Tidal Forces: Analyzing Comet Disruption by Planet w/ Mean Density > Comet

[StephenPrivitera]

A comet grazes a planet at a distance R. Show that the planet's tidal force can disrupt the comet is the planet's mean density is greater than that of the comet.

P--------R-------C

F=-GMm/r²
Now my TA advises me to expand this function about r=R using a taylor series. But c'mon, this is an introductory astronomy class. We're not supposed to know Taylor series!
But even if I do, I get:
F_g=-GMm/R²+2GMm/R³(dr)
dr=dist from center of comet
m=mass of comet
M=mass of planet

/\
|____ Is this even right?

Then I show that F_t>F_sg
where F_t=?
F_sg=self gravity=?
I'm so utterly confused.
Please feel free to include a lecture on tidal forces and taylor series! Whatever help you can give would be great!
Thanks.

 

[M98Ranger]

Hello! I understand that this topic may seem confusing, but let me try to break it down for you.

First, let's talk about tidal forces. Tidal forces are the result of the unequal gravitational pull on different parts of an object. In the case of a comet approaching a planet, the side of the comet closest to the planet will experience a stronger gravitational pull than the side farther away. This difference in gravitational pull can lead to the disruption of the comet.

Now, let's consider the equation you provided for gravitational force: F=-GMm/r^2. This equation represents the force between two objects with masses M and m, separated by a distance r. In this case, M represents the mass of the planet and m represents the mass of the comet.

Next, you mentioned using a Taylor series to expand this equation about r=R. This means that we are looking at the gravitational force at a specific distance, R, from the center of the planet. When we expand the equation, we get F=-GMm/R^2+2GMm/R^3(dr). This second term, 2GMm/R^3(dr), represents the tidal force.

To understand this better, let's look at the diagram you provided. When the comet grazes the planet at a distance R, the side of the comet closest to the planet will experience a stronger gravitational pull than the side farther away. This difference in pull creates a tidal force, which is represented by the second term in the expanded equation.

Now, to show that the planet's tidal force can disrupt the comet, we need to compare it to the self-gravity of the comet. Self-gravity is the gravitational force within an object itself. In this case, we can represent self-gravity as F_sg=-Gm^2/dr^2.

If we compare the two forces, F_t (tidal force) and F_sg (self-gravity), we can see that F_t>F_sg. This means that the tidal force is stronger than the self-gravity of the comet, and therefore, it can disrupt the comet.

I hope this explanation helps clarify the concept of tidal forces and how they can disrupt a comet. As for the use of Taylor series, it is a mathematical tool used to approximate functions and can be useful in understanding more complex relationships in physics and astronomy. If you are interested in learning more about it, I would suggest discussing it with your TA or