Pluto's diameter is approximately 2370 km, and the diameter of its satellite Charon is 1250 km .

Pluto's diameter is approximately 2370 km, and the diameter...

Pluto's diameter is approximately $2370 \mathrm{km},$ and the diameter of its satellite Charon is 1250 $\mathrm{km}$ . Although the distance varies, they are often about $19,700 \mathrm{km}$ apart, center-to-center. Assuming that both Pluto and Charon have the same composition and hence the same average density, find the location of the center of mass of this system relative to the center of Pluto.

Key Concepts

Two-body System Dynamics

In systems where two objects interact gravitationally, such as binary systems or satellites orbiting planets, understanding the location of the barycenter is essential. The distance from one of the bodies to the center of mass can be calculated using the ratio of the masses and the separation distance. This helps in analyzing their mutual orbits and gravitational effects.

Center of Mass (Barycenter)

This is the point at which the mass of a two-body system can be thought to be concentrated. In astrophysics, particularly in orbital mechanics, the barycenter is the weighted average position of the masses in the system, calculated based on both the masses and their positions. It explains the orbital motion of bodies around a common point rather than strictly around one another.

Mass Determination from Density and Volume

When two objects are composed of the same material or have the same density, their masses are proportional to their volumes. For spherical bodies, the volume is proportional to the cube of the radius or diameter. This concept allows one to deduce mass ratios simply by comparing the cubes of their diameters or radii.

Transcript

00:01 Problem 8 .53, we're trying to find where between pluto and its moon, karen, the center of mass of the two is located.

00:14 Now we're told the diameter of pluto, the diameter of keron, and we're told to assume that they have the same composition, and so their densities are the same.

00:39 Now, as far as i know, we don't actually know what the mass of, masses of pluto and karen are exactly, or you know, i even have really a good estimate for it.

00:49 So this is not just, you know, the textbook being difficult.

00:56 This is actually sort of a practical thing you might need to do if you know how big something is, which is fairly easy to figure out from observations, but you don't know how massive it is.

01:05 If you make some assumptions about, in this case, the composition of the two bodies being the same, and so their densities would be the same that you can actually figure out something about this.

01:18 I believe you can also figure out what the center of mass is through observing their orbiting around each other, but that's not what this problem is about.

01:30 So let's call this the x direction just to have something to call it, and we'll put pluto's center at zero, x equals zero, so that the x coordinate of pluto is equal to zero.

01:49 And whenever you can do something like this, it will always make your life easier because then one of the things is going to end up not mattering.

02:00 So we know that the center of mass position, in this case we only have one coordinate x to worry about, it's going to equal the mass of pluto times its exposition, which we've called its position zero.

02:26 So that's nice, like i just said.

02:30 And then the mass of karon times position of keron, which we know they're about 19 ,700 kilometers apart.

02:40 So this will be 19 ,700 kilometers because we picked this to be zero...

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