that Shahid-Saless/Yeomans eqtn 3.11, RHS, 2nd term should have sign '-' not '+'.  Also the $4(r.v)v$ term seems to generate a transverse acceleration term in $4VrVt/c^2$ which leads to an additional 133% of apsidal precession over and above the (correct) amount produced by the other terms.  But there might be a fault in my vector algebra.  
> **username_2**: Okay now you've got me curious! I'll take a look, though I'm sure I won't have any additional insight beyond what you've found already  
> **username_0**: @username_2. I went through my sums again and found that Eq.3.11 (and your equation) produce the "correct" relativistic precession.  Sorry for wild goose chase.  But it interests me that this formulation (unlike the username_1-Goldstein-Schwarzchild formulation which I linked) involves transverse acceleration as well as radial acceleration.  
> **username_0**: @username_2 I have now added reference to Shahid-Saless & Yeomans eqtn 3.11 to the answer.  That equation would make a useful extra answer to [this old question of mine](https://astronomy.stackexchange.com/questions/7900/can-general-relativity-indicate-phase-dependent-variations-in-planetary-orbital?noredirect=1&lq=1).  Would you maybe like to provide such an answer? If not I would quite like to add such myself sometime.  
> **username_2**: I've been "remiss" in my attention to this, and due to constraints that will continue another day or two. Translation: oh! I forgot about this! Please post an answer if you can. If not and I can think of something, it would happen this weekend. Thanks for the ping and reminder!  
> **username_0**: @username_2 No worries, I'll have a go at posting an answer :)  
> **username_0**: @username_2 If you are still interested in this area you may be interested in my new SE Astronomy question:[https://astronomy.stackexchange.com/questions/51649/problem-with-the-orbital-precession-predicted-by-the-relativistic-acceleration-e](https://astronomy.stackexchange.com/questions/51649/problem-with-the-orbital-precession-predicted-by-the-relativistic-acceleration-e)  
> **username_2**: I found it quite interesting! I'm your first and so far only upvoter, but I think it will catch on eventually. In a perfect world Physics SE would be a great place for it, but it's kind of a sad place sometimes and sometimes good questions go there to die. There is also [Physics Overflow](https://physicsoverflow.org/) to consider.  
> **username_0**: @username_2 Thanks for the sentiments and suggestions!  

---

