A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources the and Spacecrafts orbits Moon, Planets Calculating the 171 of by Dr. Shiu-Sing TONG
172 Calculating the orbits of the Moon, Planets and Spacecrafts I. Summary Notes Subject Area Level Keywords Prerequisites SS1 to SS3 Celestial mechanics, orbital motion, universal gravitation, Newton s laws of motion You should have some familiarity with Newton s laws of motion and the equations of uniformly accelerated motion. They should have basic knowledge of coordinate geometry and be familiar with basic algebraic manipulation. Introduction The laws of mechanics and simple numerical methods are used in this project to calculate the orbits of the Moon, planets and spacecraft. Various approaches to orbital calculation are presented, including: (i) using the method of curving-fitting to find orbital elements from observation data; (ii) using Newton s laws of motion, the law of universal gravitation and simple numerical methods to calculate orbits, and (iii) using Kepler s equation with given orbital elements to predict the position of a planet at any time. Students are guided to acquire the basic concepts and skills in orbital analysis, and then to synthesize these skills to solve more complex problems. They are eventually required to solve the problem of planning a space mission to a hypothetical planet in 3D space. The calculations are done with the Excel Solver tool so that no programming prerequisite is required. Learning Outcomes On completion of this project, you should: Have knowledge of basic Newtonian mechanics and its application in astronomy; Be able to apply simple analytical and numerical methods to solve mechanics problem; Be able to use electronic spreadsheet and analysis tools in numerical computation; Have a basic understanding of orbital motion under gravity and the parameters used to description it; Have a historical perspective on the development of celestial mechanics and appreciate the insight and efforts of great astronomers and physicists; Be careful and have perseverance in doing scientific research. II. Learning Materials An Introduction to Orbital Calculation A. Warm-up Activity I: Fitting the Orbit of Venus This activity introduces the change in apparent size and phase of an inferior planet as a result of its changing position relative to Earth and the Sun, and the Excel Solver as a tool to fit observation data with a model.
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 173 The problem Figure 1 Phases of Venus (Photo Courtesy of Wah! & Doucats) This warm up activity makes use of a series of photos of Venus taken at different times during a phase cycle (Figure 1) to find the orbital period and radius of the planet around the Sun. Venus is an inferior planet (i.e., its orbit around the Sun lies within that of Earth). It exhibits a full cycle of phases as seen from Earth. The apparent size of Venus also changes during a phase cycle. These changes can be measured in Figure 1. You are required to use a simplified geometrical model to describe the changes in phase and apparent radius with time. Fitting the observation data with Excel Solver then allows the orbital radius and period of Venus to be determined.? Question and Discussion We construct a simplified model of the inner solar system in which Venus and Earth are in uniform circular motion around the Sun. In this model, the apparent radius R of Venus is inversely proportional to its distance r from Earth. The phase of Venus can be expressed as the ratio l/r, as shown in Figure 2. Express students to express the variation of R and l/r in terms of, the orbital radius of Venus in astronomical units (AU), and, the angle subtended by Earth and Venus at the position of the Sun. Figure 2 Relationship between the phase of Venus and its position relative to the Sun and Earth.
