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Mr. K. NASA/GRC/LTP Edited: Ruth Petersen
Part 4
Pathfinder’s Path I
Preliminary Activities(Use the URL’s provided on Slide 21 to complete
the preliminary activities.)
1.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?
5. In your algebra class, discuss the conic sections. Write the equation for an ellipse with its center at the origin.
6. 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
(a,0)
(-a,0)
(0,-b)
(0,b)
x
yThe Ellipse
s1s2
P = any point on the ellipse
p
f2f1
f1 & f2 = specific points on the x-
axis
S1 + S2 = Constant
Kepler’s First law: Elliptical Orbits
Sun
Planet
rRadius Vector
vVelocity 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)
M
v
r
m
Faster
Slower
As r increases, v decreases & vice versa.
How are v and r related?
Pathfinder’s Path:
Departure: December 1996
Pathfinder’s Path:
Arrival: July 1997
Departure: December 1996
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
