# ASTRODYNAMICS. o o o. Early Space Exploration. Kepler's Laws. Nicolaus Copernicus ( ) Placed Sun at center of solar system

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1 ASTRODYNAMICS Early Space Explratin Niclaus Cpernicus ( ) Placed Sun at center f slar system Shwed Earth rtates n its axis nce a day Thught planets rbit in unifrm circles (wrng!) Jhannes Kepler ( ) Discvered rbit f Mars was nt circular Cncluded that planets mve arund the Sun in elliptical rbits with the Sun nt at the center but at a fcus Kepler's Laws Kepler's 1st Law: The rbits f the planets are ellipses with the Sun at ne fcus. Kepler's 2nd Law: The line jining a planet t the Sun sweeps ut equal area in equal times. * This means that the clser a planet is t the Sun, the faster it will travel. Kepler's 3rd Law: 1

2 The square f an rbital perid is directly prprtinal t the cube f the average distance between the Sun and a planet: * This means that the farther a planet is frm the Sun, the lnger it takes t g arund. * If yu knw the distance f a planet frm the Sun, yu knw hw lng that planet takes t g arund. Gravity Gravity is the idea that tw (r mre) chunks f matter tend t attract t each ther. The mre mass they have, the mre they will attract. Galile Galilei ( ) Used telescpes t revlutinize ur understanding f the Universe Discvered that all bjects regardless f mass fall at the same rate when drpped frm the same height (if air resistance is neglected). Isaac Newtn ( ) Wrked ut laws f mtin and gravity, revlutinizing ur understanding f the wrld. Orbits f planets and satellites are based n Newtn's laws! Newtn's Laws: 1. A bdy at rest r mving in a straight line at a cnstant speed stays that way unless a frce is applied t it. 2. A bdy's mass times its acceleratin is equal t the applied frce (F=Ma). 3. T every actin there is always ppsed an equal reactin. Newtn's Universal Law f Gravitatin: The frce f gravity between tw bdies is directly prprtinal t the prduct f their masses and inversely prprtinal t the square f the distance between them. Orbit Arund the Earth Orbit An rbit is a fixed path in space that a spacecraft fllws. Depending n the missin, the size, shape, and rientatin f an rbit will vary. Satellite A satellite is any bject in rbital mtin (fr example, the Sun, Mn, Earth, a spacecraft, r the space shuttle). The laws that determine the rbital mtin f a satellite are the same, n matter what larger bdy the satellite rbits. The Earth's gravity determines the size and shape f the rbit f a nearby spacecraft. The law f gravity requires that the spacecraft must rbit in a plane that passes thrugh the center f the Earth. 2

3 All Earth-rbiting satellites must rbit in space t avid cllisins with the trillins f air mlecules frm the atmsphere that surrund the surface f the Earth. Space begins at abut 100 miles abve the surface f the Earth, where the air is s thin that it has little effect n a spacecraft. Lw Earth Orbit Lw Earth Orbit (LEO) is a type f rbit. The space shuttle flies in a Lw Earth Orbit. The altitude in LEO varies frm 100 t 300 miles abve the Earth's surface. (Cmmunicatin satellites rbit at apprximately 25,000 miles frm Earth.) Since it takes the shuttle abut 90 minutes t cmplete ne rbit arund the glbe, it can cmplete 16 t 17 rbits in ne day. On a relative scale, the shuttle rbits the Earth very clse t its surface. The radius f the Earth is apprximately 4,000 miles, r abut 40 times larger than the distance frm the surface f Earth t where the shuttle rbits. (On a relative scale, if the Earth was the size f a basketball, the shuttle wuld rbit at a thumbnail's distance frm the ball's surface.) NOTE: An bject in Lw Earth Orbit is in an elliptical rbit. (A circular rbit is simply a special case f an elliptical rbit.) Hw t Describe an Orbit There are six Classical Orbital Elements (see next sectin) that are necessary fr us t knw abut an rbit and a satellite's place in it. These elements help us describe: Orbit size Orbit shape Orbit rientatin Orbit lcatin Elliptical Gemetry 3

