In 2004 astronomers reported the discovery of a large...

In 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term "hot Jupiter"). The orbit was just $\frac{1}{9}$ the distance of Mercury from our sun, and it takes the planet only 3.09 days to make one orbit (assumed to be circular). (a) What is the mass of the star? Express your answer in kilograms and as a multiple of our sun's mass. (b) How fast (in km/s) is this planet moving?

Transcript

00:01 Let's look at how kepler's third law applies in solar systems besides our own.

00:06 In our own solar system, kepler's third law is quite simply written as the period of a planetary or other orbit around the sun.

00:22 So this is period of orbit around the sun in years.

00:29 Is that squared is equal to the average distance from the sun cubed.

00:42 Average distance between the planet or other object and the sun.

00:49 And we say average, but we really mean the so -called semi -major axis.

00:56 So the idea is we could have an ellipse with the sun at one focus, and a is half the length of the long axis of the ellipse.

01:11 Now, newton redeveloped an expression using his universal law of gravity, and his form of the kepler's law is p squared is 4 pi squared over g, big g, gravitational constant times the mass of the central or focus object here are sun and then 8 cubed.

01:39 So using this form, we can basically say that 4 pi squared over g in solar system units is equal to 1.

01:51 And m is going to be in solar masses.

01:54 So we're taking the mass of our sun equal to one.

02:02 And what this formula will tell us is the mass of another star.

02:07 We'll go ahead and put a star next to that.

02:11 M.

02:13 The mass of a star in another solar system can be determined if we know the period of a planet as well as its semi -major axis, both in solar system units, of years in astronomical units.

02:32 The a must be in astronomical units.

02:37 So let's take a look at a case.

02:39 There was an exoplanet discovered in a fairly nearby star.

02:45 It has, the star has the name hd for the catalog number 179949.

02:55 And this is planet number b in the system, which means it is either the second one out or the second one that was found in time.

03:08 We won't worry too much about the nomenclature.

03:11 But its period is one -ninth of, not one -ninth, its orbit is one -ninth of mercury's orbit.

03:19 Its period is 3 .09 days.

03:24 And let's go ahead and express that in years, because we know those are solar system units.

03:37 And by the way, this is a type of exoplanet known as a hot jupiter.

03:43 It's a very massive planet, much bigger than the earth, and probably more like jupiter or even bigger.

03:55 But it's orbiting close to the star.