Planetary Physics Homework Help | Solving Non-Linear Differential Equations

[PlanetaryStuff]

Good afternoon folks. I am getting ready to start a Planetary Physics course next week and have been doing some of the old homework that is posted on the course site. I am struggling to figure this one out and would love some help. It's been about 5 years since first year calculus, so I am a little lost. Any guidance would be greatly appreciated.
1. Homework Statement
1. In class, we saw that the rate of growth of a planet can be written as

dM/dt = r_svR²(1+(8GrR²/v²) (1)

where M is the mass, t is time, r and rs are the densities of the solid planet and the planetesimal swarm, G is the gravitational constant, R is the planet radius and v is the relative velocity.

b) Using the relationship between mass and density for a solid body and the chain rule, rewrite equation (1) in terms of dR/dt. (1)

2. Equations
As far as any working equations for this, I assumed that we should use the fact that density = mass/volume. Since we assume the vacuum of space, the solid body would be spherical (or nearly so), thus volume could be substituted for 4/3 pir³ giving us the (useful?) equation to substitute in for both the swarm density and the density of the object density = (3m)/(4pir³)

Your answer to part b) is a first-order differential equation, but it’s non-linear, so it’s hard to solve completely.

it has been quite some time since I've worked with an equation like this in the fashion mentioned, so I am just struggling to make any headway. Any explanations would be very helpful.

 

[PlanetaryStuff]

3. My work thus far:

I have tried several different methods of rewriting equation (1) in terms of dR/dt, using leibniz notation (dy/dx = dy/du*du/dx) in the form of dR/dt=(dR/dt*dM/dt) but am missing some of the mechanics that I am forgetting. This is chiefly where I need help.

 

[SteamKing]

3. My work thus far:

I have tried several different methods of rewriting equation (1) in terms of dR/dt, using leibniz notation (dy/dx = dy/du*du/dx) in the form of dR/dt=(dR/dt*dM/dt) but am missing some of the mechanics that I am forgetting. This is chiefly where I need help.

Your first problem lies here:

dR/dt=(dR/dt*dM/dt)

If you multiply out the differentials on the RHS of the equation, you do not get dR/dt. You get (dR*dM/dt^2), whatever that is.

Not only is your calculus rusty, but your basic arithmetic is suspect as well.

 

[nrqed]

3. My work thus far:

I have tried several different methods of rewriting equation (1) in terms of dR/dt, using leibniz notation (dy/dx = dy/du*du/dx) in the form of dR/dt=(dR/dt*dM/dt) but am missing some of the mechanics that I am forgetting. This is chiefly where I need help.

If you look at it for a few seconds, you will see that dR/dt=(dR/dt*dM/dt) cannot be right (it would imply dM/dt =1!)

You know the relationship between M and R (through the density). Then just take the derivative of the two sides with respect to time. That will give you a relationship between dM/dt and dR/dt

 

[PlanetaryStuff]

Sorry, I typed out my restatement of my work incorrectly, my chain rule is correct on my paper, I just lost it in formatting.

dR/dt=(1/P)*dm/dt - (M/P² )*dP/dt is where I have landed after your suggestion nrqed, I don't have a way of checking for correctness, so if you wouldn't mind giving me a thumbs up or down I'd really appreciate it!

Thanks for being supportive SteamKing, It's great to know that when one returns from duty overseas we can look forward to such uplifting people in academia!

 

[nrqed]

Sorry, I typed out my restatement of my work incorrectly, my chain rule is correct on my paper, I just lost it in formatting.

dR/dt=(1/P)*dm/dt - (M/P² )*dP/dt is where I have landed after your suggestion nrqed, I don't have a way of checking for correctness, so if you wouldn't mind giving me a thumbs up or down I'd really appreciate it!

Well, I think that you are meant to take the density as being constant, in which case you do not need to take the derivative of P with respect to time (because it is simply zero).

You seem to have used R = m/P. This is not quite right. In your first post you seemed to be much closer to the correct expression. Recall that density is mass over volume (not mass over radius).

Thanks for being supportive SteamKing, It's great to know that when one returns from duty overseas we can look forward to such uplifting people in academia!

Congratulations for taking up quite difficult classes after what you have already been through. I am sincerely impressed. Best luck with your studies!