All of these questions are interconnected and concern a...

All of these questions are interconnected and concern a planet orbiting the star HD209548. Several years ago, a planet was discovered orbiting around this star with an orbital period of only 3.524 days. The star has a mass Mstar=1.05 times that of the Sun M⊕. The plane of the planet's orbit around HD209548 is edge-on to our line of sight. For this problem, you may assume the planet is in a circular orbit (in reality, this is a good approximation, since the eccentricity of the planet's orbit is only 0.08) and that the planet's mass is much smaller than the mass of the star. a) Find the planet's average distance (semi-major axis) from the star HD209548. Express your answer in meters. b) At what distance from the center of the star is the center of mass of the star-planet system located if the planet has a mass Mp=0.5 times that of Jupiter (MJ)? 1.0MJ? c) What is the orbital velocity of the star around the center of mass of the star-planet system for the cases given above?

Transcript

00:01 Okay, in this question, we are going to be using kepler's third law, which states that t squared, the time period of an orbit squared, is equal to 4 pi squared, multiplied by the radius of the orbit cubed over the universal gravitational constant, multiplied by the mass that that object is orbiting.

00:18 So in this scenario, the mass would be the mass of the star or the mass of the sun.

00:23 You see here that this does not depend at all on the mass of the orbiting object.

00:29 So the mass of the earth or the mass of the planet does not matter.

00:32 Now, in terms of, let's make it specific to the earth's sun system here, the time period of the earth's orbit, and then we have the radius of the earth's orbit, and then the mass of the sun.

00:41 We can isolate the mass of the sun by saying 4 pi squared, r -e -quipped over t -e -squared.

00:51 Again, let's not forget the g.

00:54 Again, where s means sun's mass of the sun, radius of the earth's, radius of the earth's orbit, cubed and then time period of the earth's orbit squared.

01:03 Now we want to find the ratio of this new star's mass in terms of, well, the ratio of the new stars mass to the sun's mass, if we're told that the new planet has a period of 600 days and a orbital radius of 1 .4 a u, so 1 .4 times earth's orbital radius.

01:22 So then, let's work out the radius of the new planet in terms of earth's radius and the radius of the time, sorry, the time period of the new planet in terms of the time period of earth's orbit, and that will give us the mass of the star in terms of the mass of the sun.

01:42 Quite a lot to get through there.

01:43 Okay, so we're already told that, i'm just going to say r, r new for the new scenario, is equal to, we have what, 1 .4 a .u, so that's 1 .4 r e.

01:56 Okay.

01:57 And then what about t new? what about the time period? well, that's going to be 600 days what's that in terms of a year all we need to do is just do 600 divided by 365 and that gets us to 1 .644.

02:13 So that's 1 .6 4 4 t e.

02:19 Now i can substitute these expressions into our new into a new expression for the mass of the new star...

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