174 Calculating the orbits of the Moon, Planets and Spacecrafts The next step is to find the dependence of on time t. Let T 1 and T 2 be the orbital periods of Venus and Earth respectively. Ask students to express in terms of T 1, T 2, and the time t after inferior conjunction (i.e., Earth, Venus and Sun on a straight line with Venus in the middle). Express the answer in terms of,, the period of Venus and time in the unit of Earth-year. Finally, the values of and are found by a least squares fit of the observation data. A typical Excel spreadsheet is shown in Figure 3. Figure 3 Excel spreadsheet for fitting the orbital radius and period of Venus The time for each photo is converted to days and finally to Earth-years. The values of R and l/r are measured for each photo. Arbitrary but reasonable initial values of and are chosen to calculate the values of, R and l/r for each photo according to our model (denoted as (theory), R(theory), and l/r(theory) on the spreadsheet). The square of the difference between the measured and calculated values of R and l/r is found for each photo. These values are summed up to and as shown in the last two columns of Figure 3. The last step involves a least square fit of the observation data with the model. The values of and are varied to minimize the sum and respectively. This is done with the Solver tool in Excel. Figure 4 shows the dialog box of the Solver tool. The Target Cell is set to the cell that contains the values of or
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 175. The Equal To option is set to Min to minimize the value of the target cell. By Changing Cells specifies the cells that contain the parameters to be varied; in this case, the parameters are, and A (see A.2-A.3). Subject to the Constraints specifies any constraints in the minimization process. Clicking on the Solve button will initiate the fitting. Figure 4 Dialog box of the Excel Solver tool? Question and Discussion Discuss the following with your classmates and teacher: 1. Compare the values of orbital period and radius of Venus you find with the accepted values. Find the percentage errors. 2. Which fitting gives a more accurate result? R or l/r? Explain. 3. What assumptions are used in our model? Are they good approximations? B. Warm-up Activity II: Fitting the Orbit of the Moon This activity introduces the equation of elliptical orbit in polar coordinates and methods of determining orbital elements from observation data. The problem This warm up activity makes use of a series of photos of the Moon taken at different times in a complete phase cycle to determine the shape of its orbit around Earth. The elliptical orbit of the Moon results in its change in distance and apparent size as seen from the Earth. Thus, measuring the apparent radius of the Moon on the photos gives the variation of its distance. The phases of Moon, after correcting for. Earth s motion around the Sun, give the variation in angular position of the Moon. The data are combined to give a polar plot of the Moon s distance against angular position, which represents the shape of the Moon s orbit. Background knowledge We first introduce the basic parameters for describing an ellipse (Figure 5(a)).
176 Calculating the orbits of the Moon, Planets and Spacecrafts The semi-major axis a is half of the major (i.e., long) axis of the ellipse and the semi-minor axis b is half of the minor (i.e., short) axis. The eccentricity e ( ) is defined as (B.1) When a = b, e = 0, the ellipse reduces to a circle. The ellipse becomes more elongated as e approaches 1. An ellipse has two foci F and F, located at equal distance ae from the centre. Kepler s first law of planetary motion states that a planet moves in an elliptical orbit around the Sun. The Sun lies on one focus of the ellipse and the other focus is empty. In the case we are studying, the Moon moves around Earth in an elliptical orbit with Earth located at one of the foci. Figure 5 (a) Basic parameters for describing an ellipse (b) Parameters for describing the orbit of the Moon The elliptical orbit can be described by the following equation in polar coordinates: (B.2) where r and are the distance and angular displacement of the Moon, respectively. The angle specifies the position of the perigee (the point closest to Earth) relative to a reference direction (+x axis in Figure 5(b)).? Question and Discussion Similar to Warm-up Activity I, the apparent radius R of the Moon on the photo is inversely proportional to its distance r from Earth: (B.3) where A is an unknown proportional constant. The apparent radii are measured on a series of photos in a complete phase cycle. Data of 1/R gives the variation of the distance with time up to a proportional constant. A typical image sequence can be found at: http://en.wikipedia.org/wiki/file:lunar_libration_with_phase_oct_2007.ogv
