"I think Isaac Newton is doing most of the driving now."

Apollo 8 Lunar Module pilot Bill Anders, when asked who was driving the capsule on the return from the Moon to the Earth, 26 December 1968.

On January 14, 2008, the Messenger spacecraft had a spectacular flyby at Mercury, passing about 200 kilometres (124 miles) above the night-side surface of the planet. While the probe is transmitting an amazing amount of exciting photos from this encounter, I would like to focus here on something more ethereal, the influence of General Relativity on Messenger's orbit.

As reported on the Planetary Society Weblog, the Mercury flyby was a case of "spectacular targeting": Messenger missed the previously planned aimpoint at Mercury by only 1.43 kilometres in altitude, and that after a flight of nearly 100 million km without firing its engines. In fact, Messenger needs some trajectory fine-tuning from time to time, and the last correction before the flyby, trajectory correction manoeuvre 19 (TCM-19), had occurred 26 days before, on December 19, 2007.

Speaking of a space probe flying to Mercury to a bunch of physics aficionados inevitably brings up General Relativity. After all, the explanation of the extra shift of the perihelion of Mercury, a tiny 43 arc seconds per century not accounted for by Newtonian gravitation, was the first big success of General Relativity. So, it seems natural to ask, if there is such a high precision in the determination of Messenger's trajectory, what is the role of General Relativity in this?

While trying to figure out how the engineers at NASA actually are handling gravitation in the trajectory calculations, I realised that a simple back-of-the-envelope calculations already yields a good estimate for the influence of General Relativity on the space probe. Just applying the formula for the relativistic perihelion shift shows that relativistic effects add up to a few kilometres for the trajectory between the TCM-19 correction and the flyby.

Comments by readers who know more about this stuff are welcome!

Actually, Messenger was, in the months before the flyby, on a quite eccentric elliptic orbit in between the orbits of Venus and Mercury. Here is part of an illustration of the orbit from the Messenger web site

The part of the Messenger orbit before the flyby (marked by the arrow on the right hand side) is shown in pale red - it's a nice elliptical orbit. Hence, it seems reasonable to apply the relativistic perihelion formula to both the Messenger and the Mercury orbits, and to look what comes out.

The angular shift of the perihelion per revolution as stemming from relativistic corrections to Newtonian gravitation is given by

Here,

*a*is the so-called semi-major axis of the orbit (that's half the longer diameter of the ellipse), and

*e*the eccentricity - for a circle,

*e*= 0, and the larger

*e*, the more elongated the ellipse. Sometimes, the quantity

*a*(1-

*e*²) is called the semi-latus rectum,

*L*(sorry, the geometry of conic sections is pretty old, hence all the Latin and Greek). Since the perihelion distance,

*p*, is related to the semi-major axis by

*p*=

*a*(1-

*e*), we can also write

The fraction

*GM*/

*c*² is half the so-called Schwarzschild radius - for the Sun,

*GM*/

*c*² ∼ 1.5 km. Since this is very small compared to Mercury's perihelion distance of 46 million km from the solar centre, the perihelion shift per revolution is a tiny angle.

However, what we actually need to know when we want to navigate a space probe very close to Mercury is not this angle, but the actual motion of the perihelion, as measured in kilometres. This motion then quantifies the offset Δ along the orbit due to relativistic effects. But it is easy to calculate: Since the angle is given in radians, we just have to multiply by the perihelion distance, and obtain

Curiously, this offset only depends on the eccentricity, and is even there if the orbit is a perfect circle!

Now, we can apply this formula to the orbits of Mercury and Messenger and plug in some numbers:

The eccentricity of Mercury is

*e*= 0.20, which yields Δ = 25 km. This is the relativistic offset of the orbit that accumulates over the 88 days of one revolution. Now, however, we are not dealing with an entire revolution. But the crucial point is that for shorter periods, we can just take the respective part of this shift. Thus, for the 26 days between the trajectory correction manoeuvre 19 (TCM-19) on December 19 and the Messenger flyby on January 14, the relativistic offset amounts to about 7 km.

The orbital elements of Messenger can be obtained from the JPL Horizons web site. This is a very cool interactive site where you can get all kinds of coordinates for nearly all the Solar System! The elliptic orbit Messenger was on around the Sun on, say, January 1, 2008 had an eccentricity

*e*= 0.38, and a period of about 140 days. The relativistic offset Δ of this orbit amounts to 22 km, and scales down, for 26 days, to about 4 km.

