Looking for help with one problem. Classical Mechanics Kibble

[haplo]

1. It is problem for kibble book, classical mechanics. I star with mass M traves through cloud of particles with density ro. Particles that collide with a star are trapped by it. Derive the expression for mass increase of the star dM/dt.

The Attempt at a Solution

Since all particles that collide with a star are trapped it's contribution is ro*Vstar*(piR^2)

However the particles that that are outside of cross sectional area are also get trapped by it. due to gravitational attraction. Thats where I had hard time to writing down conditionsat.
I was thinking of trying to figure out understand simmiar problem, suppose object of mass m, starts at (x=infinity, y=b) traveling with velocity V towards s star of mass M. So, what is the how far from the star an object should travel to get trapped by it into an orbit, but could not answer it.

Well any even smallest hint on how to solve the problem will be greatly appretiated. I have a feeling that problem is quite simple but answer eludes me.

 

[Redbelly98]

Since all particles that collide with a star are trapped it's contribution is ro*Vstar*(piR^2)

I think it's as simple as that.

 

[Dick]

I think it's as simple as that.

I agree. Adding gravity to the problem would make it way too complicated. The problem should probably say that. Hey, RedBelly, very subtle Halloweeny thing you've got going there.

 

[haplo]

Thanks for reply,
Sorry,I should have been more specific. Tthe problem states that you have to show that rate of mass accumulation is described by following equation:
dm/dt= pi*rho*v*(R^2+2*G*m*R/v^2)

where R is the radius of the star and v it's velocity. Also the problems asks you to compare the effective crossectional area to pi*R^2. From this and equation I concluded that gravitational pool of the star increases mass accumulation. . Still I couldn't understand how to derive the second term from the above equation.

BTW, this is not a homework. I have graduated with physics and while working on boring job I try to entertain myself by solving physics problems to prepare myself for grad school by keeping my skills in shape. So far I have been successful, until i got to this problem.

 

[Dick]

Working physics problems is a great way to pass the time when the job gets boring. I should know. Good. Then you don't mind a fragmentary hint. It will help you pass the time. I would assume that the velocity of the star is very slowly changing. So you can picture the problem as a stationary object in a swarm of particles moving at it with velocity v. You want to figure out how small the impact parameter needs to be before the radius of closest approach hits the radius of the star. Here's a link to a similar problem discussing these issues in the context of Rutherford scattering. http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/rutsca2.html You'll have to change repulsive scattering to attractive, and electric field to gravitational field. Etc. Have fun.

 

[Redbelly98]

Wow, so it is more complicated. But Dick gave a good hint. These particles are not trapped into orbiting the star, they actually collide with the star even for some values of y_initial>R.

Hey, RedBelly, very subtle Halloweeny thing you've got going there.

Thanks! It scared the heck out of Evo when she saw it.

 

[Phrak]

It would be more intuitive to do this in the inerial frame where the star is stationary. What is the perilium of a particle of dust approching with velocity v from infinity? By the way, I think I know you. I wax your floors at night, right? Stop it, you're scaring me Redbelly! I like the eye cut-outs.

 

[wdednam]

Hi,

Kibble and Berkshire is actually the prescribed book for my Classical Mechanics course this year and this was one of my assignment questions.

Consider a single dust particle dm. Just as Dick suggested above, you basically need to find the smallest impact parameter b for which the dust particle would collide with the star.

Try using the radial energy equation. Conservation of energy, together with conservation of angular momentum should give you the answer.

You just need to figure out at which two points you should apply the two conservation laws.

If you have any more questions I'll do my best to help you figure out other problems (without actually giving the answer completely away) as I've basically finished studying chapters 1 through 10, and 12. I wasn't able to solve all of the end-of-chapter problems though (especially those marked with an asterisk).

Good luck,

Wynand.

 

[haplo]

I figured it. By finding the distance of closest apporach and setting it to the Radius of a star R. From there I obtained the value of impact maximum parameter b for which particle will still collide.
After that the problem gets very easy. Thanks for help