Orbits: explicit r(t) and theta(t)

[nrqed]

I am sure this is a stupid question but can someone give me the equations for r(t) and theta(t) for an orbit an inverse square law (treating the central mass as being at rest). The books always seem to only give r(theta) i.e. the shape of the orbit but not the actual time evolution. Is it ever done explicitly?

thanks

 

[D H]

A more proper treatment is treating the center of mass as being at rest. If the orbiting body is of negligible mass, the central mass will essentially be at rest.

You are most likely talking about the equation

$$r=\frac{a(1-e^2)}{1+e\cos\theta}$$

where $\theta$ is the true anomaly. The time evolution in terms of $\theta$ does not have an explicit form.

An explicit form for the time evolution of the mean anomaly does exist, and it is quite simple in form:

$$M(t)=M(t_0) + (t-t_0)\dot M$$

The mean anomaly is related to the eccentric anomaly $\psi$ via Kepler's equation:

$$M(t) = \psi(t) - e\sin\psi(t)$$

The eccentric anomaly is in turn related to the true anomaly by

$$\sqrt{1-e}\tan\frac{\theta}2 = \sqrt{1+e}\tan \frac{\psi(t)}2$$

 

[tacman]

The equations for r(t) and theta(t) for an orbit under an inverse square law can be derived using Newton's laws of motion and the law of universal gravitation. First, let's define some variables:

- r = radial distance from the central mass
- theta = angle of the position vector with respect to a reference direction
- t = time

Using these variables, we can write the equations for r(t) and theta(t) as:

r(t) = r0 / (1 + e*cos(theta(t))) (1)

and

theta(t) = omega*t + theta0 (2)

where r0 is the initial radial distance, e is the eccentricity of the orbit, omega is the angular velocity, and theta0 is the initial angle.

Equation (1) represents the radial distance as a function of time, and it takes into account the shape of the orbit (determined by the eccentricity) and the initial radial distance. This equation is derived from the conservation of energy and angular momentum for a central force, such as gravity.

Equation (2) represents the angular position as a function of time, and it takes into account the angular velocity and the initial angle. This equation is derived from the conservation of angular momentum for a central force.

Together, equations (1) and (2) provide the explicit time evolution of an orbit under an inverse square law. However, these equations may not always be explicitly given in textbooks or other sources, as they can be derived from more general equations for central force motion.

I hope this helps to answer your question. Remember, there are no stupid questions when it comes to learning! Keep asking and exploring.