# How would I observe the trajectory of the planet Mercury from an inertial frame of reference?

If i were to make an observation of the orbital motion of the planet Mercury from an inertial frame of reference, would I observe the precession of Mercury's perihelion? or would I observe it moving in the way predicted by Newton's mechanics? And if I will observe the precession of Mercury's perihelion, would my frame of reference still qualifies as being inertial?

• An example of an inertial frame of reference would be that of a satellite or astronaut orbiting the Earth. Can you explain why you suspect that they might come to a different conclusion about the orbit of Mercury than an astronomer on Earth's surface would?
– Sten
Commented Aug 7, 2023 at 15:01
• What i think is since you're in an inertial frame of reference, you should see things as the laws of physics in this frame predict, so given an inertial frame, the laws of physics says that the trajectory of the planet Mercury is elliptical with no precession of its perihelion
– Jack
Commented Aug 7, 2023 at 15:15
• There are no globally inertial frames in GR. If you are falling freely your frame is locally inertial, but unless Mercury is local to you it will not obey Newton's laws. If you and Mercury are sufficiently close for a locally inertial frame to include you both then you'd just see Mercury stay motionless relative to you. Commented Aug 7, 2023 at 16:04
• @JohnRennie so the bottom line is if I were to observe the precession of Mercury's perihelion, then the frame of reference from which I am making the observation Is inevitably non inertial, right?
– Jack
Commented Aug 7, 2023 at 17:07
• "or would I observe it moving in the way predicted by Newton's mechanics?" In reality Newton's mechanics predicts the precession of Mercury. It is just not accurate enough, and GR is necessary to refine the prediction. See physics.stackexchange.com/q/696607/195949 Commented Aug 7, 2023 at 23:42

In astrophysics we recognize a hierarchy of gravitationally bound systems.

There is the Earth-Moon system. The Earth and the Moon are orbiting their Common Center of Mass. (Well, the Earth is so much heavier than the Moon that the Common Center of Mass is inside the Earth.)

Next level up is that the Sun and the planets of the Solar System are all orbiting the Common Center of Mass of the Solar System. Of course, all of the motion of the celestial objects of the Solar System is dominated by the sheer mass of the Sun, but still: Jupiter is so heavy (and so far away from the Sun) that the Common Center of Mass of the Sun and Jupiter is a bit outside the Sun.

Next level up is that the Solar System is orbiting the Center of Mass of our Galaxy, together with all of the other stars of the Galaxy.

Then there are clusters of Galaxies, and clusters of clusters, etc.

For the motions of the planets of the Solar System the relevant coordinate system is the inertial coordinate system that is co-moving with the Common Center of Mass of the Solar System.

For Mercury that is the relevant inertial coordinate system, for that is the level in the hierarchy where Mercury resides: the level of the Solar System.

The precession of the perihelion of Mercury is with respect to the inertial coordinate system that is co-moving with the Solar System's Common Center of Mass.

Theoretically all the celestial bodies of the solar system undergo some precession of their perihelion, but for all planets other than Mercury the amount is below detection level.

Since we have the fixed stars anyway we use those as reference. However, it is possible to establish what the inertial coordinate system is using only information that is available in the Solar system itself.

If you use an inertial coordinate system (co-moving with the Solar System's Center of Mass), then the motions of the planets are in accordance with the laws of motion. (Mercury too! For Mercury the relativistic effect is detectable, so for Mercury you will apply the GR law of motion, and if use an inertial coordinate system then then the motion of Mercury is in accordance with the appropriate law of motion.)

Conversely, if for whatever reason you use, say, a rotating coordinate system, then you have to apply corrections, and in the expressions for those corrections you have to enter the angular velocity of the rotating coordinate system with respect to the inertial coordinate system.

So it's a bit like that quip that is attributed to Henry Ford, when he said about the Ford Model T: "Any customer can have a car painted any color that he wants so long as it is black."

In theory of motion:
You can use any coordinate system that you want, as long as you keep the inertial coordinate system as your underlying reference.

• So, do you mean by saying "If you use an inertial coordinate system (co-moving with the Solar System's Center of Mass), then the motions of the planets are in accordance with the laws of motion. (Mercury too! For Mercury the relativistic effect is detectable, so for Mercury you will apply the GR law of motion, and if use an inertial coordinate system then then the motion of Mercury is in accordance with the appropriate law of motion.)" That relative the INERTIAL frame co-moving with the center of mass of the solar system, i wouldn't observe the precession of Mercury's perihelion?
– Jack
Commented Aug 7, 2023 at 17:03