- #1
Athenian
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- 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##).]
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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!
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!