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# THE PLANETS ASTR 105G

NMSU

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This 0 page Class Notes was uploaded by Reina Conn DVM on Sunday November 1, 2015. The Class Notes belongs to ASTR 105G at New Mexico State University taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/233217/astr-105g-new-mexico-state-university in Astronomy at New Mexico State University.

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Date Created: 11/01/15

Welcome to ASTR 105 Lab Review the Syllabus Notes for Lab 1 The Metric System Objects we deal with in astronomy cover many size scales Electron I 05 nanometers 5960 Gigameters quotmiragequot Am 45a Our Solar System I 00000000005 meters 39 45 Mmeiers V L15 sseoooooooooo meters meters 5960 Gigameters I Mike 39 539 1nquot 70quot 115 meters The metric system provides a useful notation to cover these scales Astronomical Units In many cases we have to use units Which span even larger size scales The quotAstronomical Unitquot AU is shorthand for the distance between Earth and the Sun and is useful for measuring solar system distances The quotlight yearquot is the distance light travels in a year and is useful for measuring the distance to stars l 5960 Gigameters do AU Changing Units and Scale Conversion I want to take a trip to Saturn for Spring breakto rollerblade around the rings I know I can rollerblade at 10 feet per second I want to know what this is in kilometers per hour so I can figure out how many hours this is going to take me This requires changing my units and scale conversion with my Saturn map Changing Units First we need our distance conversion factor 1 fOOt 03 meters The nice thing about the METRIC system is that conversion between metric units just involves powers of ten 1000 m 1 km There is nothing special about time in the metric system We will continue to use hours minutes seconds etc amp 0 Now I need to know the scale on this map to learn the distance around the rings Using the scale given in the bottom left corner of the image I measure with my ruler that 100000 km is equal to five inches Scale Conversion cont39d This gives us a SCALE FACTOR of 1 in 20000 km I also measured the distance around the rings with my ruler to be 140 inches To find the actual distance in km I multiply my scale factor by the distance I measured on the map Note you probably will not be getting the exact answer but you should be getting something close Changing Units and Scale Conversion fin Now that I know my speed and my distance I can calculate the time it Will take me to rollerblade around Saturn39s rings Looks like I will not be wasting my time attempting this feat further unit conversion reveals this is about 30 years amp Squares Roots and Exponents Math Fun 5 uares S uare Roots E onents The quotsquarequot of a number is just the number times itself 3 3 x 3 9 11 11x11 121 The quotsquare rootquot of a number is just the reverse which number when multiplied by itself will equal the number inside the square root symbol 12111x11 J12111 An exponent is a number taken to any quotpowerquot Just multiply the number by itself that many times 353x3x3x3x3243 Scientific Notation Scientific Notation is just a way to express a number with a lot of zeroes Distance between Sun and Earth 1 AU 149000000000 meters 1490000oo0oo meters 149 x 10 msiers All we have to do is count how many times we move the decimal point to the first nonzero number and write quotx 10 7quot Size of a hydrogen atom 000000000053 meters 000000000053 meters 53 x 103911meters If we move the decimal point to the left the blank is a positive number If we move the decimal point to the right it39s a negative num f Scientific Notation Math Multiplication and Division with Scienti c Notation Multiplication 25 x 10quot x 50 x10 Muliipiy these we 25 x 101 x 50 x101 And Ihen add these two logether 25 x 50 125 17 14 31 Division 25 x1017 50 x10 I 1 17 Then subtract Jusi divide the boiiom these wan7M irom me up 50 1o14 255005 17143 G aphingPlotting The basic equation for a line is y mx b To graph this we start plugging in values for quotxquot and see what quotyquot pops out Example y 2X 1 X0 y201 y1 X1 y211 y X2 y221 y Plotting cont d Now we take our list of values plot the points on the graph and draw a line through it Y 1 3 2 5 3 7 4 In y mx b the quotmquot is the slope of the line how quickly does it rise or fall as we move along the mods The quot1quot is the yirtercept of the line the place where it crosses the yaxis Name39 Date 5 Kepler s Laws 51 Introduction Throughout human history the motion of the planets in the sky was a mystery why did some planets move quickly across the sky while other planets moved very slowly Even two thousand years ago it was apparent that the motion of the planets was very complex For example Mercury and Venus never strayed very far from the Sun while the Sun the Moon Mars Jupiter and Saturn generally moved from the west to the east against the background stars at this point in history both the Moon and the Sun were considered planets The Sun appeared to take one year to go around the Earth while the Moon only took about 30 days The other planets moved much more slowly In addition to this rather slow movement against the background stars was of course the daily rising and setting of