Rich Mathematical Problems in Astronomy
- 1. Sandra Miller and Stephanie Smith Lamar High School Arlington, TX
- 3. 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. d r h r r – radius of the planet/moon h – height of the observer (eyes) d – distance to the horizon
- 5. d r h r r – radius of the planet/moon h – height of the observer (eyes) d – distance to the horizon d= r + h) − r 2 ( 2 d = r 2 + 2rh + h2 − r 2 d = h ( 2r + h )
- 6. Object Radius Horizon Earth 3959 mi. 3 mi. Moon 1080 mi. Mars 2106 mi. Jupiter 43,441 mi.
- 7. Object Radius Horizon Earth 3959 mi. 3 mi. Moon 1080 mi. 1.6 mi. Mars 2106 mi. 2.2 mi. Jupiter 43,441 mi. 9.9 mi.
- 9. 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. Sir Isaac Newton developed three equations that we will use to develop some interesting information about the solar system. F = ma When a force F acts on a body of mass m, it produces in it an acceleration a equal to the force divided by the mass. v a= r The centripetal acceleration a of any body moving in a circular orbit is equal to the square of its velocity v divided by the radius r of the orbit. Gm1m2 F= r2 The grativational force F between two objects is proportional to the product of their two masses, divided by the distance between them. 2
- 11. If we substitute the formula for centripetal acceleration into the F = ma equation, we have an equation for the orbital force: v 2 mv 2 F = m = r r The gravitational force that the object being orbited exerts on its satellite is GmM F= 2 r
- 12. 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. Simplify the equation and solve for v: mv 2 GmM = 2 r r
- 14. Simplify the equation and solve for v: mv 2 GmM = 2 r r GmM 2 mv = r Gm v = r GM v= r 2
- 15. 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. Rewrite the velocity equation and solve for M: GM v = r 2
- 17. Rewrite the velocity equation and solve for M: GM v = r 2 v 2r = GM v 2r M= G
- 18. Example: Use the Moon to calculate the mass of the Earth. Orbital radius: r = 3.84 × 10 8 m Period: T = 27.3 days circumference of orbit Orbital velocity: v = period of orbit
- 19. Example: Use the Moon to calculate the mass of the Earth. 2πr v= T = 2π ( 3.84 × 108 ) 24 hours 3600 seconds 27.3 1 hour 1 day = 1023 m s
- 20. Example: Use the Moon to calculate the mass of the Earth. G = 6.67 × 10 −11 N m2 kg2 v 2r M= G = 6.02 × 10 24 kg
- 21. 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 1 GmM 2 mv > 2 i r 2 r 2GM 2 v > r v> 2GM r
- 22. Calculate Earth’s escape velocity in km/s. Earth’s mass: Earth’s radius: 6.02 × 1024 kg 6.38 × 106 m v > 11.22 km s
- 23. 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. 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. Invert the problem Ask for an explanation: oral or written Examples or counterexamples Ask for prediction Break into multiple parts Original (Standard) Problem Ask for multiple representation Ask for generalization Automaticity practice Ask questions that require qualitative reasoning James Epperson, Ph.D.
- 26. The powerpoint and the worksheets will be posted on my blog at tothemathlimit.wordpress.com.