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 177 For simplicity, we make the following approximations in modeling the motion of the Moon: The Moon s orbit lies on the same plane as Earth s orbit around the Sun; and the Earth is in a uniform circular motion around the Sun. The reference direction in Figure 5(b) is taken as the direction of the Sun at New Moon, i.e., the Sun, Earth and Moon lie on the same straight line with the Moon in the middle. Measure the phase of the Moon on each photo; express it as the ratio of l/r similar to the activity on Venus (Figure 2). It is related to, the angle between Earth and the Moon subtended at the position of the Sun, by (B.4) Convert the angle can be converted to the angular displacement of the Moon by correcting for the Earth s motion around the Sun: (B.5) where t is the time after New Moon and 365.25 days is the orbital period of Earth around the Sun. Combine the data to give a plot of r versus. By using the Solver tool in Excel, fit the data using (B.2) to determine the values of e and.? Question and Discussion Discuss the following with your classmates and teacher: 1. What are the values of e and you find? 2. Compare the value of e you find with the accepted value. Find the percentage error. Is the agreement reasonable? 3. Discuss the validity of the approximations we use in the model. C. Orbital Calculation using Newtonian Mechanics In this part of the project, you are introduced to Newton s second law of motion, the law of universal gravitation and simple numerical methods that can be applied to solve problems on orbital motion. Background knowledge According to Newton s law of universal gravitation, two spherical celestial bodies attract each other with a force F equal to (C.1) where G is the gravitational constant, M and m are the masses of the celestial bodies and r is the separation between their centers. The minus sign indicates that the force is attractive. For, the central mass M is essentially at rest. The acceleration a of the mass m is given by Newton s second law of motion:
178 Calculating the orbits of the Moon, Planets and Spacecrafts (C.2) Combining (C.1) and (C.2), we have (C.3) Note that the acceleration is independent of the mass m. The acceleration is a vector and can be represented by two components in a Cartesian coordinate system (Figure 6). The components of the acceleration can be expressed in terms of the coordinates of the small mass: (C.4) A simple numerical scheme is used to calculate the motion of a planet around the Sun with given initial conditions. Divide a given time interval into N small steps, each of duration. At time (i = 0, 1, 2 N), the acceleration of the plant is given by (C.5) The unit of distance and time are chosen as AU and Earth-year respectively, so that. Figure 6 The coordinate system for writing down the equation of motion In each small time interval, we approximate the motion by the equation of uniformly accelerated motion. Quantities at the i th step can be found from those of the (i 1) th step by, to order, v v x y 1 ( i) = vx ( i 1) + ( ( ) ( )) 2 ax i + ax i 1 1 ( i) = v ( i 1) + a ( i) + a ( i 1) y 2 t ( ) t y y (C.6) (C.7) Given the initial conditions and, the equations can be used to find the position and velocity of the planet at any time.? Question and Discussion You are required to solve the following problems by applying (C.5-C.7) in Excel. The resulting orbits should be plotted with scatter plots in Excel 1. Discuss the following questions with your classmates and teacher to enhance your understanding of the results.
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 179 Task 1: Calculate the orbit with the initial conditions, and AU year -1. A convenient choice is = 0.01. A smaller will give a more accurate result but will also increase the length and time of the computation. 1. 2. 3. What is the shape of the orbit and what is its size? Find the period of the orbit by checking the time when the planet returns to its original position. What is the orbital speed of the planet in SI units? Do you think the orbit you found approximates the orbit of a planet in the solar system? Which planet? Task 2: Now keep the initial position unchanged (i.e., ) and vary the y component of the initial velocity (keep ). Try values of greater or smaller than. A convenient way of doing this is to let, where is a constant close to unity. Experiment with different values of that give you closed orbits around the Sun 2. 1. 2. What is the shape of the orbits you obtain? How does the orbit change as you change the value of smaller than 1? to (i) greater 1 and (ii) Task 3: Keep the initial position unchanged and vary both the x and y components of the initial velocity and. Observe the resulting orbit. 1. 2. How are the orbits different from those you found in Task 2? Which equation of an ellipse can be used to describe the elliptical orbit? What parameters are involved? Task 4: Now let us apply the curve-fitting method to find the change in position and speed of a planet in its orbit. First choose an orbit with known values of semi-major axis a and eccentricity e. The orbit of Mars (a = 1.524 AU, e = 0.093) would be a good choice. Plot the orbit with a scatter plot in Excel. Choose a set of arbitrary initial conditions and plot the orbit calculated by the numerical method on the same graph. For each set of data you find, convert them to polar coordinates and calculate the value of the following function: (C.8) We take = 0 for simplicity. Find the sum over the whole orbit. The sum gives a measure of the deviation of the calculated orbit from the known orbit. Using the Solver tool in Excel, vary the initial conditions (keep ) and, such that is a minimum. The calculated orbit will then coincide with the known orbit on the graph. Plot graphs of versus. They will give the change in distance and angular displacement of the planet with time. 1. How does the distance r between Mars and the Sun vary with time? 2. Does Mars move with a constant angular speed around the Sun? Why or why not? Explain in terms of Kepler s laws of planetary motion. 1 To better illustrate the resulting orbit in a scatter plot, the horizontal and vertical scales of the plot should be made the same. Excel does not provide a convenient way of doing this, but it can be done manually by setting custom plot ranges and adjusting the width and height of the plot. 2 For an initial distance of 1 AU from the Sun, initial speeds in the range v 0 < 2 2 AU year -1 give close orbits.