these objects How could all of these motions occur Because these objects were important to the cultures of the time even foretelling the future using astrology being able to predict their motion was considered vital The ancient Greeks had developed a model for the Universe in which all of the planets and the stars were each embedded in perfect crystalline spheres that revolved around the Earth at uniform but slightly different speeds This is the geocentric or Earth centered model But this model did not work very well7the speed of the planet across the sky changed Sometimes a planet even moved backwards It was left to the Egyptian astronomer Ptolemy 85 7 165 AD to develop a model for the motion of the planets you can read more about the details of the Ptolemaic model in your textbook Ptolemy developed a complicated system to explain the motion of the planets including epicycles and equants that in the end worked so well that no other models for the motions of the planets were considered for 1500 years While Ptolemyls model worked well the philosophers of the time did not like this model7their Universe was perfect and Ptolemyls model suggested that the planets moved in peculiar imperfect ways In the 1540s Nicholas Copernicus 1473 7 1543 published his work suggesting that it was much easier to explain the complicated motion of the planets if the Earth revolved around the Sun and that the orbits of the planets were circular While Copernicus was not the rst person to suggest this idea the timing of his publication coincided with attempts to revise the calendar and to x a large number of errors in Ptolemyls model that had shown up over the 1500 years since the model was rst introduced But the heliocentric Sun centered model of Copernicus was slow to win acceptance since it did not work as well as the geocentric model of Ptolemy Johannes Kepler 1571 7 1630 was the rst person to truly understand how the planets in our solar system moved Using the highly precise observations by Tycho Brahe 1546 7 1601 of the motions of the planets against the background stars Kepler was able to formulate three laws that described how the planets moved With these laws he was able to predict the future motion of these planets to a higher precision than was previously possible 52 Many credit Kepler with the origin of modern physics as his discoveries were what led lsaac Newton 1643 e 1727 to formulate the law of gravity Today we will investigate Kepler s ws and the law of gravity 52 Gravity Gravity is the fundamental force governing the motions of astronomical objects No other force is as strong over as great a distance Gravity in uences your everyday life ever drop a glass and keeps the planets moons and satellites orbiting smoothly Gravity a ects everything in the Universe including the largest structures like super clusters of galaxies down to the smallest atoms and molecules Experimenting with gravity is difficult to do You can t just go around in space making extremely massive objects and throwing them together from great distances But you can model a variety of interesting systems very easily using a computer By using a computer to model the interactions of massive objects like planets stars and galaxies we can study what would happen in just about any situation ve to know are the equations which predict the gravitational interactions of the objects The orbits of the planets are governed by a single equation formulated by Newton GM M Persist 1 A diagram detailing the quantities in this equation is shown in Fig 5 1 Here Fng is the gravitational attractive force between two objects whose masses are M1 and M2 The distance between the two objects is R The gravitationa constant G is just a small number that scales the size of the force The most important thing about gravity is that the force depends only on the masses of the two objects and the distance between theml This law is called an Inverse Square Law because the distance between the objects is squared and is in the denominator of the fraction There are several laws like this in physics and astronomy F GMleRZ gravity Figure 5 1 The force of gravity depends on the masses of the two objects M1 M2 and the distance between them R Today you will be using a computer program called Planets and Satellites by Eugene Butikov to explore Kepler s laws and how planets double stars and planets in double star 53 systems move This program uses the law of gravity to simulate how celestial objects move 0 Goals to understand Keplerls three laws and use them in conjunction with the com puter program Planets and Satellites to explain the orbits of objects in our solar system and beyond 0 Materials Planets and Satellites program a ruler and a calculator 53 Kepler s Laws Before you begin the lab it is important to recall Keplerls three laws the basic description of how the planets in our Solar System move Kepler formulated his three laws in the early 1600s when he nally solved the mystery of how planets moved in our Solar System These three empirical laws are l The orbits of the planets are ellipses with the Sun at one focus ll A line from the planet to the Sun sweeps out equal areas in equal intervals of time III A planetls orbital period squared is proportional to its average distance from the Sun cubed P2 olt 13 Lets look at the rst law and