Rich Mathematical Problems in Astronomy

1. Rich Mathematical Problems in Astronomy Sandra Miller and Stephanie Smith Lamar High School Arlington, TX

2. How far away is the horizon?

3. distance to the horizon This problem is designed to occur during a Geometry unit on circles. • A line tangent to a circle forms a right angle with a radius drawn at the point of tangency.

4. distance to the horizon • r – radius of the planet/moon • h – height of the observer (eyes) • d – distance to the horizon d r h r

5. distance to the horizon • r – radius of the planet/moon • h – height of the observer (eyes) • d – distance to the horizon d r h r

6. distance to the horizon

7. distance to the horizon

8. Mass and escape velocity

9. mass and escape velocity • This problem set is geared toward a Pre-AP Algebra I class or an Algebra II class. • By working through this packet, a student will practice • Simplifying literal equations • Creating formulas • Unit conversions • Using formulas to solve problems

10. mass and escape velocity Sir Isaac Newton developed three equations that we will use to develop some interesting information about the solar system.

11. mass and escape velocity • If we substitute the formula for centripetal acceleration into the F = ma equation, we have an equation for the orbital force: • The gravitational force that the object being orbited exerts on its satellite is

12. mass and escape velocity • Objects that are in orbit stay in orbit because the force required to keep them there is equal to the gravitational force that the object being orbited exerts on its satellite. • If we set our two equations equal to each other and solve for v, we end up with a formula that will give us the orbital speed of the satellite.

13. mass and escape velocity • Simplify the equation and solve for v:

14. mass and escape velocity • Simplify the equation and solve for v:

15. mass and escape velocity • Because the mass of the satellite m cancelled out of the equation, if we know the orbital velocity and the radius of the orbit, we can find the mass of the object being orbited.

16. mass and escape velocity • Rewrite the velocity equation and solve for M:

17. mass and escape velocity • Rewrite the velocity equation and solve for M:

18. mass and escape velocity • Example: Use the Moon to calculate the mass of the Earth. • Orbital radius: • Period: T = 27.3 days • Orbital velocity:

19. mass and escape velocity • Example: Use the Moon to calculate the mass of the Earth.

20. mass and escape velocity • Example: Use the Moon to calculate the mass of the Earth.

21. mass and escape velocity • To calculate escape velocity, we set the equation for kinetic energy to the equation for gravitational force and solve for v: Kinetic energy > Force × distance

22. mass and escape velocity Calculate Earth’s escape velocity in km/s. • Earth’s mass: 6.02 × 1024 kg • Earth’s radius: 6.38 × 106 m

23. mass and escape velocity • Now that we’ve worked through the different equations, we can calculate the mass and escape velocity of Mars as well as the mass of the Sun.

24. astronomy problems One of my favorite sites for possible astronomy-related math problems has been Space Math at http://spacemath.gsfc.nasa.gov. Unfortunately, because of cutbacks in NASA’s education budget, it will not be updated as frequently.

25. rich mathematical tasks James Epperson, Ph.D.

26. Presentation Materials • The powerpoint and the worksheets will be posted on my blog at tothemathlimit.wordpress.com.