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Announcements

- Exam 1 is in two weeks. Will cover material in textbook up through Chapter 3 plus additional material on sidereal time and Julian date
- Homework Set 3: Supplemental Problems

Kepler’s Laws of Planetary Motion

Empirical laws developed by Johannes Kepler based on the observational data of Tycho Brahe

Kepler’s Three Laws of Planetary (and stellar) Motion

1st Law: The planets move in elliptical orbits with the Sun at one focus.

2nd Law: A line drawn from a planet to the Sun sweeps out equal areas in equal times

3rd Law: The ratio of the square of the orbital period to the cube of the semimajor axis is the same for all planets

Using Kepler’s 3rd Law

So what is k? Newton eventually showed that k is related to the mass of everything inside the orbit of the planet.

M* is the mass of everything inside the orbit of the planet. Since planetary masses are so small, this is effectively the mass of the Sun

For objects orbiting other stars?

This can be solved for the mass

For a planet orbiting another star, if r is in meters and P is in seconds, this gives the mass of the star in kilograms. If you want the mass in solar masses, use r in AU and P in years then

Example Problem

The first extra-solar planet discovered orbits the star 51 Pegasi. If the semimajor axis is 0.052 AU and the orbital period is 4.23 days, what is the mass of 51 Pegasi (in solar masses and in kg)?

Example Solution 1

Since G has units of (Nm2/kg2), distances must be in meters and periods in seconds so do unit conversions first

Next, chose the equation to use.

Finally, plug in numbers to solve

Example Solution 2

If we want the mass in solar masses, use AU and years. Orbital radius is already give in AU so just convert the period into years.

Now choose the appropriate equation and plug in numbers.

Suppose we have two stars orbiting a common center?

Kepler’s 3rd Law will give the combined mass of the system. To get the individual masses, we need more information

We could use their speeds as the second piece of info

In this case it is an inverse relationship

Example

A binary system is observed for a number of years and it is found that one star appears to orbit the other at a distance of 10.0 AU every 5.00 years. From spectroscopic data it is found that one star moves at 25.0 km/s while to other star moves at 100.0 km/s. What are the masses of the two stars?

This is the position measurements for the star 70 Ophiuchi showing how one star appears to move around the other

Example Solution 1

First find the combined mass of the system using Kepler’s 3rd Law

Now use the ratio of their velocities to find the individual masses

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