talk about the nature of an ellipse What is an ellipse An ellipse is one of the special curves called a conic section If we slice a plane through a cone four different types of curves can be made circles ellipses parabolas and hyperbolas This process and how these curves are created is shown in Fig 52 Before we describe an ellipse let7s examine a circle as it is a simple form of an ellipse As you are aware the circumference of a circle is simply 27TR The radius R is the distance between the center of the circle and any point on the circle itself In mathematical terms the center of the circle is called the focus An ellipse as shown in Fig 53 is like a attened circle with one large diameter the major axis and one small diameter the minor axis A circle is simply an ellipse that has identical major and minor axes Inside of an ellipse there are two special locations called foci foci is the plural of focus it is pronounced fo sigh The foci are special in that the sum of the distances between the foci and any points on the ellipse are always equal Fig 54 is an ellipse with the two foci indenti ed LLF177 and LLFZWI Exercise 1 On the ellipse in Fig 54 are two Xls Con rm that that sum of the distances between the two foci to any point on the ellipse is always the same by measuring the distances between the foci and the two spots identi ed with Xls Show your work 2 points 54 Flguze 5 2 Fonz types of cuzves can be geneneted by sllcmg a cone Wlth a plane a allele an elhpse a paxabola and a hypexbola Stxangely these lsnn curves axe also the allowed shapes of the oxb s of plane39s estemds comets and setelhtesl Flguze 5 3 An elllpse Wlth the male and nnlnsn axes ldentl ed Demise 2 In me ellipse shown m m 5 5 Lwn pmncs Pp and Mpg azexdenu ed that are nut leaked at me we ymmnns cf Lhe in Repeat muse 1 but con m mm P and P2 axe nut Lhe m owns hpse 2 pmncs mum 5 A An ampse mm Lhe cwn m Idenu ed mum 5 5 An ellipse mm m mnr m palms Idenu ed pmble um Lhe pm 6 numan when you m you mmqu 15 nnc m you mu have m scan 15 up Ifycu TA gave you a CDROM Lh need m sett me CDROM m Lh CDROM a nn yam and open 5 as on SaLe lwas has been mstalled nn me mmqu ycu are using Lock nn Lhe daktwp 5n Mn fnu bassbuth Gemng Shavedquot Tummal SimulaLmn and Whatquot cm 55 on the Simulations button We will be returning to this level of the program to change simulations Note that there are help screens and other sources of information about each of the simulations we will be runningido not hesitate to explore those options Exercise 3 Keplerls rst law Click on the Keplerls Law button and then the First Law button inside the Keplerls Law box A window with two panels opens up The panel on the left will trace the motion of the planet around the Sun while the panel on the right sums the distances of the planet from the foci Remember Keplerls rst law states the orbit of a planet is an ellipse with the Sun at one focus The Sun in this simulation sits at one focus while the other focus is empty but whose location will be obvious once the simulation is runl At the top of the panel is the program control bar For now simply hit the Go button You can clear and restart the simulation by hitting Restart do this as often as you wish After hitting Go note that the planet executes an orbit along the ellipse The program draws the vectors from each focus to 25 different positions of the planet in its orbit It draws a blue vector from the Sun to the planet and a yellow vector from the other focus to the planet The right hand panel sums the blue and yellow vectors Note if your computer runs the simulation too quickly or too slowly simply adjust the Slow downSpeed Up slider for a better speed Describe the results that are displayed in the right hand panel for this rst simulation 2 points Now we want to explore another ellipse In the extreme left hand side of the control bar is a slider to control the Initial Velocity At start up it is set to 12 Slide it up to the maximum value of 135 and hit Go Describe what the ellipse looks like at 135 vs that at 12 Does the sum of the vectors right hand panel still add up to a constant 3 points 57 Now lets put the Initial Velocity down to a value of 10 Run the simulation What is happening here The orbit is now a circle Where are the two foci located In this case what is the distance between the focus and the orbit equivalent to 4 points The point in the orbit where the planet is closest to the Sun is called perihelion and that point where the planet is furthest from the Sun is called aphelion For a circular orbit the aphelion is the same as the perihelion and can be de ned to be anywhere Exit this simulation click on File and Exit Exercise 4 Keplerls Second Law A line from a planet to the Sun sweeps out equal areas in equal intervals of time From the simulation window click on the Second Law after entering the Keplerls Law window Move the Initial Velocity slide bar to a value of 12 Hit Go Describe what is happening here Does this con rm Keplerls second law How When the planet is at perihelion is it moving slowly or quickly Why do you think this happens 4 points 58 Look back to the equation for the force of gravity You know from personal experience that the harder you hit a ball the faster it moves The act of hitting a ball is the act of applying a force to the ball The larger the force the