# Presentation: 371574

##  Exploring Mars 

***.***

-  Exploring Mars

***Mr. K. NASA/GRC/LTP ***
***Edited: Ruth Petersen******    ***

**Part 4**
**Pathfinder’s Path I**

**Notes:**
Pathfinder’s Path - I
Orbital kinematics (the description of orbital motion without concern for dynamics) are ‘simple,’ yet difficult to comprehend, because they tend to be counter-intuitive. Exploring Mars, Part 4, examines the fundaments of orbital kinematics by exploring the path followed by the Pathfinder spacecraft from Earth to Mars.

## Preliminary Activities

***(Use the URL’s provided on Slide 21 to complete the preliminary activities.)****** ***
***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?***

**Notes:**
The persons named above played pivotal roles in the development of modern astronomy and space flight. And Mars played a pivotal role in their landmark discoveries.

## 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?***

**Notes:**
Of all the geometry developed by the Ancient Greeks, their conic sections are most essential to an heuristic understanding of orbital kinematics.

## Slide 4

**Notes:**
Tycho Brahe, a Danish astronomer, was one of the greatest observational astronomers in the Western World. Although his concept of the solar system was an alchemist’s mixture of geo- and helio-centric orbits, his observations were of impeccable accuracy, and his discoveries were first rate! Specifically, his detailed observations of Mars provided the essential basis for Kepler’s three laws of planetary motion. In his private life, Tycho was a raucous individual. Having lost his nose in a duel when young, Tycho is reported to have had a silver nose for weekdays, and a gold one for Sundays. (In this painting, a natural nose is rendered, perhaps out of respect for a truly great man.) Tycho was very much a party animal who died of a massive overdose of beer in a drinking contest.

## Slide 5

**Notes:**
With financial help from the King of Denmark, Tycho Brahe constructed an observatory on the island of Hveen in Copenhagen Sound. The observatory, which he named Uraniborg (Heavenly City), housed immense and, therefore, very accurate instruments. Remember that Tycho did not have use of a telescope; therefore, all of his observations were made with the naked eye.  His instruments were designed to enable minute angular and temporal measurements. His planetary observations, especially of Mars, were of such unprecedented accuracy, that Johannes Kepler would later use them to establish his three laws of planetary motion. These laws are still used today.

## Every planet travels in an ellipse with the sun at one focus.

***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.***

**Notes:**
Kepler was a quiet and gentle man with poor eyesight. In many ways, he was the perfect complement to the brash and often troublesome Tycho. An accomplished mathematician, professor, and court consultant, Kepler worked for many years on Tycho’s data, slowly extracting his laws of planetary motion. Among the greatest problems he had to overcome was the age-old bias, inherited from the Ancient Greeks, that the heavens were an empyrean (of pure fire and light) realm where only the perfection of the circle could be allowed. His laws are stated above: the first is the orbital law; the second, the areal law; the third, the harmonic law. The first two laws were published in 1609. The third, much more abstruse and difficult, was not published until 1619. These laws became the touchstone against which Sir Isaac Newton tested his own theoretical ideas about gravitation. They were shown, by Newton, to follow mathematically from his own laws of motion and inverse square law of gravity.

## 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.***

**Notes:**
Mars...
Of all the worlds known to people throughout the ages, Mars is, in many ways, the most significant.

## Circle    Ellipse     Parabola   Hyperbola

***Circle    Ellipse     Parabola   Hyperbola***

***The Conic Sections***

**Notes:**
The conic sections are represented algebraically by a general second-order polynomial of the form
Ax2 + By2 + Cx + Dy + E = 0
where A, B, C, D, and E are rational coefficients, and x and y are Cartesian coordinates. Heuristically, the conic sections are obtained from the sections of an equiangular cone, cut at various angles to its central axis (hence, the name!).
In everyday life, the hyperbola may be seen on the wall above and below any lamp placed nearby, provided that the lamp has a shade that wraps around the bulb and is open at the top and bottom.

## S1 + S2 = Constant

***x***

***y***

***The Ellipse***

***S******1****** + S******2****** = Constant***

**Notes:**
The ellipse is of particular interest in orbital kinematics because it is the shape of a closed, unperturbed orbit in a Newtonian gravitational field. In this figure, f1 and f2 are the focal points of the ellipse. These points are unique for each ellipse. When they coincide, the ellipse reduces to a circle!
Elliptical rooms act as whisper chambers when two individuals stand and speak, one at each of the foci.  The curve itself is constructed so that the sum of the distances s1 and s2 is a constant (s1 + s2 = C ). An ellipse may be drawn by pushing two pins into a sheet of cardboard, tying the ends of a string to the pins, and moving a pencil (point P above) so that the string remains taut. The ellipse above is represented by the expression:
(x/a)2 + (y/b)2 = 1 (See exercise #6).

