Penn State University Physics 211 ORBITAL MECHANICS 1

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1 ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there is the analysis of the two-body problem with respect to Newton s Law of Universal Gravitation, and then the examination of the specific scenario known as Kepler s problem in relation to his Three Laws of Planetary Motion. THEORY Newton s Law of Universal Gravitation provides an equivalent expression of his second law of motion,, with respect to the force of attraction between two bodies. The general form given in Eq. 1 (1) where G is the gravitational constant, G = x10-11 m 3 /kg s 2, m 1 and m 2 are the masses of the respective bodies and r 2 is the square of the distance separating the bodies. This force is a central force 2 and is therefore a conservative force, meaning that it is independent of path and equals the first derivative of the potential energy between the masses. When there exists a large difference between the masses of the objects, such that the mass of the smaller object can be neglected a specific form of the two body problem known as Kepler s problem is achieved. The actual problem to be solved though is not one of finding forces, but is instead one of finding the position or speed of the bodies as a function of time given their masses, initial positions and respective velocities. The solution to this problem is a Keplerian orbit and can be expressed with what are called the Classical Orbital Elements (COE) 3. In a simplified model of the Keplerian problem, disregarding outside forces from other bodies, atmospheric drag and relativistic effects, it is found that the only force acting on the satellite causes an acceleration directed towards the center of the inertial body, Earth in our case. Since the gravitational 1 Portions of this lab are derived from U.S. Air Force Academy s ASTRO 310 class. 2 A central force: a force being exerted by one object onto another object that is directed along the line connecting both objects. 3 Classical Orbital Element (COE) Six parameters that describe all the aspects of a specific orbit, also known as the Keplerian elements. They are: semi-major axis a, eccentricity e, inclination i, ascending node Ω, argument of perigee ω, and true anomaly ν. - 1

2 force is conservative, we conclude that the total mechanical energy of the system is a constant and the sum of the potential and kinetic energies equal that constant. The total mechanical energy is given by the expression in Eq. 2. (2) where is the semi-major axis, is the mass of the Earth, and is the mass of the orbiting body or satellite. Additionally, we can derive an expression for the tangential velocity of the satellite as given in Eq. 3. (3a) (3b) Tangential or linear velocity is the velocity vector tangent to the circular path as show in Figure 1. FIGURE I As the satellite orbits the Earth, it also has constant angular momentum resulting from the conservative gravitational force. Though today we can derive all of the constants and equations of motion for an orbiting body from Newton s Law of Gravitation, Kepler did not have such luxury and thus derived his laws of planetary motion from observation. Kepler s first law tells us that the shape of the orbit will be an ellipse, and thus we can derive a complete description of the orbit using the definition of a conic section. Specifically, given the position and velocity of a satellite at apogee 4 or perigee 5 or given the eccentricity and semi-major axis, we can determine the angular momentum and period of the orbit. Table 1 provides the related formulas for a conic section with Figure 1 showing an ellipse for reference. 4 Apogee point at which the distance between the orbiting body and Earth is at a maximum. 5 Perigee point at which the distance between the orbiting body and Earth is at a minimum. - 2

3 Table I Elliptical Parameters radius of perigee or periapsis radius of apogee or apoapsis reduced mass - ( ) Figure I Eccentricity: Semi-major axis: ( ) Parameter the Ellipse: ( ) Specific Angular Momentum: Period of the Orbit: Polar Equation of an Ellipse: ( ) ( ) Area of an Ellipse: : Orbit is elliptical, the greater the value, the more eccentric. : Orbit is circular. : Orbit is a parabola. Using the information above, you will be analyzing different orbital scenario problems first on paper, then using Satellite Tool Kit 9 (STK) 6 to verify results and simulate the scenario. STK is a software package designed specifically for simulating air and space missions. The STK engine allows a user to create a virtual world to configure and test potential or current real world situations. The allowable level detail within the software makes it the preferred application used by aerospace organizations worldwide including NASA, Boeing and Lockheed Martin to name a few. 6 Analytical Graphics, Satellite Tool Kit 9 (STK) - 3

