[SR] - Test Particle inside the Sun's Gravitational Field - Part 4

[Athenian]

Homework Statement

[Question Context: Consider the motion of a test particle of (constant) mass $m$ inside the gravitational field produced by the sun in the context of special relativity.

Consider the equations of motion for the test particle, which can be written as $$\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F},$$

OR

$$\frac{d(m\gamma \vec{v})}{dt} = \vec{F},$$

where $\vec{v}$ is the speed of the test particle, $c$ is the (constant) speed of light, and by definition, $$\gamma \equiv \frac{1}{\sqrt{1- \frac{\vec{v}^2}{c^2}}} .$$

In addition, the gravitational force is given by $$\vec{F} \equiv -\frac{GMm}{r^2} \hat{e}_r$$

where $\hat{e}_r$ is the unit vector in the direction between the Sun (of mass M) and the test particle (of mass $m$).]
------------------------------------

Three-Part Question:

1. Study the condition on the orbits which leads to a positive and finite value for the minimum and the maximum for $r(\theta)$, that is $0 < r_{min} \leq r(\theta) \leq r_{max} < \infty$ for all $\theta$ where $r_{min}, r_{max}$ are two positive and finite constants).
In this case, find the expression for the perihelion, $r_{min}$, that is the smallest value for r in the orbit, that is the minimal distance to the Sun. Find also the aphelion, $r_{max}$ , that is the largest (and finite) value for $r$ in the orbit, that is the maximal distance to the Sun.

2. What is the angle between two successive perihelia?

3. For which values of the orbit parameters does the orbit describes a trajectory which, after some revolutions, comes back to the same initial point?

Relevant Equations

Refer below $\longrightarrow$

So, here's an attempted solution:

With $r_{min}$,
$$r_{min} = \frac{1}{B + \frac{\beta}{\alpha^2}}$$

With $r_{max}$,
I get:
$$r_{max} = \frac{1}{B - \frac{\beta}{\alpha^2}}$$
or
$$r_{max} = \frac{1}{\frac{\beta}{\alpha^2}}$$

Other than this, I and the team have absolutely no idea on how to proceed with these difficult questions. Any assistance toward getting us to the correct answer will be much appreciated!

 

[PeroK]

This is about analysing the solutions. Effectively, you are trying to identify the near ellipses and what the constraints are on the initial parameters.

You might have to look up some equations for conic sections.

 

[Athenian]

Hmmm, well, my team tried our best with question 1 and this is all we could come up with ...

$$r_{min} = \frac{1}{B + \frac{\beta}{\alpha^2}}$$
----------------------------
$$r_{max} = \frac{1}{B - \frac{\beta}{\alpha^2}}$$

OR

$$r_{max} = \frac{\alpha^2}{\beta}$$

I am honestly not sure which one of the equations is the correct one for $r_{max}$ ...

Beyond that, for question 2, our team got:
$$\theta_k = \frac{2\pi}{\alpha}k$$
$$\Delta \theta = 2\pi (\alpha^{-1} -1)$$

And, finally, as for question 3, we have absolutely no idea when it's "periodic".

Any additional help or clarifications/confirmations toward the above solution would be greatly appreciated. Thank you!

P.S. This is indeed the final thread post for "Test Particle Inside the Sun's Gravitational Field".

 

[PeroK]

https://en.wikipedia.org/wiki/Eccentricity_(mathematics)

 

[PeroK]

2. What is the angle between two successive perihelia?

This is what I thought was interesting. The formula from GR for the precession of an orbit in radians per revolution is: $$\sigma = \frac{24 \pi^3 a^2}{c^2T^2(1-e^2)}$$
Where $a$ is the semi-major axis, $T$ is the period of the orbit and $e$ is the eccentricty.

You can find out more about this here:

https://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury

For this SR/Newtonian gravitational exercise, I got:$$\sigma = \frac{4 \pi^3 a^2}{c^2T^2(1-e^2)}$$
Which is exactly a sixth of the correct GR figure.

I don't think this is giving too much away as you have to derive that and at least it gives you something to shoot for.

Beyond that, for question 2, our team got:
$$\theta_k = \frac{2\pi}{\alpha}k$$
$$\Delta \theta = 2\pi (\alpha^{-1} -1)$$

This is correct. Can you express this in terms of the period, eccentricty etc.?

 

[Athenian]

I'll definitely give it a try. However, how do I determine what is my semi-major or even semi-minor axis through my past work?

For example, from what I have gathered from question 1 (according to Wikipedia), $r_{min}$ is:
$$r_{min} = \frac{\frac{b^2}{a}}{1+ \sqrt{1- \frac{b^2}{a^2}}}$$

where $\frac{b^2}{a}$ is my semi-latus rectum and $\sqrt{1- \frac{b^2}{a^2}}$ is my eccentricity.

But, despite knowing that, I am still stuck with the question "how do I find $a$ (i.e. semi-major axis) and $b$ (i.e. semi-minor axis) through my past work?

Of course, this question is also what got me to think about how would I be able to find for $a$ in the equation: $\sigma \frac{4\pi^3 a^2}{c^2 T^2 (1-e^2)}$.

Any help would be much appreciated. Thank you!

 

[PeroK]

You can use the formula for $u = \frac 1 r$ in terms of $\theta$ to get the perihelion and apehelion, surely? When is the cosine a max and a min? That should give you the semi-major and semi-minor axes.

You also have the eccentricty $e$ from the solution.

 

[Athenian]

I see. In that case, is what I'm doing below what you mean?

Using $u(\theta) = B \: cos(\alpha \theta) + \frac{\beta}{\alpha^2}$, I can take $\theta = 0$ to find the perihelion:

$$\implies r_{min} = B + \frac{\beta}{\alpha^2}$$

As for the aphelion, I can take $\theta = \pi$ to find:

$$\implies r_{max} = B cos(\alpha \pi) + \frac{\beta}{\alpha^2}$$

Something like that?

 

[PeroK]

That's sloppy.

$$\frac{1}{r_{min}} = u_{max} = B + \frac{\beta}{\alpha^2}$$
$$\frac{1}{r_{max}} = u_{min} = -B + \frac{\beta}{\alpha^2}$$

 

[Athenian]

My apologies. Dumb mistake for missing $u \equiv \frac{1}{r}$. Thank you for the corrections!