180 Calculating the orbits of the Moon, Planets and Spacecrafts D. Predicting the Position of a Planet using Kepler s Equation In this part of the project, you are introduced to: Kepler s geometrical methods of predicting the position of a planet in an elliptical orbit; The derivation of Kepler s equation using Kepler s second law of planetary motion; Simple numerical methods for solving Kepler s equation and predicting planetary positions; and Modeling a space mission of transferring a spacecraft from Earth to Mars. Background knowledge Johannes Kepler (1571-1630) thought of an elliptical orbit as a circular orbit being compressed along the direction of the semi-minor axis. Figure 7 shows an auxiliary circle drawn concentric with an ellipse, with radius equal to the semi-major axis of the ellipse. For the same value of x, the height y of the ellipse differs from that of the circle by Thus, we can think of the ellipse as the auxiliary circle being compressed by a factor of along the y direction. It follows that the area of an ellipse is also smaller than that of the circle by same factor: Figure 7 An ellipse and its auxiliary circle (D.1) Consider a planet moving along the elliptical orbit. According to Kepler s second law of planetary motion, the radius vector of the planet sweeps out area at a constant rate, covering the area of the ellipse in an orbital period T. Hence, the area is swept at a rate of (D.2) The area swept out in a time interval t is given by (D.3)
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 181 where M is an angle called mean anomaly; it is defined as (D.4) M 0 is the mean anomaly at time t = 0. The value of M 0 specifies the angular position of the planet relative to the perihelion at a certain epoch. We are now ready to derive the Kepler s equation. In Figure 8, the Sun lies on the focus S, which is at a distance ae from the centre O of the ellipse. The straight line QPD is perpendicular to the x axis with the planet located at point P of the elliptical orbit. Point Q lies on an auxiliary circle that is tangential to the elliptical orbit at A. QO makes an angle E, which is called an eccentric anomaly, with the x axis. The area swept by the radius vector of the planet after passing the x-axis is given by Figure 8 Derivation of Kepler s equation using an auxiliary circle which is smaller than the corresponding area subtended by the auxiliary circle by a factor of. Equation (D.3) then allows us to express the area in terms of the mean anomaly M: (D.4) The area can be expressed in terms of the eccentric anomaly: Using (D.4), we have (D.5) This is called Kepler s equation. Given the eccentricity of the orbit, we can solve for the eccentric anomaly E for any time t (N.B. M = 2 t/t). However, this is a transcendental equation that requires a numerical solution. The simple iteration scheme works well: (D.6)
182 Calculating the orbits of the Moon, Planets and Spacecrafts Usually a few times of iteration will give an accurate value of E. The final task is to determine the radius vector r and true anomaly (i.e., angular displacement around the Sun) of the planet from the eccentric anomaly. In Figure 8, the length SD is given by Using the equation of ellipse (B.2), we solve for r and : (D.7) (D.8)? Question and Discussion You will apply (D.4-D.8) with Excel to predict the position of a planet in its orbit at an arbitrary time. In addition, You will apply the numerical methods in Section C to model the transfer orbit of a spacecraft to Mars. Task 5: The orbital period, semi-major axis and eccentricity of Mars orbit are 1.8808 years, 1.5237 AU and 0.09332 respectively. The mean anomaly of Mars at January 1, 2000, 12:00 UT 3 is 19.4125 o. Calculate the position of Mars in its orbit at July 1 2003, 12:00 UT. Suggested Answers Time elapsed since January 1, 2000, 12:00 UT = (366 + 2 x 365 + 31 + 28 + 31 + 30 + 31 + 30) days = 1277 days = 1277/365.25 years = 3.49623 years. Thus The corresponding eccentric anomaly is given by (D.6) with e = 0.0093 and a few iterations. We obtain E = 11.9660 rad. (D.7) and (D.8) give the radius vector and true anomaly as 1.406 AU and 0.6553 rad respectively. Task 6: Now we will integrate the skills learned to model a transfer orbit from Earth to Mars. The spacecraft is assumed to move under the gravity of the Sun only. The effects of other planets are neglected. To simplify the problem, we assume that the orbits of Mars and Earth are coplanar, and that Earth s orbit is circular with a radius of 1AU 4. Assume further that at the instant of launching the spacecraft, Earth lies at the perihelion of Mars orbit (i.e., ), and Mars has an initial mean anomaly of 30 o (i.e., it is slightly ahead of Earth). Figure 9 Initial velocity of the spacecraft 3 Astronomers usually refer this time as the Julian Epoch J2000.0. UT stands for universal time. There are 365.25 days in a Julian year. 4 The orbit of Mars is actually tilted at a small angle of 1.85 o to the ecliptic. The eccentricity of Earth s orbit is 0.0167, which is much smaller than that of Mars.
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 183 The task is to find a transfer orbit that allows the spacecraft to reach Mars in 0.68 year. The solution is specified by the initial speed and angle that the initial velocity makes with Earth s orbit (Figures 9). The following procedure is used to calculate the transfer orbit. 1. 2. 3.? Given the arrival time of t = 0.68 year for the spacecraft, calculate the position of Mars at the time of arrival. The position is then expressed in Cartesian coordinates. Use the orbital elements of Mars given in task 5. Use the numerical method introduced in Section C to calculate the position of the spacecraft at the time of arrival. Use arbitrary but reasonable values of and as an initial seed for the numerical calculation. Find the distance between the spacecraft and Mars at the arrival time. Using the Excel Solver tool, vary and to minimize the distance between the spacecraft and Mars. The initial speed should be slightly greater than Earth s orbital speed, so, and for an outward transfer orbit that goes beyond Earth s orbit, we should require > 0. The constraints can be readily incorporated into the minimization process 5. Question and Discussion Discuss the following with your classmates and teacher: 1. What are the values of v 0 and you find? 2. How do you know that your minimization process is successful? 3. How does the choice of affect the accuracy of your calculation? What are the pros and cons of decreasing? 4. Discuss with you teacher on how to use Kepler s third law of planetary motion to estimate a reasonable arrival time of the spacecraft. E. Orbital Motion in Three Dimensions In this part of the project, you are introduced to: The Keplerian orbital elements that describe the size, shape and orientation of an elliptical orbit relative to the ecliptic (Earth s orbit plane); The transformation equations that give the Cartesian coordinates of a planet in 3D; and Modeling a space mission of transferring a spacecraft from Earth to a hypothetical planet in 3D. 5 The constraints can be added to the Subject to the Constraints option in the Excel Solver dialog box. See Figure 4.
184 Calculating the orbits of the Moon, Planets and Spacecrafts Background knowledge We first introduce the Keplerian orbital elements that fully describe an elliptical orbit and the initial position of a planet (Figure 10). The plane of reference is taken as the Earth s orbital plane (ecliptic), on which the reference direction is taken as the Vernal Equinox. Three orbital elements specify the orientation of a planet s orbit relative to the ecliptic. Inclination i defines the angle of tilt of the orbit relative to the ecliptic. The position at which the planet crosses the ecliptic while travelling northwards is called the ascending node. The longitude of ascending node is the angle between the direction of the ascending node and that of the Vernal Equinox 7 measured on the Earth s orbital plane. The argument of perihelion is the angle between the perihelion and ascending node measured on the orbital plane of the planet. The size and shape of the orbit are defined by the semi-major axis a and eccentricity e. The mean anomaly at epoch M o defines the position of the planet on the orbit at a specific time. Figure 10 Orbital elements The position of a planet in 3D can also be expressed in terms of a Cartesian coordinate system centered on the Sun. The x axis of this coordinate system is taken as the direction of the Vernal Equinox; the z direction is taken perpendicular to the ecliptic. Given the orbital elements i, and, the planet s position can be transformed from the polar coordinates on its orbit to the Cartesian coordinates (x,y,z) in 3D space using the following equations: (E.1) 6 A natural extension of this project would be asking students to examine the relative position of Earth and Mars in a given time interval, say within a few years, suggest possible launching times for the spacecraft and calculate the corresponding transfer orbits. Due to time limitations, however, this was not included in the trial run.
