- #1
Aleolomorfo
- 73
- 4
Homework Statement
A binary stellar system is made of one star with ##M_1=15{M}_\odot## and a second star with ##M_2=10{M}_\odot## revolving around circular orbits at a relative distance of ##d=0.001pc##. At some point ##M_1## explodes in a supernovae leaving a neutron star of mass ##M_{NS}=1.4{M}_\odot##. The new binary system ##NeutronStar-M_2## is still bounded? If the answer is positive, are they still on circular orbits? What would be the difference if the initial mass of ##M_1## were ##11{M}_\odot##?
Homework Equations
The Attempt at a Solution
I have first written the initial potential gravitational energy: ##E_{grav_i}=-\frac{GM_1M_2}{d}##. Then I can write the final one: ##E_{grav_f}=-\frac{GM_{NS}M_2}{a}## with ##a## the new separation. I have thought of using the conservation of energy:
$$E_i=E_f$$
$$E_i=-\frac{GM_1M_2}{d}+\frac{1}{2}\frac{M_1M_2}{M_1+M_2}v^2_i$$
$$E_f=-\frac{GM_{NS}M_2}{a}+\frac{1}{2}\frac{M_{NS}M_2}{M_{NS}+M_2}v^2_f$$
In order to find ##v^2_i## I have used the virial theorem: ##v^2_i=\frac{(M_1+M_2)G}{d}##; and substituing in the intial energy I have found: ##E_i=\frac{1}{2}\frac{GM_1M_2}{d}##. I can make something similar with the final situation and imposing the conservation of energy I have found just a relation to calculate ##a## and nothing else. I think what I have done is correct (I hope at least), but I have the sensation that this is not the way to solve the problem. Thanks for the help.