# Orbits problem

## Homework Statement

This is a conceptual question. Does the Earth orbit the sun or does the sun orbit the Earth? I know this is silly of me to ask. After all, everyone learns at a young age the the Earth -obviously- orbits the sun.

We can represent the Earth orbiting the sun by defining the sun to be at the the origin of coordinates, (0,0). Then if the Earth's orbit were perfectly circular, its orbit can be described on a plane by

r = |r|(cos t, sin t)

Suppose we instead choose the Earth to be the center of coordinates. Furthermore, we allow the orientation to be preserved. For example, from the Earth, the positive y' direction is in the same direction as the original positive y direction, and the positive x' direction is in the same direction as the original positive x direction.

Consider the situation at t=0.

In the original setup, the Earth is on the x axis, |r| distance away from origin in the positive x direction. Under the transformation, the sun is on the x' axis, |r| distance away from the origin in the negative x' direction.

In the original setup, the Earth is moving in the positive y direction at t=0. Under the transformation, the sun is moving in the negative y' direction at t=0.

It suffices to show this for t=0. In every case, the transformation only reverses the sign, but everything else is preserved (the radius, the orbiting velocity, even the area that the orbit sweeps out). We can parametrize the sun's orbit relative to the Earth as

r' = |r|(cos (t-π), sin (t-π))

The sun is really (also) orbiting the Earth!

Generalizing this further, the shape doesn't have to be a circle. It can be an ellipse too. I haven't worked out the details, so I'm not going to claim other shapes (such as the hyperbola and the parabola).

Given that this transformation works and that the situations are equivalent, why is heliocentrism historically significant? Why did silly people (like Galileo) defend this notion with their lives? Why do most people learn about heliocentrism in elementary school, but not about the fact that heliocentrism is mathematically the same as geocentrism? Why is Galileo largely praised for defending heliocentrism, and why do most elementary school students learn about this like it's such an important fact (or at least, that's the impression I got)?

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## Answers and Replies

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Why should the earth orbit the sun.
Why not sun orbit the earth

No good at mathematics.
The sun has greater pulling force than the earth.
Now who has greater dictating power on others(jupiter, mars, neptune..)?
Just like a car has greater pull than the friction.
Just like a bird colliding with a train.

Why should the earth orbit the sun.
Why not sun orbit the earth

No good at mathematics.
The sun has greater pulling force than the earth.
Now who has greater dictating power on others(jupiter, mars, neptune..)?
Just like a car has greater pull than the friction.
Just like a bird colliding with a train.
Any two body problem depends on the mass of both objects, even if one is negligible. This includes both force and momentum equations. In fact, such equations are always symmetrical to both bodies.

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Now youre talking about mass.
Is the mass of earth equal to mass of the sun?
Then they are like binary stars orbiting each other.

D H
Staff Emeritus

The sun is really (also) orbiting the Earth!
Yes, you can look at things this way.

There's a problem, however. What happens when you look at the orbits of other planets from the perspective of this Earth-centered point of view? Those other orbits take on a very complex shape from this perspective. When you look at the planets as orbiting the Sun, all of the orbits are more or less ellipses.

An even better (simpler) point of view is to look at both the Earth and the Sun as orbiting their center of mass. Now you are looking at things from the perspective of an inertial frame. Since the mass of the Sun is about 330,000 times that of the Earth, treating the Sun as the center is fairly close to this inertial point of view.

Yes, you can look at things this way.

There's a problem, however. What happens when you look at the orbits of other planets from the perspective of this Earth-centered point of view? Those other orbits take on a very complex shape from this perspective. When you look at the planets as orbiting the Sun, all of the orbits are more or less ellipses.

An even better (simpler) point of view is to look at both the Earth and the Sun as orbiting their center of mass. Now you are looking at things from the perspective of an inertial frame. Since the mass of the Sun is about 330,000 times that of the Earth, treating the Sun as the center is fairly close to this inertial point of view.
What about the Tychonic system, proposed by Tycho Brahe? In the Tychonic system, the sun and and moon rotate around the Earth, and the rest of the planets rotate around the sun.

