# Presentation: 870863

##  Exploring Mars 

***.***

-  Exploring Mars

***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)

***Tycho Brahe (1546 - 1607)***

## Uraniborg - Tycho’s Famous Observatory

## Johannes Kepler (1571 - 1630)

***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. 

***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

***Circle    Ellipse     Parabola   Hyperbola***

***The Conic Sections***

## x

***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***

***f******1***

***f******2***

***s******1***

***s******2***

***p***

***P = any point on the ellipse***

## Kepler’s First law: Elliptical Orbits

***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”

**“*****Vis Viva”***

***v***

***r***

***Conservation of Energy: 1⁄2mv******2****** - GMm/r = K***

***M***

***m***

***v = *****{*****2(K + GMm/r)/m *****}****1/2**

## “Vis Viva” (Continued)

**“*****Vis Viva” (Continued)***

***v***

***r***

***M***

***m***

***Faster***

***Slower***

***As r increases, v decreases.***

***How are v and r related?***

## Pathfinder’s Path:

***Pathfinder’s Path:***
***Start***

***Departure: December, 1996***

-  Hohmann Transfer Ellipse

-  Earth's Orbit

-  Mars' Orbit

## Pathfinder’s Path:

***Pathfinder’s Path:***
***Finish***

***Arrival: July, 1997***

## Circle    Ellipse     Parabola   Hyperbola

***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; 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

***x***

***y***

***(a,0)***

***(-a,0)***

***(0,b)***

***(0,-b)***

***(f****** ******,0)***

***(f,0)***

***s******1***

***s******2***

***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/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