## Kepler’s First law: Elliptical Orbits

***Kepler’s First law: Elliptical Orbits***

***Sun***

***Planet***

***The sun is located at one of the two foci of the ellipse.***

**Notes:**
Kepler’s first law states that objects moving in a Newtonian gravitational field travel along conic section curves in space, provided that no other forces disturb them. In our solar system, the sun and planets all exert a gravitational influence. The sun, of course, predominates. For heliocentric orbits, the perturbing forces usually take the form of gravitational pulls and tugs from the planets themselves, especially giant Jupiter. Planetary moons move around their primaries just as the planets move around the sun. In fact, it was his observations of Jupiter’s moons (Io, Callisto, Ganymede, and Europa) that prompted Galileo to support the Copernican notion of a sun-centered solar system. Jupiter, he asserted, provided a model, in miniature, of the solar system at large.

## “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**

**Notes:**
The idea of living force (vis-viva) became fashionable for a while in post-Newtonian thinking. Living force was a conceptual precursor of the modern concept of energy. Natural philosophers spoke of the ‘conservation of force’ – a concept later brought into clearer focus as the ‘conservation of energy.’ For a planet moving about the sun without perturbation (variation caused by an external body), the total energy K shared between the planet and the sun is conserved. This energy has two components:
Kinetic energy of the planet: 1⁄2mv2 
Potential energy: - GMm/r .
The potential energy has a negative sign because it is a binding energy whose maximum value is taken to be zero at r = , and less than zero everywhere else. Although this idea may seem somewhat strange, it enables an accurate mathematical dynamical representation of motion in the gravitational field. As they progress in their studies, physics students eventually become so acclimated to the concept of negative energy, that they seldom give it a second glance. If the total energy K is less than zero, the planet is bound to the primary and the orbit is a closed conic. If K = 0, the orbit becomes open and is a parabola. K = 0 is the usual condition for calculating escape velocity. If K is greater than zero, the orbit is hyperbolic, and the object will have a net positive energy at r = .

## “Vis Viva” (Continued)

**“*****Vis Viva” (Continued)***

***M***

***Faster***

***Slower***

***As r increases, v decreases & vice versa.***

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

**Notes:**
The vis-viva equation and Kepler’s areal law tell us the same thing: the closer an object moves to its primary in an elliptical orbit, the faster it moves. One may think of the orbit as either falling toward or rising away from the primary without actually bumping into it. The complete motion of a planet may be calculated from the vis-viva equation and Newton’s laws of motion without direct reference to Kepler’s laws (although Kepler’s laws are much more fun just because they are just so elegant!).

## Pathfinder’s Path:

***Pathfinder’s Path:***

-  Earth's Orbit

-  Mars' Orbit

**Notes:**
Each object moves around its primary in a conic section. Thus, Earth and Mars move in elliptical orbits around the sun. Spacecraft and moons move in elliptical orbits around planets, and so on...  To get from one planet to the other, one must find a suitable means of traveling between their respective orbits. One such means is Walter Hohmann’s transfer orbit (Germany, 1925), a minimum energy (and therefore cost–effective) orbit if one is not particularly concerned about travel time. One ‘simply’ accelerates the vehicle to the suitable velocity, then coasts all the way to the destination. Unpiloted space vehicles make use of Hohmann transfer orbits on a regular basis. The Apollo missions to the moon also made use of Hohmann orbits. Piloted interplanetary spacecraft, however, will use thrust to shorten the travel time and, therefore, ensure greater astronaut safety. If KE is the total energy of the Earth in its orbit, and KM is the total energy of Mars in its orbit, then the total energy of the Hohmann orbit KH satisfies the inequality:
KE < KH < KM.

## Pathfinder’s Path:

***Pathfinder’s Path:***

***Arrival: July 1997***

***Departure: December 1996***

**Notes:**
Pathfinder used the Earth’s orbital velocity around the sun (~67,000 mph eastward) and its rotational velocity at Cape Kennedy (~1,000 mph eastward) to gain a running start. Onboard thrusters fired in earth orbit gave the vehicle an additional 6,000 mph eastward to start it on its way towards Mars. The spacecraft then began to coast. Seven months later, it made a soft landing on the night side of Mars, using Mars’ gravitational field to assist it in its approach and landing. Of course, timing was everything. Since mission controllers wanted the planets close at arrival (for purposes of data transmission). And since everything was moving, they had to select a launch time when Earth and Mars were in suitable relative positions. A 28-day launch window was available. If this window had been missed, the mission would have been delayed for almost two years, awaiting the next favorable planetary alignment.

## 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***

**Notes:**
The circle is a special case of the ellipse. Recall that the equation for an ellipse is:
(x/a)2 + (y/b)2 = 1
If a = b, then this expression reduces to:
x2 + y2 = a2 
which  is a circle of radius a. For energy considerations on closed and open orbits, please refer back to slide 11.

## 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.)***

**Notes:**
Read carefully! The answers are (more or less) already in the preceding text!

## 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?***

**Notes:**
Ditto...

## 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.***

**Notes:**
This one requires some skill, but the average high school algebra student ought to be able to do it!

## 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***_

**Notes:**
Just in case, though, here’s one way of doing it...

## 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:***

**Notes:**
If you have questions, please feel free to e-mail me. My address is on the last slide.

##  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