If Earth accelerates, what reference frame is its acceleration relative to?

Question

It should be a very fundamental thing, a very simple question. But there's something I want to understand.

We know that when we throw an apple vertically upwards, it experiences a force of gravity due to the Earth, and in turn, the Earth also experiences a force acting on it, equal in magnitude. That's what Newton's 3rd law says would happen. Although the acceleration of apple towards the Earth is much largerer (because of its must smaller mass compared to Earth's) than the rate at which Earth accelerates towards the Apple. Earth's acceleration is negligible, but it is not zero.

Here's what I want to understand. Since motion is always relative, what if I am asked this question : Apple's and Earth's accelerations are with respect to which observer (or which reference frame)?

I could say that the Apple accelerates with respect to the Earth's reference frame. Because when we consider the acceleration of the Apple relative to Earth, we assume that Earth is at rest even if it is moving (relative motion). Similar to

$a_{AB}$ $=$ $a_A$ $-$ $a_B$

Here $B$ is observing $A$ and $B$ is treated to be at rest (relative to Earth) even though $B$ has its own acceleration, its acceleration is added to $A$ with a negative sign.

So I could say that the Apple is accelerating at whatever rate it is accelerating at, $relative$ to $Earth's$ reference frame. What about Earth's acceleration? I can't say that the earth is accelerating relative to the Apple because in that case its acceleration would be equal to Apple's acceleration with a minus sign. So the Earth accelerates relative to which frame? Also, is it correct to say the Apple is accelerating relative to Earth's frame (just want to confirm).

Answer 1

Apple's and Earth's accelerations are with respect to which observer (or which reference frame)?

In any inertial frame, you will measure the same acceleration of the Earth and the same acceleration of the apple.

One example is the one in which the Earth is initially at rest (neglecting the Earth's motion about the sun, etc).

Another could be the frame in which the sun is at rest (neglecting its motion about the galactic center and due to whatever other gravitational forces it experiences)

Answer 2

The third law asserts that whenever two objects are exerting a force upon each other, such as gravity, the two objects are both accelerating towards the common center of mass.

(And this of course generalizes to any number of mutually force exerting objects.)

In other words, the common center of mass of the objects involved will remain in inertial motion.

Therefore with any collection of objects the natural choice of reference frame to express their acceleration is the common center of mass.

In astronomy the choice of reference frame is guided by the desired level of accuracy. The higher the required level of accuracy, the wider the required scope.

In the case of the Earth-Moon system the Moon and Earth are both revolving around their common center of mass. Actually, the Earth is so much heavier than the Moon that the common center of mass is not somewhere in between, the common center of mass is inside the Earth. Anyway, the Earth center of mass is not in inertial motion. Therefore: for a minimal level of accuracy the common center of mass of the Earth and Moon must be used.

If higher accuracy than that is needed you have to move to the common center of mass of the Solar system as a whole. Jupiter is so heavy that the common center of mass of the Solar system is a bit outside the Sun. To calculate the motion of the Moon you need to express all of the motions of the celestical bodies of the Solar system with respect to the Solar system's center of mass. From that you can evaluate all the gravitational pulls that have a noticable effect, and then proceed to evaluate the future motions of the celestial bodies.

Moving to larger and larger scales:

The Solar system is in orbit around the center of mass of our Galaxy.

Our Galaxy and the Andromeda Galaxy are exerting a gravitational pull on each other, so they are being accelerated towards their common center of mass. Given the distance between the two galaxies, and their masses, it can be predicted after how many billions of years a process of galaxy merging will commence

Answer 3

One way to think of it is that Earth and Apple are accelerated in reference to their common center of gravity.