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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 (1546 - 1607)
Uraniborg - Tycho’s Famous Observatory
Johannes Kepler (1571 - 1630)
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, s1 + s2 = const. (x/a)2 + (y/b)2 = 1
(See follow-up exercise #6).
The Ellipse
f1 f2
s1s2
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: ½mv2 - 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 108 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; a = 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, f1 and f2 (the foci), the sum of the distances from f1 and f2 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) (f,0)
s1s2
P(x,y)
Solution to #6: The Setup
Solution to #6: The Algebra
Given: s1 + s2 = k (f - x)2 + y2 = s1
2 … (eq. i) (f + x)2 + y2 = s2
2 … (eq. ii)1.) Let (x,y) = (a,o). This gives k = 2a, and
s1 = 2a - s2
2.) Let (x,y) = (0,b). This gives s1 = s2 = (f2+b2)1/2, and f2 = a2 - b2
3.) Result 2.) eq. ii givess2 = a + (x/a)(a2 - b2)1/2
4.) Result 2.) and 3.) eq. ii gives(x/a)2 + (y/b)2 = 1Be careful: The algebra gets messy!
From geometry:
Johannes Kepler:
csep10.phys.utk.edu/astr161/lect/history/kepler.html
www.vma.bme.hu/mathhist/Mathematicians/Kepler.html
Tycho Brahe:
http://www-groups.dcs.st-andrews.ac.uk/~history/
Mathematicians/Brahe.html
Hohmannn Transfer Orbits:
http://www.jpl.nasa.gov/basics/bsf-toc.htm
joseph.c.kolecki@grc.nasa.gov
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