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 185? Question and Discussion Task 7: Apply (D.4-D.8) and (E.1) to calculate the orbit of a hypothetical planet with the following orbital elements: a = 1.6 AU, e = 0.3, = 40 o, i = 20 o, = -80 o, M o = 25 o. The orbit is highly elliptical and tilted at a large angle to the ecliptic to allow for better visualization. Plot the orbit using a 3D graph-plotting software. Show Earth s orbit and the position of the Sun on the same graph. A 3D visualization software package called TOPCAT 8 can be downloaded from: http://www.star.bris.ac.uk/~mbt/topcat/ If you have time, vary the orbital elements i, and and visualize how the orbit is changes. Task 8: Now you are ready to integrate what you have learned to model a transfer orbit from Earth to a hypothetical planet in 3D. Assume that the planet takes the same orbital elements as in task 7. In 3D space, the spacecraft has one more dimension to move. Equations (C.6) and (C.7) can be easily generalized to 3D by adding the z components. The direction of the initial velocity is now specified by the angles and (Figure 20): (E.2) Figure 11 Initial velocity of the spacecraft. The y axis is tangential to Earth s orbit. Minimize the distance between the planet and the spacecraft at the arrival time by varying v 0, and subject to the constraints,, and.? Question and Discussion Discuss the following with your classmates and teacher: 1. Can you obtain the transfer orbit? What is the preset transfer time you choose? 2. Discuss with you teacher to see what other challenging cases could be investigated with the methods you learn. 7 The Vernal Equinox is the position on the celestial sphere where the celestial equator and the ecliptic meet, with the Sun travelling northwards along the ecliptic. In fact, the position of the Vernal Equinox changes slowly due to the precession of Earth s axis. This effect is neglected in the present calculation. 8 TOPCAT is an interactive graphical viewer and editor for tabular data designed by the Astrophysics Group of the Department of at the Bristol University. It is specially designed for visualizing astronomical data.
186 Calculating the orbits of the Moon, Planets and Spacecrafts III. Activity Guidelines Number of Sessions This project can be completed in 8 sessions, each lasting about 2 hours. However, the actual teaching schedule will depend on the mathematical background of the students and their ability to handle numerical calculations. Session 1 Your teacher will show you photos of the phases of Venus, and introduce Warm Up Activity I Learn IT tools for measuring lengths on the photos Learn the basic operation of Excel and the Solver tool that is needed for the calculations in this project Session 2 Learn a model of circular orbits for Earth and Venus, and work out a formula relating Venus orbital radius and period with the observed quantities Use Excel to fit the orbit Venus with given observation data; discuss the result and possible sources of error Session 3 Your teacher will show you a series of photos showing the complete phase cycle of the Moon, and introduce Warm Up Activity II Learn the equation for an elliptical orbit, and the basic parameters for describing the orbit Derive a model for fitting the observation data; do the fitting and discuss the result Session 4 Your teacher will introduce Newton s second law of motion, the law of universal gravitation and a numerical method for orbital calculation Use the numerical method in Excel to obtain simple orbits; experiment with different initial conditions and discuss the results Use the fitting method you have learned to find the change in position and orbital speed of a planet, e.g. Mars, during an orbital period; discuss the result in relation to Kepler s second law of planetary motion If time allows, find the period and semi-major of the orbits you obtain, and verify Kepler s third law of planetary motion Session 5 Learn Kepler s geometrical method for predicting the position of a planet, and derive Kepler s equation from Kepler s second law of planetary motion; you should fill out the intermediate steps leading to the result Learn the iteration scheme for solving Kepler s equation; try the method with Excel and discuss the result Session 6 Integrate the numerical methods with Kepler s equation to model the transfer of a spacecraft from Earth to Mars Discuss the result and the choice of initial conditions based on Kepler s laws
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources 187 Session 7 Learn the Keplerian orbital elements that completely describe an elliptical orbit and the initial position of the planet Learn the visualization software TOPCAT Calculate the orbit of a hypothetical planet and plot the orbit with TOPCAT; experiment with different orbital elements Session 8 Integrate what you have learned to model the transfer of a spacecraft from Earth to a hypothetical planet in 3D space Discuss the result with your teacher and summarize what you have learned in this project IV. References and Online Resources References Duffett-Smith, P. (1985). Astronomy with your personal computer. Cambridge: Cambridge University Press. Montenbruck, O., & Pfleger, T. (2000). Astronomy on the personal computer (4th ed.). Berlin: Springer. Pollard, H. (1966). Mathematical introduction to celestial mechanics. Englewood Cliffs, N.J.: Prentice Hall. Online Resources Alcyone ephemeris: an accurate and fast astronomical ephemeris calculator http://www.alcyone.de/ Elements playground: a multimedia website for introducing Keplerian orbital elements http://wlym.com/~animations/ceres/calculatingposition/elements.html Kepler s laws calculator http://people.bridgewater.edu/~rbowman/isaw/keplercalc.html Planet positions using elliptical orbits: introducing the algorithms for calculating the positions of planets http://www.stargazing.net/kepler/ellipse.html