formula for circular orbital velocity?

[tony873004]

Does anyone know the formula to compute orbital velocity of a circular orbit when the orbiting body is not considered a massless particle? Using the formulas for circular orbital velocity of a massless or nearly massless particle compared to the primary (such as a spaceship) does not give good results once the orbiting body has appricable mass compared to the primary body.

Thanks in advance

[Evil Dr Ganymede]

Not off-hand... but do you have the book "Solar System Dynamics" by Carl Murray and Stan Dermott (published by Cambridge University Press, 1999)? That's a very good resource (if somewhat advanced later in the book) for orbital dynamics - it might be in there. My copy's at the office so I can't check it now.

[tony873004]

I do have that book! In fact it's an Amazon link on my Gravity Simulator webpage. If anybody buys it through my link I earn a couple of bucks. So far... no purchases . But I can't find the formula in there. Most of the math in that book is over my head anyway .

An example of "over my head":
Formula 2.129 on page 52 states:
h = (Y*Zdot - Z*Ydot, Z*Xdot - X*Zdot, X*Ydot - Y*Xdot)

so what does h equal?? You can't have commas in a formula!
Actually this isn't the only place where I've seen commas in a formula though, so I'm sure I'm wrong. But I have no formal education in higher mathamatics, so this representation eludes me. And I'd love to know what they're talking about in that section. It relates to converting orbital elements to state vectors. I've got a set of formulas in Gravity Simulator that do the same thing quite well, except my formulas get the Mean Anomony wrong when going from vectors to elements. I get it right going the other direction, from elements to vectors though. But I want it right in both directions. And I know good formulas are staring me in the face here if I could just get past understanding the commas.

[granthutchison]

Add the two masses together before inserting them in the equation for orbital period, and use the separation of the two masses as the orbital radius in the formula - this'll give you how long they take to revolve around the sytem's barycentre. Then use the radial distance of your mass from the barycentre to calculate the velocity that corresponds to that period.

Grant

[granthutchison]

h = (Y*Zdot - Z*Ydot, Z*Xdot - X*Zdot, X*Ydot - Y*Xdot)

It looks like a vector. Think of the commas as separating the x, y and z components of the vector.

Grant

[granthutchison]

Add the two masses together before inserting them in the equation for orbital period, and use the separation of the two masses as the orbital radius in the formula - this'll give you how long they take to revolve around the sytem's barycentre. Then use the radial distance of your mass from the barycentre to calculate the velocity that corresponds to that period.

In fact, you could streamline the above by going through the mean motion (n) in radians per second, rather than the period. Mean motion is given by:

n^2 = G*(M+m)/(a^3)

where M is the primary mass, m is the secondary mass, and a is their mean separation. Multiply n by the body's radius of gyration around the barycentre, and you'll have its velocity.

The orbital radius of mass M is:

R = a*m/(M+m)

and of mass m is:

r = a*M/(M+m)

Multiplying (and remembering to square the radius term) gives you:

V^2 = (G*m^2)/(a * [M+m])

and

v^2 = (G*M^2)/(a * [M+m])

where V is the primary velocity and v is the secondary velocity.

Grant

[tony873004]

Thanks, Grant. Your formulas work perfect! (lol, I'm starting to think that should be my signature line)

My 30 billion km seperation between 1 & 0.05 solar mass objects was only varing by +/- 500,000 km. That's pretty darn circular!

[granthutchison]

Thanks, Grant. Your formulas work perfect!

Ah, the unreasonable effectiveness of mathematics ...

Grant