Thus, taken all together, the relativistic effects on the Messenger trajectory over the period of 26 days since the last correction manoeuvre can add up to an uncertainty of a few kilometres at the moment of the Mercury flyby.

Since the height of the flyby was about 200 km above ground, this is probably not critical - Newton is safe enough a pilot. However, the trajectory was calculated to a much higher precision, and since an "experimental" error of 1.43 kilometre of the actual versus the calculated trajectory could be measured, it's clear that all the calculations are done with a strong assisting hand by Einstein!

Calculating orbits around a central mass in General Relativity amounts to calculate geodesics in the Schwarzschild metric. As it comes out, for large enough distances from the centre, the motion of a mass corresponds very well to the motion according to Newtonian gravity, but in addition to the 1/

*r*potential of Newton, there is an extra term, proportional to 1/

*r*³. This extra term acts as a perturbation to the Newtonian elliptic orbits, and yields a shift of the perihelion of the ellipse.

Actually, this 1/

*r*³ term is taken into account in the calculation of the Messenger trajectory (thanks to Amara's inquiry)

In general, these days General relativity

- enters the very definition of the barycentric coordinate system of the Solar System,
- is taken into account to calculate the ephemerides (the coordinates as a function of time) of the planets and other Solar System bodies, and
- is taken into account to calculate trajectories of space probes

I've found a lot of theoretical background and useful information on the role of General Relativity in the calculation of trajectories in the "Monograph 2: Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation" by Theodore D. Moyer at the web site of the JPL. It shows, for example, this impressive set of relativistic equations of motion (page 4-19) -

The equation, describing the acceleration of mass

*i*due to gravity, makes use of the so-called PPN formalism of Will and Nordtvedt (1972). The first term is the Newtonian term, and the first term in the curly brackets the first correction by General Relativity, which yields the perihelion shift. All other terms are much smaller (some of them depend on the velocity, that's what is called gravitomagnetism and yields the Thirring-Lense effect...)

However, for actual calculations, not all these terms are really taken into account - it all depends on the precision that is required (and that can be actually measured by spacecraft telemetry)

## 22 comments:

Hi Stefan,

“I realised that a simple back-of-the-envelope calculations already yields a good estimate for the influence of General Relativity on the space probe.”

Stefan, you have to tell me where you buy your envelopes for I simply must have some of those:-)

Seriously though it’s great that you took this a step further by demonstrating what’s involved and actually cranking out some calculations. It certainly would form the basis of a good question after a GR introduction course. This is the sort of thing that makes GR relevant without having to go to neutron stars and black hole scenarios right off the bat. This Perihelion precession of Mercury was the dark matter of the times between Newton and Einstein. Ironically they to where looking for something that couldn’t be seen to explain it all. Not that I’m saying that this applies to the dark matter problem, just pointing out the similarities. What Einstein showed was that there wasn’t something missing to observe but rather something(s) missing to consider. In the current situation I believe as many that the answer will not be found changing GR yet may require us to expand our understandings of matter beyond that of the Standard Model.

Best,

Phil

P.S. I never did consider this solar wind thing as the joke yet rather that it could serve to throw another fly into the ointment;-)

Hi Stefan,

Just a follow up to the earlier comments I made about the importance of making GR relevant to the public at large and how that might be done. These articles you have written and the related discussion is as I have said serves as an example of how GR really counts as being practical and relevant. When thinking about this I remembered another such example that was resultant of a discussion that took place on the old CompuServe Math/Science Forum, probably more the 12 years ago or more. What it entailed constituted as to be a question posed in the forum of a scenario that one might actually encounter. This scenario was as follows;

Suppose that you are driving back from a birthday party from which you have taken with you one of the helium filled balloons. Seeing that you didn’t want the balloon to simply float all over the interior of the car you tied the connecting string to the head rest besides you as to have it float freely just an inch or two off the ceiling of the car. At a stop light you accelerate rapidly when the light turns green. Now the question. Which way upon acceleration would the balloon move in relation to the car’s interior? As a second and related question, how would you describe the resultant movement in terms of physics? I was pleased to discover that I was able to answer both parts correctly when many did not. I was going to supply the answer and indicated relevance yet have decided it might prove fun if I leave it open. I am certain this is far too easy considering the caliber of the group around here, yet never the less it could prove interesting.

