Question

Q: Generally speaking astronomers don’t know the distances to objects they are studying. So, for example, in a binary star system, we measure the angular separation between the two stars vs. time leading to an orbit measured in angular units instead of in length units. If we knew the system’s distance (i.e., parallax), then we could convert the angles and know the actual size of the orbit in kilometers or AU.

Rewrite the form of Kepler’s Third law using a parallax angle (p) and semi-major axis (a; in arcsec).

Use your equation to calculate the total mass (M_tot) of the following binary star system.

Measurements of the binary star:

period (P) = 25 years

angular semi-major axis (a) = 0.67 arcsec

parallax angle (p) = 0.050 arcsec

Thanks so much for the help!

Answer 1

Solution :

Kepler's Third law for planetary motion states that, " The square of the orbital period, P of a planet is proportional to the cube of semi-major axis, a of the planet's orbit."

Mathematically, it can be written as : P² a³

where, M₁ and M₂ are masses of the binary stars. ( and M₁ + M₂ denotes the total mass of the system)

Please note that the masses (M₁ and M₂) are in terms of solar mass (), time period (P) is in years, and semi- major axis (a) is in astronomical units (A.U.)

Now, to convert semi-major axis to angular semi-major axis measured in arc seconds, we have,

where, a" = angular semi-major axis measured in arc seconds
and, p = parallax measured in arc seconds

Thus, Kepler's third law can be written as :

or, in order to find the masses. it can be simplified as -

TO FIND THE TOTAL MASS OF THE BINARY SYSTEM :

Period (P) = 25 years
Angular semi-major axis (a) = 0.67 arcsec
Parallax angle (p) = 0.050 arcsec

Thus, total mass of the binary system, (M₁+M₂) is :

Plugging in the values,

Therefore, the total mass of the system is