. Mr. K. NASA/GRC/LTP Part 4 Pathfinder’s Path I

Preliminary Activities 1. Use the URL’s provided at the end of this lesson. Who were Tycho Brahe and Johannes Kepler? What did they contribute to modern astronomy and space exploration? 2. Write down Kepler’s three laws of planetary motion. Why are these laws significant today? 3. What role did Mars play in the discovery of Kepler’s law of planetary orbits? 4. Why is Mars significant today?

4. In your algebra class, discuss the conic sections. Write the equation for an ellipse with its center at the origin. 5. What role do the conic sections play in planetary and spacecraft orbits?

Tycho Brahe ( )

Uraniborg - Tycho’s Famous Observatory

Johannes Kepler ( ) Three Laws of Planetary Motion Every planet travels in an ellipse with the sun at one focus. The radius vector from the sun to the planet sweeps equal areas in equal times. The square of the planet’s period is proportional to the cube of its mean distance from the sun.

Kepler’s study of Mars’ orbit lead him to the discovery that planetary orbits were ellipses. Actually, we know now that orbits can be any conic section, depending on the total energy involved.

Circle Ellipse Parabola Hyperbola The Conic Sections

x y (a,0) (-a,0) (0,b) (0,-b)  p, s 1 + s 2 = const.  (x/a) 2 + (y/b) 2 = 1 (See follow-up exercise #6). The Ellipse f1f1 f2f2 s1s1 s2s2 p P = any point on the ellipse

Kepler’s First law: Elliptical Orbits v r Sun Planet Radius Vector Velocity Vector The sun is located at one of the two foci of the ellipse.

“Vis Viva” v r Conservation of Energy: ½mv 2 - GMm/r = K M m v = {2(K + GMm/r)/m } 1/2

“Vis Viva” (Continued) v r M m Faster Slower As r increases, v decreases. How are v and r related?

Pathfinder’s Path: Start Departure: December, 1996

Pathfinder’s Path: Finish Arrival: July, 1997

Circle Ellipse Parabola Hyperbola The Conic Sections - Revisited Closed orbits: Planets, moons, asteroids, spacecraft. Open orbits: Some comets Parabolic velocity = escape velocity

Follow-Up Activities 1. Earth orbits the sun at a mean distance of 1.5 X 10 8 km. It completes one orbit every year. Compute its orbital velocity in km.sec. 2. The Pathfinder required a greater velocity than Earth orbital velocity to achieve its transfer orbit (why?). Since additional velocity costs NASA money for fuel, can you explain why we launched the spacecraft eastward? (Hint: When viewed from celestial north, the Earth and planets orbit the sun counter-clockwise.)

3. The equation for an ellipse with its center at the origin is (x/a) 2 + (x/b) 2 = 1 Under what mathematical condition does the ellipse become a circle? (Check with your algebra teacher if necessary). 4. Plot the ellipse choosing different values of a and b. (a b). What do you observe? 5. In the Vis-Viva equation for velocity, how does the velocity vary around a CIRCULAR orbit?

6. Extra Credit:The ellipse is defined as a locus of points p such that for two points, f 1 and f 2 (the foci), the sum of the distances from f 1 and f 2 to p is a constant. Use this definition and your knowledge of algebra to show that the equation of an ellipse follows: i.e., that (x/a) 2 + (y/b) 2 = 1 where a and b are the x and y intercepts respectively.

x y (a,0) (-a,0) (0,b) (0,-b) (f,0) s1s1 s2s2 P(x,y) Solution to #6: The Setup

Solution to #6: The Algebra Given: s 1 + s 2 = k (f - x) 2 + y 2 = s 1 2 … (eq. i) (f + x) 2 + y 2 = s 2 2 … (eq. ii) 1.) Let (x,y) = (a,o). This gives k = 2a, and s 1 = 2a - s 2 2.) Let (x,y) = (0,b). This gives s 1 = s 2 = (f 2 +b 2 ) 1/2, and f 2 = a 2 - b 2 3.) Result 2.)  eq. ii gives s 2 = a + (x/a)(a 2 - b 2 ) 1/2 4.) Result 2.) and 3.)  eq. ii gives (x/a) 2 + (y/b) 2 = 1 Be careful: The algebra gets messy! From geometry:

Johannes Kepler: csep10.phys.utk.edu/astr161/lect/history/k epler.html epler.html Tycho Brahe: andrews.ac.uk/~history/Mathematicians/Bra he.html Hohmannn Transfer Orbits: