By: Andrea Lee & Erica-Lee Lampert Planetary Rotations Johannes Kepler The Comet Apophis Why do these calculations matter? Our Calculations Conclusions References Johannes Kepler was born in the Holy Roman Empire of German nationality in the city of Weil der Stadt (Today a part of Italy) in 1571. After observing the orbits of planets, Kepler created the laws of Planetary Rotation. ("Kepler: Johannes Kepler"). The comet Apophis is a comet that orbits our Sun and intersects the revolution of the Earth around the Sun. The Apophis was discovered by Roy A. Tucker, David J. Tholen and Fabrizio Bernardi. There have been numerous theories regarding whether or not the two satellites will ever collide. Astronomers are able to determine whether comets will collide with the earth by using mathematical equations ("Apophis Risk Assessment Updated"). Step-by-Step Calculations For starters, these equations help us to understand why Earth travels in an ellipse, how we are able to experience our different seasons and, have 365 days a year. Without this orbit, Earth would continue to travel in a straight line into no where.

They also help us to know when our seasons occur and when to expect summer, winter, fall and spring.

The orbit of the moon helps tell us the tides of our oceans and if we will experience a tidal wave or not.

Most importantly, these orbital equations help us to understand when or if Apophis is going to hit the Earth. If we know the orbit of Apothsis , we know if we need to take precaution and find a way from stopping it from entering our atmosphere.

As for who uses them, astronomers use them to determine if objects such as meteors or debris are going to crash into our Earth. When they know this, they can warn us if a comet or such, will hit based on its orbital path and can help evaluate the location of impact and eventually evacuate a place or town if need be.

Thus, astronomers also use these to tell if Apophis will hit us or when it will hit us. like the one that may hit the Earth in 2014. We would be unable to determine when or if a satellite would impact Earth without the discovery of these equations ("Summary of Important Concepts"). By using the equations in our presentation, astronomers

and scientists are able to determine whether or not comets, like Apophis, are a threat to our planet. Without the use of math, scientists could not be certain about the likelyhood of an impact. While determining the path of Apophis is most important, astronomers are also able to determine metor paths, seasons, changing in the tides, information for voyages to the moon, and weather patterns based on these equations as well. "Algebra II: Ellipse." Algebra II: Ellipse. Cliffnotes, n.d. Web. 19 Mar. 2013.

"Apophis Risk Assessment Updated." Apophis Risk Assessment Updated. N.p., 28 Feb. 2013. Web. 10 Mar. 2013

Dictionary.com. Dictionary.com, n.d. Web. 20 Mar. 2013

"Discovery Circumstances." Discovery Circumstances. NASA, n.d. Web. 16 Mar. 2013.

"Eccentricity an Ellipse." Eccentricity an Ellipse - Math Open Reference. Math Open Reference, 2009. Web. 19 Mar. 2013

"Gravity and Orbits." PhET. N.p., n.d. Web. 20 Feb. 2013.

"International Astronomical Union." Near Earth Asteroids. IAU, 06 Mar. 2013. Web. 16 Mar. 2013.

"Kepler: Johannes Kepler." Kepler: Johannes Kepler. N.p., n.d. Web. 25 Feb. 2013.

"Kepler's Laws Calculator." Interactive Planetary Orbits. N.p., n.d. Web. 20 Feb. 2013.

"Simulations." Physics. N.p., n.d. Web. 10 Mar. 2013.

Stump, Dan. "The Motion of Planets." Mathematics of Motion. N.p., n.d. Web. 20 Feb. 2013.

"Orbit of a Comet." Orbit of a Comet. N.p., n.d. Web. 19 Mar. 2013.

"Semi-major / Semi-minor Axis of an Ellipse." Semi-major / Semi-minor Axis of an Ellipse - Math Open Reference. Math Open Reference, 2009. Web. 19 Mar. 2013.

"Summary of Important Concepts." Astronomy 114. UMass, n.d. Web. 20 Feb. 2013. The equations for the orbits of both the comet Apophis and Earth is as follows... The arrows represent the intersection between the orbit of Apophis and Earth. These intersections happen every 23 years.

("Apophis Risk Assessment Updated"). The first law stated that:

Planets move in ellipses (not circles) around the sun in one focus point of the ellipse.

The second law stated that:

An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time.

The third law stated that:

The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun

("Kepler's Law Calculator"). The formula for Kepler's third law looks like this: The variable "p" stands for

planetary period and the variable "a" stands for the semimajor axis in AU

("Kepler's Law Calculator"). 2 ) / [1+e cos ] r=a(1-e The Shapes and Graphs of Planetary Orbits http://phet.colorado.edu/sims/my-solar-system/my-solar-system_en.html Step one: Finding the Semi Major Axis and the Major Axis ("Semi-major / Semi-minor Axis of an Ellipse"). Major Axis = a+b where "a" and "b" are the distances from each focus to any point on the ellipse.

Minor Axis= where "a" and "b" are the distance from each focus and the f is the foci.

Step Two: Find the Eccentricity of the orbit ("Eccentricity an Ellipse").

e=

Step Three: Find the measurement of the Ellipse ("Algebra II: Ellipse").

Step Four: Find the True Anomaly ("Algebra II: Ellipse").

Plug in your "a" (semi major aixs) valve and "e" (eccentricity) valve into the equation and now you have the orbit of a planet or a comet!:) Planetary Orbits:

Planets tend to orbit around the Sun in a path known as an ellipse. An ellipse is the set of points in a plane where the sum of the distances from two fixed points in that plane stay constant. The two points are each called a focus (foci). The midpoint of the segment joining the foci is called the center of the ellipse. An ellipse has two axes of symmetry. The longer one is called the major axis, and the shorter one is called the minor axis. The two axes intersect at the center of the ellipse ("Algebra II: Ellipse"). Thank You! :) x= Point on the "x" axis

y= Point on the "y" axis

a= end point of the major axis (-a)

b= the end point of the minor axis (-b) Comets, like Apophis, go around the Sun in a highly elliptical orbit. They can spend hundreds and thousands of years out in the depths of the solar system before they return to Sun at their perihelion. Like all orbiting bodies, comets follow Kepler's Laws - the closer they are to the Sun, the faster they move thus, thus why we used a planet's orbit to determine the Apophis' path.

As comets come closer to the Sun however, the warming of its surface causes its materials to melt and evaporate producing the comet's characteristic tail. Comet tails can be as long as the distance between the Earth and the Sun. This is why some comets tend to crash into surrounding planets, just as the Apophis could in a few years. Even if the comet does not directly impact our surface, the fragmented parts of their tails may ("Orbit of a Comet"). Vocab you may need to know:

Semi Major Axis: Half the distance across an ellipse measured along a line through its foci.

Foci: Two points inside an ellipse: At both end points.

Major Axis: The major axis of an ellipse is its longest diameter, a line that runs through the center and both foci.

Eccentricity: Deviation of a curve or orbit from circularity.

True Anomaly: The polar angle of an object in a Ksest approach. epler orbit, measured from in the orbital plane.

("Dictionary.com"). All Images from Google Images ("Simulations") ("Simulations") We chose to investigate the use of mathematics in space by investigating how math is used within planetary orbits as they revolve around the sun and each other. We attempted to apply what we learned in class and compared and contrasted that information within the equations we discovered that astronomers and scientists use when exploring planetary rotations. Sample Problem: Step One: add 4 and 5 together to get the Major axis

4+5=9

Major axis is 9

Step Two: add the valves of “a” and “b” together

4+5=9 and square that number

9X9= 81

After you square nine, you take your valve of “f” of 7 and square that as well

7X7= 49

You subtract 81 from 49

81-49= 32

Then you take the square root of 32

= 5.66 which equals your Semi Axis

Step Three: you want to find the eccentricity of the orbit.

For this, you will want to square the “a” valve and the “b” value:

4X4= 16 (b)

5X5= 25 (a)

Then you will want to subtract those two numbers

25-16= 9

You take the square root of 9

3 and divide it by your “a” value to give you

1.8 As your e]Eccentricity

Step 4: find your ellipse measurement

You take your “y” ,which I will say equals 2, and your “x” which equals 2 and square them

2X2=4

2X2=4

Then you square you “a” value and “b” values again

5X5=25

4X4=16

Then you divided your “x” value by your “a” and “y” by your “b” value

4/25 = .16

4/16= .25

You add those together and you get

.16 + .25 = .41 as your Ellipse orbit.

Step Five: then you take all your numbers and plug it into the equation on the left and get your finally True Anomaly! Which would be,

5.66(1-1.8)2/ (1-1.8cos60)

= 36.224 a(5) b(4) _____________________________7__________________________ F F a(5) b(4) 60 (x,y)

(2,2)

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