The reason I ask is that on the atomic scale, an electron can either be seen as orbiting the proton or vice versa. The Bohr formulation of quantum mechanics involved an electron rotating around a proton as the model for the hydrogen atom. What if, instead, it was postulated that the proton orbited around the electron? Rather than the electron being represented by a delocalized wave around a proton, instead, the proton is represented by a delocalized wave around the electron?

Would this result in a working model for quantum mechanics?

D H
Staff Emeritus

What about the Tychonic system, proposed by Tycho Brahe? In the Tychonic system, the sun and and moon rotate around the Earth, and the rest of the planets rotate around the sun.
It doesn't work, at least not to any reasonable degree of accuracy. It also doesn't account for things such as annual paralax. Brahe's main objection to Copernicus' notions was that the Earth is a hulking, lazy body, unfit for motion. He had no idea how much more massive the Sun is than is the Earth.

The reason I ask is that on the atomic scale, an electron can either be seen as orbiting the proton or vice versa.
No, it can't.

The Bohr formulation of quantum mechanics involved an electron rotating around a proton as the model for the hydrogen atom.
The Bohr model is an obsolete model. It is not how electrons orbit.

It doesn't work, at least not to any reasonable degree of accuracy. It also doesn't account for things such as annual parallax. Brahe's main objection to Copernicus' notions was that the Earth is a hulking, lazy body, unfit for motion. He had no idea how much more massive the Sun is than is the Earth.
I'm not sure what you mean. How can there be an accuracy difference if mathematically if the two situations are equivalent? Shifting reference frames shouldn't shift the magnitudes of any measurement (especially relativistically?).

The Bohr model is an obsolete model. It is not how electrons orbit.
The way I learned quantum mechanics is:
1. Blackbody Radiation → quantization of energy → Bohr theory
2. Because Bohr's theory was inadequate, the Schrodinger equation was written to model the electron as a wave around the nucleus rather than an actual orbiting particle.

I'm NOT a grad student learning advanced quantum theory (for example, I haven't learned Heisenberg's or Dirac's theories), but I'm pretty sure that the Schrodinger wave equation is current with modern physics. In the Schrodinger theory, the nucleus is treated like a point particle, and the electron is described by a probability distribution around this point particle. Of course, this also means that the electron doesn't truly orbit the nucleus in the classical sense.

I'm under the impression that postulating an electron distribution around a point particle nucleus came from the historical artifact that the electron was classically viewed as "orbiting" a point particle nucleus rather than the other way around. This leads me to question what happens if we instead fix the electron as a point particle and then describe the proton around the electron using a probability distribution.

First, is this possible? Why or why not? Second, what would such a hypothetical "proton orbital" look like?

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D H
Staff Emeritus

I'm not sure what you mean. How can there be an accuracy difference if mathematically if the two situations are equivalent? Shifting reference frames shouldn't shift the magnitudes of any measurement (especially relativistically?).
You specifically mentioned Tycho Brahe's model. It is not mathematically equivalent to our real solar system. Instead you need to add fictitious forces that account for the acceleration of the Earth toward the Sun and the planets. Another set of problems are the aberration of light and stellar parallax. How do you explain that with a Tychonic model? The primary influence of Brahe's work was that it served as a springboard for Kepler and his laws.

I'm under the impression that postulating an electron distribution around a point particle nucleus came from the historical artifact that the electron was classically viewed as "orbiting" a point particle nucleus rather than the other way around. This leads me to question what happens if we instead fix the electron as a point particle and then describe the proton around the electron using a probability distribution.

First, is this possible? Why or why not? Second, what would such a hypothetical "proton orbital" look like?
No.

You are ignoring the fact that a proton is 1836 times as massive as is an electron. It just doesn't make sense to look at things from the perspective of the electron being fixed. You are ignoring that except for hydrogen, neutral atoms have more than one electron. Which electron is fixed? Finally, you are ignoring that the concept of force doesn't carry much weight in quantum mechanics. The fictitious forces that make it possible to have a geocentric point of view for the motions in the solar system don't carry over to quantum mechanics.