∙ Detecting planets around other stars. Roughly 500 planets...

$\bullet$ Detecting planets around other stars. Roughly 500 planets have so far been detected beyond our solar system. This is accomplished by looking for the effect the planet has on the star. The star is not truly stationary; instead, it and its planets orbit around the center of mass of the system. Astronomers can measure this wobble in the position of a star.(a) For a star with the mass and size of our sun and having a planet with five times the mass of Jupiter, where would the center of mass of this system be located, relative to the center of the star, if the distance from the star to the planet was the same as the distance from Jupiter to our sun? (Consult Appendix E.) (b) If the planet had earth's mass, where would the center of mass of the system be located if the planet was just as far from the star as the earth is from the sun? (c) In view of your results in parts (a) and (b), why is it much easier to detect stars having large planets rather than small ones?

Transcript

00:01 So this is actually a really interesting problem where you're thinking about how planets are detected by considering what the center, where the center of mass would be between a star and a planet, given the sizes of those things.

00:21 So to start off, we're going to be using center of mass.

00:26 So generally, center of mass, i'm going to just write it for a two -body system here is going to be m1x1 plus m2 x2 divided by the total mass m1 plus m2 when we're just thinking about spherical masses like planets and suns.

00:47 So this is the formula we're going to need for this problem.

00:52 And before we got started, i also thought it would be helpful to actually look up all of the values that are going to be important in the problem.

01:04 And i know this says to reference appendix e.

01:08 I don't know what textbook you're working out of.

01:11 So i've just looked up these values myself.

01:15 They might be slightly different from the values that are listed in your appendix e because the these are, of course, very difficult things to measure.

01:27 And different sources will round by different amounts.

01:33 So my answers might have more or fewer decimal places than yours would be in appendix e.

01:41 So just keep that in mind as i'm working through the problem that my numbers might be slightly different because i'm not using the numbers that are listed in your textbook.

01:52 I'm looking, sorry, i'm using numbers that i looked up and found online.

01:57 So first we need the mass.

02:00 We'll need the mass of the sun.

02:03 And i found the mass of the sun in kilograms to be a whopping 1 ,988 ,500 times 10 to the 24th kilograms.

02:21 Wow.

02:23 We're also thinking about jupiter.

02:26 It also has a very large mass, although not anywhere near as large as the sun at 1 ,898 .13 times 10 to the 24 kilograms.

02:42 And then the mass of earth.

02:46 Earth is puny in comparison to the sun and to jupiter at a measly.

02:55 5 .972 times 10 to the 24 kilograms.

03:00 That's still quite large if you ask me.

03:03 And then we want a few different distances.

03:07 So first, i'm going to look up the radius of the sun.

03:11 This is not something that we actually need for the problem in order to do the calculations, but i think that it might be helpful in comparing where the center of mass is to where the sun ends, essentially.

03:30 So we can think about is the center of mass inside the sun, outside the sun, is it close to the center of the sun, or is it near the edge? so the radius of the sun in meters is 6 .957 times 10 to 8 meters.

03:51 And then lastly, we need the distance from jupiter to the sun and the distance from earth to the sun.

04:01 So i'm going to label these as d.

04:04 So d subj is the distance between the sun and jupiter.

04:09 And i found that distance to be 7 .79 times 10 to the 11 meters.

04:18 And for earth, i found that distance to be 1 .5199 times 10 to be 11 meters...

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