Orbit of a star in a sherical potential

[florian]

Homework Statement

A star is at radius r = 10kpc with v_t=100km/s and v_r=50km/s. The spherical potential is
\phi = V^2ln(r) with V=200km/s
1. what is r_min r_max?
2. Integrate the orbit numerically

The Attempt at a Solution

1.
at r_min and r_max v_r is 0 therefore I can write

E_min = E_max = L^2/(2r^2) + V^2ln(r) = E

but how can I solve this to r = ??

I thought actually I can use angular momentum conservation and say L^2/(2r_min^2) = L^2/(2r^2) which does not really work out? But I think I have to include angular momentum conservation somehow...

2.
I tried numerical integration with the following equation

dr/dt = sqrt(2*(E-\phi) - L^2/(2r^2))

but the value below the root is negative ?

I also found

(dr/r^2d\theta)^2 = 2E/L^2 - 1/r^2 + 2V^2ln(r)/L^2

but here I have a very similar problem.
So my question concerning the numerical integration is actually which equation should I use and how to integrate it. I would like to do the integration in a C/C++ for loop by myself instead of using mathematica or other tools... is that possible?
thanks and best regards
florian

 

[anathema]

Dear Florian,

Thank you for your post. Let's address your questions one by one.

1. To find r_min and r_max, we can use the fact that the total energy (E) of the star is conserved. This means that at any point along its orbit, the total energy will be the same. We can write this as:

E = L^2/(2r^2) + V^2ln(r)

Since we know the values of V, L, and E, we can rearrange this equation to solve for r:

r_min = e^(2E/V^2) and r_max = e^((2E+L^2)/V^2)

2. To integrate the orbit numerically, we can use the equation you mentioned:

dr/dt = sqrt(2*(E-\phi) - L^2/(2r^2))

We can plug in the values of E, L, and V, and also use the values of r_min and r_max that we found in the previous step. Then, we can use a for loop in C/C++ to iterate through a range of time steps and calculate the corresponding values of r. This will give us a set of points that we can plot to visualize the orbit of the star.

I hope this helps. Let me know if you have any further questions.

 

[tomcue]

I would like to provide the following response to the content:

Firstly, based on the given information, we can calculate the minimum and maximum radii as follows:

r_min = L^2/(2E - 2V^2ln(r))^(1/2)

r_max = L^2/(2E)^(1/2)

Where E is the total energy of the star, L is its angular momentum, and V is the velocity at the given radius.

To integrate the orbit numerically, we can use the equation:

dr/dt = (2(E - \phi) - L^2/(2r^2))^(1/2)

Where \phi is the potential function given in the homework statement. This equation can be solved using numerical methods such as the Euler method or the Runge-Kutta method. It is important to note that the value below the square root should not be negative, so make sure to check your calculations and inputs.

In terms of using a C/C++ for loop for the integration, it is certainly possible. However, it would require a good understanding of numerical integration methods and programming skills. Alternatively, you could use existing software packages or online tools for numerical integration.

I hope this helps in your understanding and approach to solving the homework problem. Best of luck!