Calculate the Masses of two binary stars

[victoriafello]

Homework Statement

I am trying to understand the following example question in my course book, The answer is actually given but i don't understand how they got to it, i would like someone to give me an idea how this has been worked out

In the Sirius Binary system the orbital period is 50 years and the semi major axis of the relative orbit is 20AU. Calculate the masses of sirius A and B expressing your answer in solar masses

Homework Equations

M+m = 4 Pi^2 a^3 / GP^2

The Attempt at a Solution

M+m = 4Pi^2 (20 x 1.50 x 10^11 m) ^3 / (6.67 x 10 ^-11 N m^2 Kg^-1) x (50 x 3.16 x 10^7 s) ^2

I really don't understand as i thought from the equation that it would be 4 Pi^2 x 20^3 / 6.67 x 10^-11 x 50 ^2

What am i missing ?

 

[tiny-tim]

Welcome to PF!

Hi victoriafello ! Welcome to PF!

(have a pi: π and try using the X² tag just above the Reply box )

M+m = 4 Pi^2 a^3 / GP^2

The Attempt at a Solution

M+m = 4Pi^2 (20 x 1.50 x 10^11 m) ^3 / (6.67 x 10 ^-11 N m^2 Kg^-1) x (50 x 3.16 x 10^7 s) ^2

I really don't understand as i thought from the equation that it would be 4 Pi^2 x 20^3 / 6.67 x 10^-11 x 50 ^2

hmm … they seem to have converted everything to SI units …

AU to m, and year to s …

which gives them an answer in kg.

But the problem asks for an answer in solar masses, so you should be able to use AU and year, and get the answer from the given equation just by dimensional considerations (and without using G)

 

[victoriafello]

Thanks, I thought that if the units were in AU and Yrs then i could use the equation in the form

(M+m) / Msun = (a/AU)^3/(P/yr)^2

So this would make the equation

20^3/50^2 = 3.2

Is this correct ? I am still a bit unsure about the first part of the equation is this answer now in solar masses or so i need to convert ?

 

[tiny-tim]

Thanks, I thought that if the units were in AU and Yrs then i could use the equation in the form

(M+m) / Msun = (a/AU)^3/(P/yr)^2

So this would make the equation

20^3/50^2 = 3.2

Is this correct ? I am still a bit unsure about the first part of the equation is this answer now in solar masses or so i need to convert ?

Yes, that looks right to me …

mass is proportional to axis cubed and inversely proportional to period squared, so the result is a multiple of the mass of the Sun+Earth, in other words in solar masses