Building Planetary Orbits (the Feynman way)


 Brice Godfrey Wade
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1 Building Planetary Orbits (the Feynman way) Using a method of computing known as finite differences we can quickly calculate the geometry of any planetary orbit. All we need are a starting position and an initial velocity then we just put Sir Isaac in the driver s seat. You get to create your own planet and see if the orbit is stable. Follow these instructions stepbystep: 1. Open Excel or Google Sheets. Note: this approach was first popularized by the late great Richard Feynman. In his day, students did all these calculations by hand! To derive maximum benefit from the spreadsheet: ask if you need help with syntax (especially entering and copying formulas). The central body (mass M) is at the origin of an (x, y) coordinate system. To model a solar system, we will use astronomical units (the mean radius of the earth s orbit, 1 AU = 1.5x10 11 m) as our distance unit, years as our time unit and one solar mass as the mass unit. This sets G = 4 2, which makes the allimportant Kepler s 3 rd Law ratio GM/4 2 = M in solar mass units. The sun is 1 solar mass. Our planet starts on the x axis with an initial velocity in the +y direction. We ll model an earth in circular orbit around the sun. Note that the starting velocity of AU/year will give the expected period for our earth. 2. Enter the following initial data table at the top of your spreadsheet: Note the velocity is given in components x and y. A B C D E Numerical calculation of planetary orbits Initial values X0 1 delta t 0.01 Y0 0 GM 39.5 Vx Use the diagram at right to Vy write the vector components of acceleration for the orbiting body of y mass m (from the vector gravitational force, y = r sin ax Mm ay F ˆ g G r): Fg 2 r r v0 a x = a y = M x = r cos (x0,y0) x
2 Be sure to use the correct algebraic signs or its goodbye, planet! 4. In your spreadsheet s rows beneath your input table, enter column headings time, x position, y position, distance, ax, ay, vx, vy. 5. In the next row, enter the initial time (0) and the corresponding x, y values from the initial data table as cell references (for example, under the column heading x position, type =the cell that holds the value of x0. Do the same for the y position. Important: Whenever you use a reference to a cell containing a constant (such as x0, y0, G or dt), be sure you the absolute cell reference format $col$row. 6. In the next column, calculate the distance from the origin to the current x,y position, using the builtin square root function, =sqrt( ). 7. Enter your formulas for the acceleration x and y components in the next two columns in the first row. 8. We re treating acceleration as a constant during each time interval: we should be using a small enough time interval so we can get away with this. So the velocities in row 1 (at time=0) will not be the initial vx and vy components from the input table. Enter the formula for vx: =v0x + ax*dt/2, where these symbols are the cells with the appropriate values. This gives us the velocity at the midpoint of the time interval. Remember that all formulae start with =. Do the same for vy. 9. In the second row first column, enter a formula that adds the quantity dt to the time in the row immediately above. Recall that while entering formulas, after typing an equals sign, you can use the arrow keys (up, down, left, right) to point to cells you want inside the formula. The formula for the new time is thus = [up arrow] + $row$col, where row col is the location of dt. 10. Positions in the second row are equal to the prior position (in the rows immediately above) plus dt times the average velocity during the time interval, =[up arrow] + dt*[prior velocity component]. Radius and the two acceleration components can be calculated with the same formulas as in the first row  use copy down. 11. The velocity components in the second row are the prior velocities plus the corresponding component of acceleration ax or ay * dt (no ½ this time). You should now have two rows filled; looking something like this (your values may not be exactly the same):
3 Time x y r ax ay vx vy Copy the entire second row down so that you have 100 rows filled in. Voila! Check the last row to see if your satellite has made it all the way back to its starting location. If not, adjust delta t so that it just does; this is one way to determine the period of your planet s orbit. Note: the smaller your delta t, the more rows you need, but the more accurate your result. 13. Create a graph of y vs. x using the scatter plot type of graph. Other nice graphs to make are x and y vs. time and speed (from vx and vy) vs. distance r. Earth orbit Find the minimum, maximum and average radial distances from the r column, using the conveniently named =min( ), =max( ) and =average( ) functions. These give perihelion, aphelion and semimajor axis. Use the semimajor axis to calculate orbital period. 15. Add columns for kinetic, potential and total mechanical energies. distance, AU distance, AU Save your spreadsheet so that you have a copy you can get to at school and at home. Make sure your partners have their own copies. Problems a). Use the spreadsheet to calculate the ratio r 3 /T 2, where r is the average radial distance, for your earth.