faster the ball moves and generally the farther it travels In the equation for the force of gravity the amount of force generated depends on the masses of the two objects and the distance between them But note that it depends on one over the square of the distance 1R2 Letls explore this inverse square law77 with some calculations 0 If R 1 What does 1R2 0 If R 2 What does 1R2 0 If R 4 What does 1R2 What is happening here As R gets bigger what happens to 1R2 Does 1R2 de creaseincrease quickly or slowly 2 points The equation for the force of gravity has a 1R2 in it so as R increases that is the two objects get further apart does the force of gravity felt by the body get larger or smaller ls the force of gravity stronger at perihelion or aphelion Newton showed that the speed of a planet in its orbit depends on the force of gravity through this equation V CMsun MplaneLXQT 1a 2 where r is the radial distance of the planet from the Sun and a is the mean orbital radius the semi major axis Do you think the planet will move faster or slower when it is closest to the Sun Test this by assuming that r 05a at perihelion and r 15a at aphelion and that a1l Hint simply set CMsun Mplanet 1 to make this comparison very easyl Does this explain Keplerls second law 4 points 59 What do you think the motion of a planet in a circular orbit looks like ls there a de nable perihelion and aphelion Make a prediction for what the motion is going to look likekhow are the triangular areas seen for elliptical orbits going to change as the planet orbits the Sun in a circular orbit Why 3 points Now lets run a simulation for a circular orbit by setting the Initial Velocity to 10 What happened Were your predictions correct 3 points Exit out of the Second Law and start up the Third Law simulation Exercise 4 Keplerls Third Law A planet7s orbital period squared is proportional to its average distance from the Sun cubed P2 olt a3 As we have just learned the law of gravity states that the further away an object is the weaker the force We have already found that at aphelion when the planet is far from the Sun it moves more slowly than at perihelion Keplerls third law is merely a re ection of this factithe further a planet is from the Sun a the more slowly it will move The more slowly it moves the longer it takes to go around the Sun P The relation is P2 olt as where P is the orbital period in years while a is the average distance of the planet from the Sun and the mathematical symbol for proportional is represented by K To turn the proportion sign into an equal sign requires the multiplication of the 0L3 side of the equation by a constant P2 C x as But we can get rid of this constant C by making a ratio We will do this below In the next simulation there will be two planets one in a smaller orbit which will represent the Earth and has a 1 and a planet in a larger orbit where a is adjustable 60 Start up the Third Law simulation and hit Go You will see that the inner planet moves around more quickly while the planet in the larger ellipse moves more slowly Letls set up the math to better understand Keplerls Third Law We begin by constructing the ratio of of the Third Law equation 12 C x 13 for an arbitrary planet divided by the Third Law equation for the Earth C X lt3 PE C x aE In this equation the planets orbital period and average distance are denoted by Pp and ap while the orbital period of the Earth and its average distance from the Sun are PE and am As you know from from your high school math any quantity that appears on both the top and bottom of a fraction can be canceled out So we can get rid of the pesky constant C and Keplerls Third Law equation becomes P 1 e 1 lt4 13 But we can make this equation even simpler by noting that if we use years for the orbital period PE 1 and Astronomical Units for the average distance of the Earth to the Sun aE 1 we get 1 P2 3 TPaTP or Pga 3 5 Remember that the cube of 1 and the square of 1 are both 11 Lets use equation 5 to make some predictions If the average distance of Jupiter from the Sun is about 5 AU what is its orbital period Set up the equation P a 535x5x5125 6 So for Jupiter P2 125 How do we gure out what P is We have to take the square root of both sides of the equation vPZ P V 125 112 years 7 The orbital period of Jupiter is approximately 112 years Your turn If an asteroid has an average distance from the Sun of 4 AU what is its orbital period Show your work 2 points 61 In the Third Law simulation there is a slide bar to set the average distance from the Sun for any hypothetical solar system body At start up it is set to 4 AU Run the simulation and con rm the answer you just calculated Note that for each orbit of the inner planet a small red circle is drawn on the outer planet7s orbit Count up these red circles to gure out how many times the Earth revolved around the Sun during a single orbit of the asteroid Did your calculation agree with the simulation Describe your results 2 points If you were observant you noticed that the program calculated the number of orbits that the Earth executed for you in the Time window and you do not actually have to count up the little red circles Letls now explore the orbits of the nine planets in our solar system In the following table are the semi major axes of the nine planets Note that the average distance to the