## Thursday, December 30, 2010

### The Effects Of Special Relativity On Planetary Orbits.

General relativity affects the orbits of planets in ways Newtonian gravity cannot account for. Interestingly, Lemmon and Mondragon explore if special relativity can account for the same behavior predicted by general relativity.   They find that qualitatively it can, but quantitatively it comes up a little short and so the full general relativistic treatment is still needed.

First a reminder: Precession:  Let's remind ourselves the effects coming from general relativity.  The first is the precession of the orbit demonstrated for Mercury in the image above.  The dotted curve shows what is expected from Newtonian gravity alone: an obit that stays fixed in an elliptical shape forever.  The solid line shows how this changes with general relativity: the orbit moves or precesses over time around the sun.

It may help to consider the gyroscope on the right: if you were to put a red dot on the edge of the spinning disk, and watched above, you would not see that it only traces out a circle as it spins, but a precession pattern like the one in the plot above.

Second reminder: A shrunken orbit:  The second effect general relativity makes is that of shrinking the orbit, illustrated in the plot above again for Mercury.  This is an image of Mercury's potential energy.  Planets want to minimize their potential energy and therefore, like a ball on a hill, "roll to the bottom" of their potential.  As can bee seen, the bottom of the potential for Newtonian gravity is at a larger radius than for general relativity.   Therefore, general relativity forces planets to have a smaller radius.

Note: Both of the effects above have been verified experimentally and are major reasons why general relativity was embraced in the first place.

But is full general relativity really needed? Turning back to the paper, the authors decide to work out these effects in special relativity alone.  First, we start with precession.  Using special relativity alone you get:

Where G is the gravitational constant, c is the speed of light, a is the semimajor axis and e is the eccentricity.  This rate of precession is equivalent to 7.16 arcseconds per century.  This should be compared to precession predicted by general relativity which is 43 arcseconds per century.  Therefore, special relativity undershoots by a factor of 6.

And what about the radius? For special relativity the change in radius is a similar story and becomes:
where 1/2ε is the correction. General relativity gives a correction of 3ε, and so therefore again special relativity comes up short by the same factor of 6.

Always the same factor of 6 huh? I guess so, and I don't off of the top of my head know why it should always been a factor of six.  Anyways, despite being off by this factor, it is very interesting that special relativity predicts the same qualitative behavior as general relativity which is absent in Newton.  Furthermore, since calculations in special relativity are significantly simpler than for general relativity, this special relativistic calculation is ideal if you are just trying to give a qualitative picture of what effects are relativistic.  Perhaps a good one for undergraduates?

Anyone want to take a stab at why it is always a factor of 6???
You can in the comments. :)

Tyler J. Lemmon, & Antonio R. Mondragon (2010). First-Order Special Relativistic Corrections to Kepler's Orbits Submitted to American Journal of Physics arXiv: 1012.5438v1

1. JS,

What do I know of relativity? Except, it is about six degrees of freedom!

Or, there is third six missing, to finish the dreaded number, and proving all this is the work of the horned guy!

Anyway, you are on a roll for bringing up the stuff far away in the past!

Have a Happy New Year!

2. Ancient1,

Yes, I wouldn't be surprised if it has something to do with degrees of freedom somehow. Or there there are 6 isometries (3 bootes and 3 relations) in Minkowski space of special relativity, or something like this.

Or maybe it is proof that relativity is evil and that there is a third 6 still around to go with the other two. :)

3. I have absolutely no idea why 6, but I liked the post. Last semester I took Dynamics and Control of Spacecraft and we looked a lot at orbital mechanics. At least in practice, most of the work done in astrodynamics seems to use Newtonian physics. I'd be interested to see such a class taught using relativity. But maybe it's overkill for the practical reality of putting a spacecraft into orbit. In that class, when we designed controllers you simply assumed the controller would properly respond to disturbances, noise, etc. Perhaps to engineers the differences in Newtonian physics and relativity are just considered noise that can be taken care of with a robust controller. Dunno. (astronautics isn't my specialty FWIW)

4. jmb275,

I've been told that for GPS to work as accurately as many need, the effects from general relativity need to be included. So I wouldn't be surprised if some relativistic corrections would be helpful for aeronautics as I would think using something like GPS for positioning to pinpoint accuracy could be useful.

5. This comment has been removed by the author.

6. Oops, I didn't read your comment right, so I deleted what I wrote. As for GPS, the benefit to using relativity is (as I understand) only a matter of a few meters. Considering that most aircrafts are larger than a few meters, it might not be worth the computational effort or cost. I work primarily with small UAVs and the costs are definitely greater than the benefits considering the low-power, lightweight restrictions on such aircraft.

Also, it might be that relativistic effects are taken into account by the GPS processing unit itself (as opposed to the main CPU). For example, you can buy COTS GPS units that have military grade accuracy (which I suspect incorporate relativistic effects). Generally, however, to my knowledge, this is not done by the main CPU in an aircraft.

7. check out http://toe.sytes.net:65333/planetary%20precession.htm