4 b). Check the same ratio for Mars orbit, using x 0 = 1.67 AU, y 0 = 0 (Mars aphelion, point where it is furthest from sun) and V x0 = 0 (you ll have to find the value of V y0 that gives 1.38 AU for Mars perhihelion (shortest distance to the sun) hint: is this velocity component more than or less then that of earth?) c). Determine the eccentricity of the Mars orbit you ve created: ra rp eccentricity, where r a is the longer aphelion radial distance from sun to planet and r p ra rp is the perihelion. Kepler found the eccentricity of Mars as d). Move the x 0 to 1.25 AU and see what happens with the same earth initial velocity of 6.28 AU/yr. Calculate the eccentricity of that orbit. Find the velocity that will make this orbit circular. e). Go back to x 0 = 1 and find the lowest value of vy 0 that produces an open (parabolic) orbit. This represents the velocity needed to escape the solar system when launched from earth. f). Build a spreadsheet that results in a single plot showing the orbits of Earth and Mars. Set the dt so that the slowest planet goes around once. g). Extra credit: Show Earth s orbit and Halley s comet: perihelion = AU, aphelion 35.1 AU Extra extra credit (for the truly daring few): Create a new spreadsheet that includes the gravitational effects of the sun and the earth on the moon orbiting the earth. Note that the gravitational force of each object acts upon all others  we ll allow the simplification that the sun is stationary. You ll need the earth s mass: M e = 5.97x10 24 kg, the average radius of the moon s orbit, r m = 3.84x10 8 m and the moon s orbital period = 27.3 earth days. And you will need to take into consideration what to do with earth days in the AU, year, G = 4 2 system we adopted. Suggestion: start the moon and earth on the x axis (what phase of the moon would that be?). Use the velocity of the earth around the sun, as already determined; find the moon s velocity. Good luck!
5 Building Planetary Orbits part 2 1. Mars anyone? The standard means of getting from one planet to another is to use a transfer orbit. To get to Mars, we start in a circular Earth orbit and increase speed so that we enter an elliptical orbit around the Sun. As shown in the illustration, the perihelion of the transfer orbit is the launch point near Earth and the aphelion is the target at Mars; however, both are moving around the Sun. Thus the position shown for Earth is at the start of the transfer and the position shown for Mars is at the end of the transfer. Mars orbit a = 227.9x10 6 km (1.52 AU), e = 0.09 Mars mass = 6.4x10 23 kg, Mars radius = 3.4x10 6 m a. Assume a 600 km altitude Earth orbit to start. What is the velocity for this orbit? b. What velocity do we need at perihelion of the transfer orbit? How much delta v does this require? c. How long does it take to make the transfer? Hint: it s half of an ellipse. d. What velocity do we have when we arrive at Mars? e. What velocity do we need for circular orbit around Mars at an altitude of 100 km? How much delta v does this require? Verify your calculated orbit with your spreadsheet. f. For a simplifying assumption, take Mars orbit to be circular. Once you know how long the transfer takes, at what point on its orbit should Mars be when you start the transfer? You must meet with it when you both arrive at aphelion. g. A clever demo of the success of this maneuver would use the spreadsheet to model Mars elliptical orbit and the motion of the spaceship on the same chart. 2. Shoot the moon The Apollo missions went from an Earth orbit 330 km above the surface (conveniently known as low Earth orbit or LEO) to a lunar orbit 60 km above the lunar surface. Let s run a few numbers to simulate this trip, assuming the Moon is stationary and ignoring the space ship s acceleration due to lunar gravity. Earth radius = 6.37x10 6 m Lunar radius = 1.73x10 6 m Mass = 5.97x10 24 kg Lunar mass 7.35x10 22 kg Note: do not be alarmed by the figure 8 in the illustration; we re just finding an ellipse.
6 Determine: a. The velocity and orbital period of LEO b. The velocity necessary to leave LEO and enter an elliptical trans lunar orbit, arriving at the appropriate distance from the Moon. This requires perigee = LEO radius and apogee = the earth moon distance (3.84x10 8 m) + the radius of the lunar orbit. c. The time needed to reach the Moon on this path. d. The spaceship speed when it reaches apogee
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