Sun77 a that we have been using above is actually a quantity astronomers call the semi major axis77 of a planet 1 is simply one half the major axis of the orbit ellipse Fill in the missing orbital periods of the planets by running the Third Law simulator for each of them 3 points Table 51 The Orbital Periods of the Planets a Notice that the further the planet is from the Sun the slower it moves and the longer it takes to complete one orbit around the Sun its year Neptune was discovered in 1846 and Pluto was discovered in 1930 by Clyde Tombaugh a former professor at NMSU How 62 many orbits or what fraction of an orbit have Neptune and Pluto completed since their discovery 3 points 54 Going Beyond the Solar System One of the basic tenets of physics is that all natural laws such as gravity are the same everywhere in the Universe Thus when Newton used Keplerls laws to gure out how gravity worked in the solar system we suddenly had the tools to understand how stars interact and how galaxies which are large groups of billions of stars behave the law of gravity works the same way for a planet orbiting a star that is billions of light years from Earth as it does for the planets in our solar system Therefore we can use the law of gravity to construct simulations for all types of situationsieven how the Universe itself evolves with time For the remainder of the lab we will investigate binary stars and planets in binary star systems First what is a binary star Astronomers believe that about one half of all stars that form end up in binary star systems That is instead of a single star like the Sun being orbited by planets a pair of stars are formed that orbit around each other Binary stars come in a great variety of sizes and shapes Some stars orbit around each other very slowly with periods exceeding a million years while there is one binary system containing two white dwarfs a white dwarf is the end product of the life of a star like the Sun that has an orbital period of 5 minutes To get to the simulations for this part of the lab exit the Third Law simulation if you haven7t already done so and click on button 7 the Two Body and Many Body77 simulations We will start with the Double Star77 simulation Click Go In this simulation there are two stars in orbit around each other a massive one the blue one and a less massive one the red one Note how the two stars move Notice that the line connecting them at each point in the orbit passes through one spotithis is the location of something called the center of mass In Fig 56 is a diagram explaining the center of mass If you think of a teeter totter or a simple balance the center of mass is the point where the balance between both sides occurs If both objects have the same mass this point is halfway between them If one is more massive than the other the center of massbalance point is closer to the more massive object Most binary star systems have stars with similar masses M1 z M2 but this is not always the case In the rst default binary star simulation M1 2M2 The mass ratio77 1 in this case is 05 where mass ratio is de ned to be 1 MzMl Here M2 1 and M1 2 so 1 MzMl 12 05 This is the number that appears in the Mass Ratio77 63 MIXI MZXZ Figure 56 A diagram of the de nition of the center of mass Here object one M1 is twice as massive as object two Therefore M1 is closer to the center of mass than is M2 In the case shown here X2 2X1 window of the simulation Exercise 5 Binary Star systems We now want to setup some special binary star orbits to help you visualize how gravity works This requires us to access the Input77 window on the control bar of the simulation window to enter in data for each simulation Clicking on lnput brings up a menu with the following parameters Mass Ratio Transverse Velocity Velocity magnitude and Direction Use the slide bars or type in the numbers to set sverse Velocity l0 Velocity magnitude 00 and Direction 00 For now we simply want to play with the mass ratio Use the slide bar so that Mass Ratio 10 Click Ok This now sets up your new simulation Click Run Describe the simulation What are the shapes of the two orbits Where is the center of mass located relative to the orbits What does q 10 mean Describe what is going on here 4 points Ok now we want to run a simulation where only the mass ratio is going to be changed Go back to Input and enter in the correct mass ratio for a binary star system with M1 40 and M2 10 Run the simulation Describe what is happening in this simulation How are the stars located with respect to the center of mass Why Hint see Fig 56 4 points Finally we want to move away from circular orbits and make the orbit as elliptical as possible You may have noticed from the Keplerls law simulations that the Transverse Velocity affected whether the orbit was round or elliptical When the Transverse Velocity 10 the orbit is a circle Transverse Velocity is simply how fast the planet in an elliptical orbit is moving at perihelion relative to a planet in a circular orbit of the same orbital period The maximum this number can be is about 13 If it goes much faster the ellipse then extends to in nity and the orbit becomes a parabola Go back to Input and now set the Transverse Velocity 13 Run the simulation Describe what is happening When do the stars move the fastest The slowest Does this make sense