.

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