"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-e²) 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/c² is half the so-called Schwarzschild radius - for the Sun, GM/c² ∼ 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/r³. 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/r³ 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)