Regards,

Phil

You demonstrated the power of a sound physics education. Suitably armed, one can pose a simple question, find the appropriate context to answer it, then turn the crank. No appeal to authorities required.

Really great post! Thanks.

On a less elegant plane... how is the space probe kept cool when it is so close to the sun? Add deep UV and solar wind to have a nighmare material survival environment.

OK, about Mercury first let me repeat the following cute Halloween pictures link:

My girlfriend and I went to a Halloween party as Venus and Mercury, complete with toga and put-up hair, winged golden helmet and sandals, etc. I have to say, she can get away with playing Venus! Go to Link.

The GR angle is fascinating of course. I am still a bit confused over whether gravitomagnetism can be represented as a vector field (in weak gravity approximation) analgous to magnetism, and what formulas work if the relative velocities are very high (e.g. 0.6 c etc.)

Also, re Ken Nordtvedt: he wrote a much and wrongly IMHO forgotten paper (K. Nordtvedt, “The equivalence principle and the question of weight,” Am. J. Phys. 43 (3), 256-257 (1975) about how gravitational fields reduce the apparent weight of objects suspended by long cables. It is like a red or blue shift for weight. IOW, let's say I have an elephant held by a long cable suspended in a gravitational field. The force at my end, holding, is less than the (locally defined force!) mg where the elephant is. (That issue of locality confused his critic Ø. Grǿn! I defended and expanded his explanation to arbitrary weak fields in a 1988 paper in

Physics Essays.) The difference is the same as the red/blue shift f'/f = U'/U, where U is photon energy.This cute effect, which in an extreme case would allow me to hold up an elephant (well, pretending negligible mass of cable) is not talked about in popularizations etc. and is all but forgotten. However, it seems to me it must have important implications.

Hi Stefan - nice back of the envelope computations, thanks! So, the take-home-message is an error of a few kilometers by not taking into account GR effects due to the Sun.

The long equation you show is the same (in a slightly different format) that I have from JPL technical memo 33-451, but the equation itself is referenced to "Moyer, T. D. JPL internal document Jan. 4, 1968".

Best.

Hi Phil,

tell me where you buy your envelopesactually, the frist one was the sheet of paper with the address label that comes with the Scientific American ;-)

... serves as an example of how GR really counts as being practical and relevant.Well, I am not convinced if Messenger is a good example. After all, Newton does the driving, with Einstein for fine-tuning.

That you need General Relativity for precision may be plausible, but it is still quite removed from everyday experience. In fact, I had hoped that the GR effect on the trajectory was bigger by a factor of 20-40 at least. Then, one could have argued that neglecting General Relativity results in a high risk to have an impact instead of a flyby, someting that is obviously spectacular and easy to visualise ;-)

Hi Uncle,

yes, shielding against heat and particle radiation seems to be a big issue that contrains the design of the spacecraft.

Unfortunately, I don't know how to estimate the influence on the orbit...

Best, Stefan

Hi changcho,

So, the take-home-message is an error of a few kilometers by not taking into account GR effects due to the Sun.Yes ...unless I have made some stupid mistake ;-).

But it's of course perfectly in line with the Schwarzschild radius argument you had mentioned before - depending on the length of the free-fall trajectory, the difference between a Newtonian and fully relativistic path sums up to a few Schwarzschild radii, and to 3π = 10 Schwarzschild radii for one not too eccentric, complete orbit.

The long equation you show is the same... referenced to "Moyer, T. D. JPL internal document Jan. 4, 1968".Ah, thanks, that's interesting! Well, GR hasn't changed that much since the 1960's... But then they

couldhave used Einstein to calculate the Apollo trajectories to the Moon ;-)Best, Stefan

Hi Stefan, another great interesting post. I wonder what the consequence of Newton's actual predictions?..ie where would one end up in relation to actual going to Mercury?

Where one ends up via Newton, and where one ends up via Einstein, if one uses a "google-earth", google-mercury type camera? Comparing the differing photographic evidence would be fascinating, the difference of maybe seeing a written message such as S.O.S or seeing no messsage at all?..best pv.