4 PART I Newton s Law of Universal Gravitation and Ballistic Trajectories In this part of the experiment you will use STK to place a missile at the North Pole approximately km above the Earth s surface. Using Newton s Law of Universal Gravitation and the related derived formula for tangential velocity (Eq. 3b) where r is the radius of the earth added to the given altitude of the missile ( ( ) ( ) ) determine what minimum velocity that is required for the missile to complete one revolution around the Earth. Calculations: (Remember to check the units of any constants and make sure they are consistent.) Tangential Velocity Required for a Complete Revolution: Period of One Revolution: PROCEDURE 1. Open STK. 2. On the startup screen, click Create New Scenario (Figure 1). Click Create a New Scenario FIGURE 1 3. In the New Scenario Window, name the scenario and set the analysis time. (Figure 2) a) Do not use spaces in the scenario name. You can use the underscore _ in place thereof. b) The analysis start and stop time determine how long of a period you need to generate and collect information for. In this activity, 6 hours will be more than sufficient. Change the end time so it is exactly 6 hours later than the start time. (Reference Figure 2). - 4

5 Type a name for the scenario. The location to save the scenario and related files. Leave as it. Description is optional. Start time. End time. Change this to be 6 hours after the start time. FIGURE 2 4. Add and configure a FACILITY OBJECT to the scenario a) Left click on the arrow next to the Satellite Object ( ) on the default toolbar to open the Object Catalog list (as shown to the right). b) Double click on the Facility icon it will close the window and a Facility will appear in the Object Browser under and attached to the Scenario icon. c) Right click on the icon and select Rename. d) Rename the Facility North_Pole the underscore between North and Pole is important STK does not allow spaces in its object names. e) Double click on the icon in the Object Browser this will open the Properties Browser showing the properties for the object. f) Type in the Latitude, Longitude (pick any value -180 to 180 deg), and Altitude (use 0 km) for the North Pole (See Figure 3 and note the values in the figure are NOT for the North Pole. Change them!). g) Click Apply at the bottom of the Properties Browser, click OK deg 0.0 deg FIGURE 3-5

6 5. Verify in the 3-D window that your facility is now at the North Pole a) Place the mouse cursor in the 3-D window. Holding down the left mouse button, rotate the Earth to verify your facility is at the North Pole. (If it is not there, repeat the preceding step.) b) Click on the View From/To icon ( ). c) Select the North_Pole in the View From frame and click OK. (See Figure 2) d) Move the view around using the left mouse button. Zoom in and out by clicking and holding the right mouse button. e) After verifying, click the Home View ( ) button on the vertical tool bar. 6. Add and configure a Missile to the Scenario Figure 4: Select North_Pole Select the North_Pole a) Pull down the Object Catalog ( ) from the default toolbar. Double click on the Missile to add a missile to the scenario. b) Double click on the Missile to open the Properties Browser to view the missile s Property Pages. c) Select 3D Graphics/Trajectory Property Page. Change the Ground Track Lead Type to None, change the Trajectory Lead Type to None and set the Trajectory Trail Type to All. Then click Apply. (See Figure 5) Change to None Change to None Change to All Select 3D Graphics / Trajectory Figure 5: Ground Track/Trajectory Type - 6

7 d) Select the Basic/Trajectory Property Page. e) Type in the Launch Latitude Geodetic and Launch Longitude of the North Pole. f) Type in a Launch Altitude of km. g) Change the Impact Latitude Geodetic to Launch Elevation. h) Set the Elevation to 0 0 a horizontal launch. i) Set the Launch Azimuth to the angle from true north measured clockwise to the launch direction. j) Set the Fixed Delta V to 5.0 km/sec. k) Click Apply at the bottom of the Properties Browser. Do Not click inside the 2-D window. This will cause the missile launch location to change. If you accidentally do, reset the Latitude/Longitude/Azimuth/Elevation in the Properties Browser. Instead of selecting the 2-D window by clicking on its banner, select the 2-D tab at the bottom of the Workspace. 7. Evaluate the effects of launch velocities a) Arrange your Workspace windows so you can see the 3-D and 2-D window and the Basic Trajectory Property Page (see Figure 6). Do Not click inside the 2D window. Time Step Figure 6: Suggested Workspace Window Arrangement b) You can change the time step or speed of your scenario and also step forward or backward frame by frame: You may want to slow down the simulation using the Time Step Controls ( ) to get a more accurate reading. You can also use the scenario Step Forward ( ) or Step Reverse ( ) buttons move the scenario forward or backward one time step for a more accurate reading. The Reset button ( )will restart the scenario, and the Pause button ( ) will pause the scenario. c) Press Start ( ) to run the animation until the 3-D trajectory disappears. Record the data in Table 2 below. d) Change the Fixed Delta V value from 5.0 km/sec to 8.5 km/sec in 0.5 km/sec intervals. e) Remember to click Apply after each change and to Reset the animation before replaying the scenario. - 7

