A pair of identical stars is separated by 25.0 AU, and the stars orbit each other with a period

A pair of identical stars is separated by 25.0 AU, and the...

Key Concepts

Kepler's Third Law

Kepler's Third Law relates the square of the orbital period of two objects to the cube of the semi-major axis of their orbit. It is fundamental in understanding how the period of an orbit depends on the distance between the orbiting bodies and, when combined with Newton's law of gravitation, allows us to determine the masses of the objects involved in the orbit.

Binary Star Systems

Binary star systems consist of two stars that orbit a common center of mass. In such systems, especially when the stars are of equal mass, the dynamics are symmetric, and the orbital characteristics such as period and separation can be used to deduce the individual masses of the stars.

Center of Mass

The center of mass is the point where the total mass of a system can be considered to be concentrated. In a binary star system with identical masses, the center of mass lies exactly halfway between the two stars, which simplifies the analysis of their orbital motion.

Mass Determination Using Orbital Parameters

The determination of stellar masses in a binary system relies on measurements of orbital parameters such as the orbital period and the separation between the stars. By applying Newtonian mechanics in conjunction with Kepler's Third Law, the sum of the masses can be deduced, and in systems with identical stars, this sum is evenly divided to obtain the mass of each star.

Gravitational Constant and Unit Conversions

Accurate mass calculations in astrophysics often require the use of the gravitational constant (G) and careful unit conversions to ensure consistency across the parameters. This is particularly important when orbital periods, distances, and masses are given in different units, necessitating conversion to a common system for effective application of physical laws.

Transcript

00:01 Question number 52 asks, what is the mass of two stars in a binary system who are separated by 360 million kilometers, and circling a common center of mass between them every 5 .7 earth years? such a detailed question calls for a little illustration.

00:19 So i will draw two stars at the top and bottom of the screen here.

00:27 They are separated by a distance that we will call it.

00:31 D, and directly between the two stars is a common center of mass, which they both orbit.

00:38 Now each star orbits at some velocity v, we're assuming they share the same orbit around the center of mass, and each tugs on the other with a gravitational force f sub g.

00:52 F sub g is calculated using the universal law of gravitation, where f sub g is equal to g times m1 times m2 over d squared.

01:09 Now we're assuming for our problem that each star has the same mass, and they each experience the same rotational force as they do the same gravitational force.

01:22 The rotational force, f sub r, is equal to m times v squared over r.

01:31 Now, r is equal to half the distance between these two stars, or d over two.

01:39 If we wanted to, we could simply substitute d over two in for r, to get our rotational force equal to mass times velocity squared over d over two.

01:51 As gravitational force and rotational force both affect each of these two stars in a similar fashion, we can set f sub g equal to f sub r.

02:04 Therefore, g times m1 times m2 over d squared can be equal to m times v squared over d divided by 2.

02:20 There is another assumption we're making on the part of our problem.

02:24 We're assuming that both these stars share a circular orbit around this.

02:29 Center of mass...

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