Hi Stefan,

“actually, the first one was the sheet of paper with the address label that comes with the Scientific American ;-)”

I get the same thing with Discover magazine. Show’s one the practical difference between the two

“Then, one could have argued that neglecting General Relativity results in a high risk to have an impact instead of a flyby, something that is obviously spectacular and easy to visualise ;-)”

You are correct that it wouldn’t have crashed if they had ignored GR. To do that one simply needs to confuse metric input with imperial;-)

Seriously though in that Amara has now confirmed that they do take it into account they obviously consider it has some value. Perhaps with fuel always being such a large concern on any such mission this might serve as their motivation rather then avoiding failure. That is less error, less correction, thus less fuel.

Regards,

Phi

Very nice post. These simple results illustrate both interesting physics and its "every-day" applications.

One small point though, the eccentricity of the circle is e=0.

"for a circle, e = 1, and the smaller e, the more elongated the ellipse"

Vagelford, right, this should read

"for a circle, e = 0, and the bigger e, the more elongated the ellipse"

Minor error, rest of the post is very good.

Hi Vagelford, changcho,

the eccentricity of the circle is e=0.oops, of course, that is true - I have corrected the error, and changed the original text from the wrong version "for a circle, e = 1, and the smaller e, the more elongated the ellipse"...

Thank you very much for your careful reading :-), and sorry if I have confused some readers with the mistake.

I hope that the main argument is not flawed, though!

Best, Stefan

Hi Stefan,

“Auch das Beatles”

As perhaps related information NASA announced today that they will be celebrating their 50th anniversary of its first deep space mission (explorer 1) with an attempt to call the ETs, well actually more like serenade them. On this coming Monday Feb 4th at 7:00 pm Eastern time they will point the Deep Space network of antennas at Polaris and broadcast the Beatles song “Across the Universe”. Of course with Polaris being 431 light years away it might be sometime before we can expect the return call.

Regards,

Phil

Interesting post.

I heard that GPS calculations also take in account GR. Does anyone know if that is true?

Michael:

http://relativity.livingreviews.org/Articles/lrr-2003-1/

Dear Bee,

thanks for the pointer!

Section 12 of Neil Ashby's paper, on Global Navigation Systems, has a curious statement about the future European Galileo Navigation System:

Information released in 2006 by the GALILEO project states that relativistic corrections will be the responsibility of the users (that is, the receivers).That's only fair to all those die-hards who are sure they can dismiss General Relativity :-)

And it may offer the cool opportunity to "measure" the relativistic effects by switching off the corrections in the receiver, and watch drift away the coordinates of your desk :-)

Best, Stefan

Stefan,

"Information released in 2006 by the GALILEO project states that relativistic corrections will be the responsibility of the users (that is, the receivers)."

Interesting, however I beleive the reason for this is that they will be the only ones to be able to supply the proper correction information to the users. This I beleive is tied in with their intending to charge for the service. From the same page you cited they also say:

"GALILEO is a project of the European Space Agency, intended to put about 30 satellites carrying atomic clocks in orbit. In contrast to GPS which is free to users, the GALILEO system ultimately will be funded by user fees."

I wonder how they will convince users to pay when the current U.S. system is free?

Regards,

Phil

Stefan - more about GR and its effects on orbits.

In 2006, J. Williams of JPL gave the Brower award lecture at the AAS's Division of Dynamical Astronomy (DDA) meeting; the topic was lunar laser ranging.

Regarding the topic of your post, the main point (see his p. slide 21) is that the effect of GR on the lunar orbit is on the order of cm.

Michael - if you have a subscription to Physics Today, check out the May 2002 issue where they discuss GR and the GPS. Unfortunately, the article itself is not available to non-subscribers, only the article's abstract.

Hi, just to revisit this one more time: E. Lakdawalla of the Planetary Society goes into this and actually got quantitative information from the people doing the navigation for MESSENGER (KinetX) - they use the JPL software.

Check out her excellent post here

Hi changcho,

yes, professional bloggers are faster ;-)... The reader Emily is mentioning who asked the question was indeed me, and I've got the same mail from KinetX' Tom Taylor, but haven't condensed that yet in a blog post... my version will follow over the week...

Best, Stefan

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