4 An ellipse lks like an val, r squashed circle. The lngest line drawn frm ne end f the ellipse (thrugh the center) t the ther side is called the majr axis (2a). Every ellipse has tw fci (F and F'), and the distances between each fcus and the center f the ellipse are equal (c). In a circle, the tw fci lie n tp f each ther. The pint n the semi-majr axis clsest t the Earth is called the perigee, while the pint n this axis farthest frm the Earth is called the apgee. Defining the Classical Orbital Elements I. Semi-majr Axis The size f the rbit is described by the semi-majr axis (a), which is ne-half the distance acrss the majr (lng) axis f the rbit. II. Eccentricity Eccentricity (e) specifies the shape f an rbit and is given by the rati f the distance between the tw fci and the length f the majr axis. e = 2c/2a The eccentricity f a circular rbit is zer, and can range frm zer t less than ne fr an ellipse. III. Inclinatin (i) [degrees] The angle between the plane f the equatr and the rbital plane. 4

5 IV. Right Ascensin f the Ascending Nde (Ω) The angle between the Sun and the intersectin f the equatrial plane and the rbit n the first day f spring in the Nrthern Hemisphere. The day is called the vernal equinx. Lking dwn frm abve the Nrth Ple, the right ascensin f the ascending nde is psitive cunter-clckwise. VI. V. Argument f Perigee (w) [degrees] The angle between the ascending nde and the rbit's pint f clsest apprach t the Earth (perigee). True Anmaly (v)[degrees] The angle between the perigee and the vehicle in the rbit plane. 5

6 Satellite Grund Tracks The Six Classical Orbital Elements allw us t describe what an rbit lks like in space. What we need t knw next is what part f the Earth the satellite is passing ver at any given time. A grundtrack shws the lcatin n the Earth that the spacecraft flies directly ver during its rbital path arund the Earth. T understand grundtracks, we need t knw the fllwing: The grundtrack fllws what is called a great circle rute arund the Earth. A great circle is any circle which cuts thrugh the center f the Earth. All grundtrack drawings use the latitude/lngitude system. Latitude: Latitude measures hw far nrth r suth a pint lies ff the equatr. The equatr is at zer degrees latitude, the Nrth Ple is at 90 degrees nrth latitude and the Suth Ple is at 90 degrees Suth latitude. Lngitude: Lngitude measures hw far east r west a pint lies frm an imaginary line that runs frm the Nrth Ple t the Suth Ple thrugh Greenwich, England. This line is called the Prime Meridian. The lngitude varies frm 0 degrees at the prime meridian t 180 degrees west and 180 east. Grundtrack drawings appear n a Mercatr prjectin f the Earth's surface. This type f map allws the entire surface f the rund wrld t appear n a rectangle. The prjectin f a spacecraft rbit n a flat map (Mercatr prjectin) lks like a sine wave. This is the spacecraft's grundtrack. If the Earth did nt rtate, the grundtrack f an rbit wuld cntinuusly repeat. But, the Earth spins eastward n its axis at nearly 1,000 mph at the equatr while the spacecraft rbits arund it. 6

7 Hw des this affect the grundtracks? Even thugh the spacecraft rbit stays fixed in space, the Earth rtates t the east. S t a pint fixed n the Earth, the spacecraft appears t shift t the west during successive rbits. The highest latitude reached by a satellite rbit is equal t its inclinatin. T illustrate this cncept, the fllwing grundtracks represent rbits with the same perid, but A has an inclinatin f 10 degrees, B has an inclinatin f 30 degrees, C has an inclinatin f 50 degrees, and D has an inclinatin f 85 degrees. Field f View Field f View (FOV) is the angle that describes the amunt f the Earth's surface the spacecraft can see at any given time. The higher the spacecraft is abve the Earth, the mre yu can see. Thus, the size f the rbit determines the spacecraft's field f view. The area f cverage n the Earth's surface is called the swath width. 7

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