Example 2 (N84/P2/Q7) Binary Stars

Two stars of equal mass M move with constant speed v in a circular orbit of radius R about their common centre of mass O. What is the net force on each star?

A $\frac{G M^{2}}{R^{2}}$ B $\frac{G M^{2}}{4 R^{2}}$ C zero D $\frac{2 M v^{2}}{R^{2}}$ E $\frac{M v^{2}}{2 R}$

activity:

select from the down-drop menu "M1=M2=1,circular_orbit"

click play and observe the motion, noting down the magnitude of the forces (green arrows) on each mass M1 and M2.

let assume in the simulation that $M_{1} = 1 s . m . = 1.988 x 1 0^{30} k g$ and $M_{2} = 1 s . m . = 1.988 x 1 0^{30} k g$

the distance between the masses is

R_{1} + R_{2} = 2 a u = (2) (1.496 x 1 0^{11}) m

the force is therefore, $F_{1} = \frac{G M_{1} M_{2}}{{(R_{1} + R_{2})}^{2}} = \frac{(6.67 x 1 0^{-11}) (1.988 x 1 0^{30}) (1.988 x 1 0^{30})}{{((2) (1.496 x 1 0^{11}))}^{2}} = 2.94 x 1 0^{27} N$

therefore net force on each M1 star is $F_{1} = \frac{G M^{2}}{4 R^{2}}$

similarly, the net force on M2 star is $F_{2} = \frac{G M^{2}}{4 R^{2}}$

in astronomy terms, notice the force on M1, $F_{1} = 9.87 \frac{s m^{2}}{a u^{2}}$ $\frac{}{}$

and

F_{2} = 9.87 \frac{s m^{2}}{a u^{2}}

Two-Body Orbits JS Model

The Two-Body Orbits JavaScript Model shows the motion of two objects (e.g., binary star or moon-planet system) interacting via Newton's law of universal gravitation. It is designed to teach physics, Earth science, and environmental science topics by showing the spatial path of objects around their common center of mass (barycenter). Default units are chosen for Earth obit about our Sun so that the distance unit is one astronomical unit and the time unit is one year.

Credits:

The Two-Body Orbits JS Model was developed by Wolfgang Christian using the Easy Java Simulations version 5 (EJS 5) modeling tool. Although EJS is a Java program, EJS 5 can create stand alone JavaScript programs that run in almost any PC or tablet browser.

Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.