Question

Procyon was determined to be is 11.4 ly from Earth via parallax. In addition, astronomers found that Procyon is a binary system. Procyon B, the companion, has a maximum angular separation from Procyon A (semimajor axis) 4.35 as (arc second).

(a) What is the distance between Procyon A and Procyon B? Answer in km and in AU. What planet(s) in our solar system is (are) closest to this distance from the Sun?
(b) Procyon B is observed to have an orbital period of 40.8 yr. What is the total mass of Procyon A and B system? Answer in km and in solar masses.

Answer 1

a)

We have been provided with the following information:

Distance from Earth to Procyon=11.4 ly

In km,this distance=1.07×10¹⁴ Km

And in AU, this distance=719020.8 AU

(since 1 AU=1.5×10⁸ Km)

The angular separation between Procyon A and B=4.35 arcseconds

Now we can set up the geometry of the problem as follows.

Look at the triangle formed by Earth, Procyon A and the midpoint between Procyon A and B i.e. ∆QPR

It is clear from the geometry of the triangle, that,

..........(1)

But we know PR= 1.07×10¹⁴ Km and

Let us plug these values in equation (1)

Now we solve above equation for QR which gives us the distance from Procyon A to the midpoint between Procyon A and B.

So obviously the distance between Procyon A and B is 2QR i.e. 2×1.12×10⁹ Km=2.25×10⁹ Km.

In terms of AU, this distance=15.04 AU

So the distance between Procyon A and B is 2.25 ×10⁹ Km or 15.04 AU.

b)

We can use the Kepler's third law for solving this problem.

For binary stars, Kepler's third law is written as follows:

.......(2)

Where, P is the Period in years=40.8 years and 'a' is the distance between Procyon A and B=15.04AU and M_{​​​​total} is the total mass of Procyon A and B system in Mo(i.e Solar masses)

For this choice of units,the constant=1

So equation (2) can be written as follows:

No now we solve the above equation for total mass,

Let us plug in the values

Thus the total mass of Procyon A and B System is 2.04 Mo.

1 Mo=1.989×10³⁰ Kg.

Therefore total mass of Procyon A and B in kg is about 4.06×10³⁰ Kg