8 f) Record your observations in Table 2 Remember, if you click in the 2-D map window without first clicking the Zoom In button, the launch location will be changed to the cursor location. You may want to zoom into a particular area of the 2-D map using the Zoom Control ( ) buttons. Click on the Zoom In button and then draw a box around the area you want to enlarge. Click the Zoom Out button to reset the 2-D map. To get the Time of Flight, watch the Current Scenario Time window in the default tool bar at the top of the screen or use the time readout at the bottom of the screen. To get the Lat/Long of a location on the 2-D map, place your cursor over the location and look to the bottom of the screen for the Lat/Long readout. Latitude, Longitude Table 2: Simulation Data Record Launch Delta V (Km/Sec) Time of Flight (Approximate to impact or 1 revolution) Impact Location (Approximate Latitude and Longitude) 4.5 km/sec 16 min 5 sec N Latitude, 4 0 W Longitude 5.0 km/sec Shape of trajectory/orbit (Circular, Elliptical, Parabolic, Hyperbolic) Earth intersecting elliptical or ballistic trajectory 5.5 km/sec 6.0 km/sec 6.5 km/sec 7.0 km/sec 7.5 km/sec 8.0 km/sec 8.5 km/sec Questions: 1. At what Fixed Delta V does the missile make one complete circular orbit around the Earth? 2. How does this Delta V compare to your theoretical calculation? 3. What is the gravitational force and centripetal acceleration acting on the missile that makes the first complete revolution? (Neglecting the mass of the missile and using in place of GM E m s.) 4. How does the time for one revolution compare to your period calculation? - 8

9 Part II Kepler s First Law Johannes Kepler was a German astronomer who is best known for his laws of planetary motion. Kepler derived his three laws of planetary motion over a 16 year period from observation data collected by Tycho Brahe, a Danish astronomer, whom Kepler was an assistant to. Newton, who was born 12 years after Kepler s death, demonstrated that Kepler s laws of planetary motion are the consequences of the gravitational force between the planets and the sun. Kepler s laws of planetary motion are: 1. Each planet in the Solar System moves in an elliptical orbit with the Sun at one focus. 2. The radius vector drawn from the Sun to any planet sweeps out equal areas in equal time intervals. 3. The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit. Using Kepler s first law and the definition of an ellipse along with our knowledge of central and conservative forces, we can determine the shape and attributes of a satellites orbit about Earth. Recall that our Earth-satellite system operates under the conservative force of gravity, therefore the sum of the kinetic and potential energy is constant as give in Equation 4. (4) Where E is the total energy of the system, is the kinetic energy and is the potential energy. Gravitational potential energy is dependent on position and therefore where is the distance from the center of the Earth to the satellite we can express the gravitational potential energy of the system as a function of position, Eq. 5. (5) 1. Given a satellite s perigee and apogee radius 7, determine the constants of motion from Newton s laws and the geometry of an ellipse. Calculate the semi-major axis, eccentricity, specific angular momentum and period of this satellite. (Using the information provided in Table 1.) a. Radius at Perigee: (km) b. Radius at Apogee: (km) Constant a Semi-major Axis e Eccentricity h Specific Angular Momentum T - period Table 3: Constants of Motion 2. What is gravitational potential energy of the system? 3. What shape is the orbit? Highly elliptical, moderately elliptical, circular? Values 7 As measured from the center of the Earth. - 9

10 Procedure 1. Open STK and open the Scenario named COE_DEMO1 2. Configure the Satellite object in your Scenario a) Double click on the Satellite icon to open the Properties Browser. b) On the Basic Orbit property page, change the Coord Type to Cartesian c) You will see six boxes for the three components of the R R iˆ R ˆj R kˆ vectors. Enter the values from step 1 above in Table 4. x y z and V V iˆ V x y ˆj V kˆ z OPTION VALUE Table 4: Basic Orbit Properties X R x = 0 Y R y = (radius at apogee) Z R z = 0 X Velocity V x = Y Velocity V y = 0 Z Velocity V z = 0 Click Apply. d) Now change the Coord Type to Classical Do the semi-major axis and eccentricity agree with those you calculated in mission planning? Record STK s values in Table 5. Click OK to close the Properties Browser for the Satellite. Table 5: STK Calculated COEs CLASSICAL ORBITAL ELEMENT a Semi-major Axis e Eccentricity VALUES (4 decimal places) Select the 3-D window you should see part of the satellite s orbit, the three principal vectors, and the fundamental plane. If all of the vectors do not show up immediately, run the scenario for a short time and they should appear. 3. Run the scenario a) Note the current start time in the toolbar at the top of your window. Start time: b) Run the scenario and pause it after one complete orbit. You will need to use the Pause button ( ) on the toolbar as well as the Decrease and Increase Time Step buttons ( button ( ). ), and/or Reverse - 10

