How do I find the orbital angular momentum of one star in a binary star system?

Question

As a hobby, I've been trying to study binary stars on a physics level, and I'm pretty confused as to how to approach problems since I've never really done this sort of math before. I'm using this formula:

L_i=M_ir_i²ω to find the AM of star i

I know that ω=2π/P where P is the orbital period and that r_i=r(M_other/M_total).

However, I don't know how to actually use this problem when given real masses and periods because I don't know which units to use!

Looking at wikipedia, the period of orbit of Sirius is 50.1284 ± 0.0043 yr and the mass of Sirius A is 2.063±0.023M☉ and Sirius B is 1.018 ± 0.011M☉. Lets say I simplify these values to 50yr, 2M☉, and 1M☉. Now what do I do? Do I need to convert these numbers to other units before I start? And also, what should unit should I use for r???? And does yr mean earth years or something else entirely???

Answer 1

The 50.1284 yr is measured in Earth Years, so about 18.310 Earth days, or 1.58*10^9 seconds.

Generally, units are converted to SI before combining them, but not always.

In theory, you can use whatever units you have, as long as you keep them consistent. Angular momentum has dimensions of [mass*length^2/time], so you can mix any mass, length, and time units you want.

For Example, let's say you had a period of 50 yr, a mass of 2 solar masses (2M_S), and an r_i of 15 AU (1 AU is the distance from the earth to the sun). Then, your angular momentum would be 2π*2M_S*(15 AU)^2/(50yr)

or 18π [solar-mass*AU^2 per year]. 18π is a pretty easy number to work with, and all of the astronomical scale is in the units.

If you converted all of these to SI Units, you would get a period of 1.58*10^9 s, a mass of 3.98*10^30 kg, and a r_i of 2.24*10^12 m. Computing gives an angular momentum of 2π*3.98*10^30 kg*(2.24*10^12 m)^2/(1.58*10^9 s) = 7.97*10^46 [kg*m^2/s]. Here, the units are the standard SI units, but the number is absurdly large.

Use whichever system makes the most sense to you. The only important thing is that you keep your units consistent.

For the value of "r" to use, that is the separation between the two planets. Unfortunately, this is not a fixed number because the Sirius system is elliprical, so the distance changes throughout the orbit. The value of ω also changes throughout the orbit, the 2π/P number just gives you the average angular speed. The angular momentum "L" is a constant, but the r and ω can change, as long as a change in one is balanced by a change in the other.

You would want to use the value of r and ω at the same point during the orbit (for example, the minimum distance combined with the maximum angular speed) to get the angular momentum, and then that angular momentum will be a fixed constant throughout the orbit.