Binary Stellar System: 15M⊙ & 10M⊙ Revisited

[Aleolomorfo]

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.

 

[gneill]

What exactly are $v_i$ and $v_f$? There are two bodies, so won't they have different (linear) velocities?

You should be able to use the fact that the initial scenario has mutual circular orbits to determine the KE's of the two stars. Then you'll need to make (and state) an assumption about the initial velocity of the supernova remnant (the neutron star).

 

[Aleolomorfo]

What exactly are $v_i$ and $v_f$? There are two bodies, so won't they have different (linear) velocities?

You should be able to use the fact that the initial scenario has mutual circular orbits to determine the KE's of the two stars. Then you'll need to make (and state) an assumption about the initial velocity of the supernova remnant (the neutron star).

So I have to split the kinetic energies:
$$E_i=-\frac{GM_1M_2}{d}+\frac{1}{2}M_1v^2_1+\frac{1}{2}M_2v^2_2$$
I need to find the velocities, but first I need to find the distance of each star from the CM and I can use the property of the CM: $R_1M_1=R_2M_2$. WIth $d=R_2-R_1$ I can find $R_1$ and $R_2$. Then since they are revolving around circulat keplerian orbits I can state $F_{grav}=F_{cent}$:
$$\frac{GM_1M_2}{R^2_1}=M_1\frac{v^2_1}{R_1}$$
$$v^2_1=\frac{GM_2}{R_1}$$
Analogously for $v^2_2$.
I can make the assumption that the velocities just after the explosion are the same before it (I think it is reasonable), and so:
$$E_f=-\frac{GM_{NS}M_2}{a}+\frac{1}{2}M_{NS}v^2_1+\frac{1}{2}M_2v^2_2$$
I can use the conservation of energy to find $a$. With $a$ I can calculate the new escape velocity for the neutron star $v_{escape}=\sqrt{\frac{2GM_{NS}}{a}}$. If it is less than the velcocity of the neutron star the system is bound, otherwise it is not.
If what I have done is right (I am not so sure) I can find if the system is bound or not, but how can I found if the orbits are circular?

 

[gneill]

So I have to split the kinetic energies:
$$E_i=-\frac{GM_1M_2}{d}+\frac{1}{2}M_1v^2_1+\frac{1}{2}M_2v^2_2$$
I need to find the velocities, but first I need to find the distance of each star from the CM and I can use the property of the CM: $R_1M_1=R_2M_2$. WIth $d=R_2-R_1$ I can find $R_1$ and $R_2$.

Shouldn't that be $d = R_1 + R_2$? The R's are the orbit radii to the center of mass.

Then since they are revolving around circulat keplerian orbits I can state $F_{grav}=F_{cent}$:
$$\frac{GM_1M_2}{R^2_1}=M_1\frac{v^2_1}{R_1}$$
$$v^2_1=\frac{GM_2}{R_1}$$
Analogously for $v^2_2$.
I can make the assumption that the velocities just after the explosion are the same before it (I think it is reasonable), and so:
$$E_f=-\frac{GM_{NS}M_2}{a}+\frac{1}{2}M_{NS}v^2_1+\frac{1}{2}M_2v^2_2$$
I can use the conservation of energy to find $a$. With $a$ I can calculate the new escape velocity for the neutron star $v_{escape}=\sqrt{\frac{2GM_{NS}}{a}}$. If it is less than the velcocity of the neutron star the system is bound, otherwise it is not.
If what I have done is right (I am not so sure) I can find if the system is bound or not, but how can I found if the orbits are circular?

Overall it looks like a good plan. You should be able to use the concept of Total Mechanical Energy, $\xi = KE + PE$, to determine whether or not the system is bound. If $\xi$ is negative then the system is bound.

To judge whether the new orbits will be circular, consider the required centripetal force and gravitational force again for the new arrangement.

 

[Aleolomorfo]

Shouldn't that be d=R1+R2d=R1+R2d = R_1 + R_2? The R's are the orbit radii to the center of mass.

Yes, you are right.

You should be able to use the concept of Total Mechanical Energy, ξ=KE+PEξ=KE+PE\xi = KE + PE, to determine whether or not the system is bound. If ξξ\xi is negative then the system is bound.

So I can calculate the toal energy after the explosion (Can I make the extra assumption that even the separation $d$ remains unchanged? On the contrary I have to calculate the new separation using the conservation of energy or some other methods which do not come to mind):
$$E_f=-\frac{GM_{NS}M_2}{d}+\frac{1}{2}M_{NS}v^2_1+\frac{1}{2}M_2v^2_2$$
I can calculate $v_1$ and $v_2$ from the virial theorem as I said before. Substituing the values in this formula, if it is negative is bound, right?

To judge whether the new orbits will be circular, consider the required centripetal force and gravitational force again for the new arrangement.

Do I have to see if the centripetal force and the gravitational force are equal? But they are not the same for every orbit, indipendently from the shape?

Thank you very much for your help and hints.

 

[gneill]

So I can calculate the toal energy after the explosion (Can I make the extra assumption that even the separation $d$ remains unchanged?

Yes. This will be true up to the moment after the supernova expelled mass shell passes beyond the second star. The speeds and separation should remain constant until then.

I can calculate $v_1$ and $v_2$ from the virial theorem as I said before. Substituing the values in this formula, if it is negative is bound, right?

As stated above, the speeds will be the same as those before the explosion until the system has a chance to evolve in time. So you don't need to recalculate them to determine whether or not the new system is still bound.

Do I have to see if the centripetal force and the gravitational force are equal? But they are not the same for every orbit, indipendently from the shape?

Yes you need to look at that. No they are not the same for every orbit shape. Only for circular orbits are they always equal. That's why, for example, elliptical orbits are not circular