11 Look at the Current Scenario Time window ( ) on the default toolbar at the top of the screen. Is the time one orbital period after the initial time? If not, check your calculations for the period. 4. Display the COEs and Earth Coordinate Inertial (ECI) data for the satellite a) Open the Satellite s property page and go to 3-D Graphics/Data Display. b) Check the Show box of the Classical Orbital Elements (Figure ). Select check-box next to Classical Orbit Elements Figure 7: Configure the 3-D COE Display c) Click Apply. d) Switch to the 3-D window you should see a list of the six COEs. e) Back in the 3-D Graphics/Data Display property page, check the Show box next to J ECI Position Velocity. f) In the position box, set the X Origin: to 700. g) Click Apply. h) Switch to the 3-D window. You should see the COEs displayed on the left and the R and V vectors displayed on the right. (Figure 8) Figure 8: Position the 3-D J2000 Position/Velocity Vector Display 8 J2000 J2, J4 and TwoBody are the propagation engines that perform the calculations to determine and animate the satellite in the scenario. This propagation engine performs a simplified analytical solution. Other propagation engines perform numeric solutions and account for the various other forces that may be acting on a satellite. - 11

12 5. Run the scenario a. Click the Start ( ) button and observe the orbit in three dimensions from various angles. Which COEs change and which remain constant? Which R and V value(s) is/are changing? 6. Reporting. a) Click the Pause( ) button. Right click on the Satellite icon to open the Properties Drop Down Menu (Figure 9). Scroll down to Report & Graph Manager and move the mouse pointer over the arrow ( ) to the right. Select Apogee and Perigee Report. Print this report. Figure 9: Report & Graph Manager b) Right click on the Satellite icon to open the Properties Drop Down Menu (Figure 9). Scroll down to Report & Graph Manager and move the mouse pointer over the arrow ( ) to the right. Select Angular Momentum Report. You only need the data for a short period of time, therefore locate the STOP time directly above the report data in the toolbar (Figure 10). Refresh Button Highlight the value you want to change. 15 to 30 minutes of data is sufficient. Figure 10: Adjust Report Period - 12

13 Change the hour by highlighting it and advance it two hours from the start time. Now select the day by highlighting it and correct the value so it matches the day of the start time. Click the refresh button to generate the new report data. Part III - Explore Real World Orbits 1. Close the current STK scenario. Click on File, Close. Click NO when prompted to save changes. 2. Open a scenario. Click File, Open. Select the ORBITS.SC scenario in the Open Dialogue and click OK. 3. This scenario contains satellites that are currently in orbit and operational today. Run the scenario and observe. a. You can change the viewing perspective in the 3D window by clicking on the View From/To button. (Reference Figure 11) Figure 11: Change the 3D perspective. What different artificial satellites and space vehicles are orbiting? What are their orbits? Circular, elliptical, LEO 9, MEO 10, GEO 11, HEO 12? Can you determine which one of these satellites is the research satellite operated by PSU from our Mission Operations Center 13 at University Park? 9 LEO Low Earth Orbit km altitude (from the surface of the Earth) 10 MEO Mid Earth Orbit 20,100 km altitude (from the surface of the Earth) 11 GEO Geosynchronous Earth Orbit 35,786 km altitude (from the surface of the Earth) with a period equal to 23 hrs. 56 min 12 HEO High Earth Orbit 39,500 km altitude (from the surface of the Earth) 13 Mission Operations Center at PSU: https://www.swift.psu.edu/ - 13

14 Questions: 1. How do your calculated values from Table 2 compare with those as reported by STK? 2. What do you notice about the specific angular momentum value with respect to position and velocity? We show below the derivation of the area swept out by the radius vector r of the satellite using the vector equation for angular momentum where L is the vector form of angular momentum, r is the position vector, v is the velocity vector and p is the linear momentum. (6) (6a) (6b) Note, in the simplified model we assume m sat=1kg. (6c) ( ) (7) 3. How is Eq. 7 related to Kepler s Second Law? - 14

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