Angular momentum in close binaries

[Kurdt]

Does anybody know how the angular momentum of close binaries was calculated as:

[tex] L=M_sM_p(\frac{GD}{M})^\frac{1}{2} [/tex]

This is something which is plaguing me so any input would be appreciated.

[tex]M=M_s+M_p[/tex], D= total separation between the two stars.

Thanks guys

 

[Labguy]

Does anybody know how the angular momentum of close binaries was calculated as:

[tex] L=M_sM_p(\frac{GD}{M})^\frac{1}{2} [/tex]

This is something which is plaguing me so any input would be appreciated.

[tex]M=M_s+M_p[/tex], D= total separation between the two stars.

Thanks guys

Don't know if that formula always holds as there is almost always angular momentum loss in close binaries with either filled Roche Lobes or actual mass transfer. Too many types (variables) of binary stars to lump into one category. Take a look at: http://www.rri.res.in/ph217/binary.pdf and:
http://www-astro.physics.ox.ac.uk/~podsi/lecture11_c.pdf for a lot on this.

 

[Kurdt]

Yeah I know the gravitational waves leak angular momentum away from the system and also stellar winds from the secondary trapped in magnetic loops. The momentum equation should hold as the radius of orbit changes when the angular momentum is leaked away thus allowing the secondary to fill its Roche lobe and maintain contact with the L1 point. I will have a look at the links though, thanks for replying.

 

[Kurdt]

Ahh. Forgot I posted this. I managed to work it out and for anyone who was interested here's the derivation.

First we start with summing the [tex] \mathbf{v}\times \mathbf{r} [/tex] for each star in the system. This gives.

[tex] \mathbf{L_{orb}} = M_sr_sv_s\mathbf{e}_k+M-pr_pv_p\mathbf{e}_k[/tex]

now [tex] v_s=r_s\Omega [/tex] and similarly for the primary also we can substitute [tex] r_s=\frac{M_p}{M}D [/tex] into the equation and again a similar relation is found for the primary to yield.

[tex]\mathbf{L_{orb}} =(M_s\frac{M^2_p}{M^2}D^2+M_p\frac{M^2_s}{M^2}D^2)\Omega\mathbf{e}_k[/tex]

and manipulation gives

[tex]\mathbf{L_{orb}} =\frac{M_sM_p}{M^2}D^2\Omega(M_s+M_p)\mathbf{e}_k[/tex]


[tex](M_s+M_p)=M[/tex] and [tex]\Omega=(\frac{GM}{D^3})^{1/2}[/tex]

So

[tex]\mathbf{L_{orb}} =\frac{M_sM_p}{M^2}D^2(\frac{GM}{D^3})^{1/2}\mathbf{e}_k[/tex]

[tex]\mathbf{L_{orb}} =M_sM_p(\frac{GD}{M})^{1/2}\mathbf{e}_k[/tex]

 

[Pinkline Jones]

Kurdt,

The angular momentum of close binaries was calculated by a modest young diesel mechanic working in a sheltered workshop in Yorkshire. The young man - Winthrop Spencer Flibberdigit has since gone on to write some fascinating books released in sanskrit by "Absolutely No Frills and Assoc. Publishers". Here's just a short list of his books to date:

"The Peripatetic Life of the Sandwich Island Penguins"

"Low Orbital-Decibel ratios of Semi-Diametric Rolling Hub Caps"

"Desperate and Dateless Conversations - How My Electromagnetic Spectrum Theorem sent my Lover into a Coma"

"Dangers of The Spiral Galaxy - Fifty Bucks to Clean the Cab"

My pleasure...

DR PINKLINE JONES a.l.s.c.

 

[Kurdt]

Thank you for your input Pinkline I shall certainly investigate these publications when I next visit the library. I have several exams in the coming three weeks though so I doubt I will be able to spare any time soon. The titles definitely look interesting.

 

[Kurdt]

Would you believe it! I managed to find one of his publications in my own personal library. "Dangers of The Spiral Galaxy - Fifty Bucks to Clean the Cab" is a most interesting journey through the derivation of angular momentum in the andromeda galaxy all on a wild night out. I particularly like the climax when he reaches his conclusions in the taxi on the way home. Also the last chapter about his abduction by aliens is well worth the read alone.

 

[Pinkline Jones]

LOL Kurdt - I'm glad you found the book and happy to be of assistance wherever I can in cyber world. But obviously I have to fine you for possessing a sense of humour - that's a very dangerous attribute to have in this mixed up world - people look at you funny.

PINKLINE JONES

 

[Kurdt]

Well I may never have to show it again as I think I have learned a valuable lesson to read peoples posts thoroughly first time. So if you don't tell i won't.