Theoretical Radial Velocity Curves

[stardust8297]

Okay... so here's the problem I'm working on. Suppose you're observing a celestial body in space that is following a (relatively small) elliptical orbit around a larger distant celestial body (let's say it's at a distance d such that d>>a, where a is the semi-major axis of the orbit). For simplicity let's treat both the Earth and the larger celestial body as fixed, so the only moving object is the smaller body.

I would like to find a way of expressing the radial velocity (i.e. the component of the object's velocity that lies along the line of sight from the earth) as a function of time, expressed in terms of the angle of inclination i, the semi-major axis a (or perhaps a sin i as one parameter), the eccentricity e, the period P, the longitude of the ascending node capital omega, and the argument of the perihelion lower-case omega.

I am relatively familiar with all of the equations relating each of the parameters to each other, and I can even calculate for a specific time t what you would expect the radial velocity to be (with given initial conditions), but if it's possible I would like some kind of an equation for it (again, "it" being the radial velocity) even if it contains inverse trig functions, so that I can plot my own radial velocity curves to look at, without a large amount (or rather, any amount) of programming knowledge. In other words, I'd like to be able to do it with Mathmematica, if possible.

I don't expect anyone to hand me an equation or anything, but any information, insight, references, etc. relating to this matter would help.

I've been reading some literature on binary star systems, which seems to include some of this kind of stuff. It's been helpful, and I've seen a number of places where people have plotted these kinds of curves, but they don't do the greatest job of explaining how it was done...

Thanks to anyone who has read this far!

 

[AndyW]

Thank you for your interesting question. I am happy to provide you with some information and insight on this matter.

First, let's define some terms for clarity. The radial velocity is the component of an object's velocity along the line of sight from the observer. In the case of a celestial body in orbit, this would be the velocity towards or away from the observer, as seen from Earth.

To calculate the radial velocity of a celestial body in an elliptical orbit, we need to consider the various parameters you mentioned: the angle of inclination (i), the semi-major axis (a), the eccentricity (e), the period (P), the longitude of the ascending node (Ω), and the argument of the perihelion (ω).

One useful equation for calculating the radial velocity is the radial velocity formula, which is given by:

Vr = K * [cos(E) - e],

where Vr is the radial velocity, K is a constant related to the mass of the larger celestial body, and E is the eccentric anomaly, which can be calculated using Kepler's equation:

E = M + e * sin(M),

where M is the mean anomaly, which can be calculated using the period and time (t):

M = 2π * t / P.

Now, to express the radial velocity as a function of time, we can substitute the equations for E and M into the radial velocity formula and get:

Vr = K * [cos(2π * t / P + e * sin(2π * t / P)) - e].

To plot this equation in Mathematica, you can use the Plot function and specify the values of the parameters (i, a, e, P, Ω, and ω) to see how the radial velocity changes over time. You can also plot multiple curves by changing the values of the parameters.

I hope this information helps you in your research. If you have any further questions, please don't hesitate to ask.

 

[sir-pinski]

Hi there,

Thank you for sharing your problem and explaining it in detail. It sounds like you are working on a fascinating problem with many variables to consider.

To start, let's define some terms for clarity:
- Radial velocity: the component of an object's velocity that is directed along the line of sight from the observer (in this case, Earth)
- Semi-major axis (a): half of the longest diameter of an ellipse, representing the average distance between the two foci of the ellipse
- Eccentricity (e): a measure of how "elongated" an ellipse is, with 0 being a perfect circle and 1 being a parabola
- Period (P): the time it takes for an object to complete one full orbit around another object
- Inclination (i): the angle between the orbital plane and the reference plane (in this case, the plane of Earth's orbit)
- Longitude of the ascending node (Ω): the angle between the reference direction (such as the vernal equinox) and the ascending node of the orbit (where the orbit crosses the reference plane from below to above)
- Argument of the perihelion (ω): the angle between the ascending node and the point on the orbit closest to the central body (perihelion for orbits around the Sun)

With these terms defined, we can start to think about how to express the radial velocity as a function of time. One way to approach this is to use the equation for the radial velocity of an object in an elliptical orbit, which is:

v_r = K * [cos(E) - e]

Where K is a constant and E is the eccentric anomaly, which can be calculated using Kepler's equation:

M = E - e * sin(E)

Where M is the mean anomaly, which is related to the orbital period and time by:

M = 2π * t / P

Putting all of these equations together, we can express the radial velocity as a function of time:

v_r = K * [cos(2π * t / P - e * sin(2π * t / P)) - e]

This equation includes all of the parameters you mentioned, except for the angle of inclination. To incorporate that, we can use the sine rule for ellipses:

sin(i) = sin(ω) * sin(Ω + ν)

Where ν is the true anomaly, which can be calculated using