Whywhy not 4 points The nal exercise explores what it would be like to live on a planet in a binary star systeminot so fun In the Two Body and Many Body77 simulations window click on the 65 Dbl Star and a Planet button Here we simulate the motion of a planet going around the less massive star in a binary system Click Go Describe the simulationiwhat happened to the planet Why do you think this happened 4 points In this simulation two more windows opened up to the right of the main one These are what the simulation looks like if you were to sit on the surface of the two stars in the binary For a while the planet orbits one star and then goes away to orbit the other one and then returns So sitting on these stars gives you a different viewpoint than sitting high above the orbit Lets see if you can keep the planet from wandering away from its parent star Click on the Settings window As you can tell now that we have three bodies in the system there are lots of parameters to play with But lets con ne ourselves to two of them Ratio of Stars Masses and Planetistar Distance The rst of these is simply the q we encountered above while the second changes the size of the planets orbit The default values of both at the start up are q 05 and Planetistar Distance 024 Run simulations with q 04 and 06 Compare them to the simulations with q 05 What happens as 1 gets larger and larger What is increasing How does this increase affect the force of gravity between the star and its planet 4 points 66 See if you can nd the value of 1 at which larger values cause the planet to stay home while smaller values cause it to eventually crash into one of the stars stepping updown by 001 should be adequate 2 points Ok reset 1 05 and now lets adjust the Planetistar Distance In the Settings window set the Planetistar Distance 01 and run a simulation Note the outcome of this simulation Now set Planetistar Distance 03 Run a simulation What happened Did the planet wander away from its parent star Are you surprised 4 points Astronomers call orbits where the planet stays home stable orbits Obviously when the Planetistar Distance 024 the orbit is unstable The orbital parameters are just right that the gravity of the parent star is not able to hold on to the planet But some orbits even though the parents hold on the planet is weaker are stablekthe force of gravity exerted by the two stars is balanced just right and the planet can happily orbit around its parent and never leave Over time objects in unstable orbits are swept up by one of the two stars in the binary This can even happen in the solar system In the Comet lab you can nd some images where a comet ran into Jupiter The orbits of comets are very long ellipses and when they come close to the Sun their orbits can get changed by passing close to a major planet The gravitational pull of the planet changes the shape of the comet7s orbit it speeds up or slows down the comet This can cause the comet to crash into the Sun or into a planet or 67 cause it to be ejected completely out of the solar system You can see an example of the latter process by changing the Planetistar Distance 04 in the current simulation 55 Summary 35 points Please summarize the important concepts of this lab Your summary should include 0 Describe the Law of Gravity and what happens to the gravitational force as a as the masses increase and b the distance between the two objects increases Describe Keplerls three laws in your own words and describe how you tested each one of them 0 Mention some of the things which you have learned from this lab 0 Astronomers think that nding life on planets in binary systems is unlikely Why do they think that Use your simulation results to strengthen your argument Use complete sentences and proofread your summary before handing in the lab 56 Extra Credit Derive Keplerls third law P2 C x 13 for a circular orbit First what is the circumference of a circle of radius a If a planet moves at a constant speed 1 in its orbit how long does it take to go once around the circumference of a circular orbit of radius a This is simply the orbital period P Write down the relationship that exists between the orbital period P and a and 1 Now if we only knew what the velocity v for an orbiting planet was we would have Keplerls third law In fact deriving the velocity of a planet in an orbit is quite simple with just a tiny bit of physics go to this page to see how it is done httpwwwgoednetnscalarryorbitskeplerhtml Here we will simply tell you that the speed of a planet in its orbit is v GM112 where G is the gravitational constant mentioned earlier M is the mass of the Sun and a is the radius of the orbit Rewrite your orbital period equation substituting for 1 Now one side of this equation has a square root in itiget rid of this by squaring both sides of the equation and then simplifying the result Did you get P2 C x 13 What does the constant C have to equal to get Keplerls third law 5 points 57 Possible Quiz Questions 1 Brie y describe the contributions of the following people to understanding planetary motion Tycho Brahe Johannes Kepler Isaac Newton 2 What is an ellipse What is a focus What is a binary star ypw 5 Describe what is meant by an inverse square law 6 What is the de nition of semi major axis 68

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