Calculus Without Tears
Orbits and the Juno Space Probe
Motivating the Secondary School and Undergraduate Math Curriculum
The only way to motivate the mathematics curriculum is to model and analyze interesting physical phenomena. Two typeof physical systems are studied in the CWT series, mechanical and electrical. In this page we extend the examples in the CWT series to two dimensional orbits (not possible in the books because trigonometry is required).
The Equation that Characterizes the Trajectory of a Falling Object
First, the math - the force of gravity F on an object of mass m near earth is given by
Newton's 2nd Law of Motion tells us how the force of gravity affects a falling object,
Combining the equations above gives the equation for the acceleration of a falling object
Orbits are two dimensional, and in two dimensions it's necessary to resolve forces and accelerations into their x and y components. If the magnitude of the force of gravity on an object at coordinates (x,y) is g, then the x component of gravity is g*cos(a) = g*x/R and the y component is g*sin(a) = g*y/R where (R, a) are the polar coordinates of (x,y).
That's it for the math.
Computational Calculus, i.e. Using MATLAB/FREEMAT/OCTAVE to Calculate Orbits
We will model a falling object above the earth. The coordinates of the center of the earth are (0,0), and the coordinates of the object are (x,y). We will model a trajectory of duration T seconds and divide it into N subintervals, numbered 1 to N, of lenght dt = T/N. As the trajectory is computed we'll keep lists (i.e. MATLAB vectors) of the time, the x and y positions, and the x and y velocities at the start of each subinterval, in vectors named t, x, y, vx, and vy respectively.
The basis of the model is the code that advances time, position, and velocity from the start of one subinterval to the start of the next next. It is given below.
The Space Station Orbit
The BASIC LOOP code determines the trajectory of the orbit once the initial position and velocity are given. So, to simulate a satellite in low earth orbit we only have to provide the initial position and velocity and start the program.
We'll start the satellite at 300 km above the north pole, with an initial velocity of 77502 m/s determined purely by trial and error to get an approximately circular orbit. The code for the simulation is as follows:
The graph shows the calculated orbit and the outline of the earth for reference.
The code for drawing the earth is:
A Geosynchronous Orbit
The figure shows a geostationary orbit at 36000km. A geostationary orbit is a circular orbit above the equator, with a period of one day. Thus a satellite in a geostationary orbit appears to a ground observer to be in a fixed position in the sky. The code for the simulation is as follows:
The code for the simulation is as follows:
An Asteroid Fly By
The figure shows an asteroid fly-by with a parabolic orbit. The code for the simulation is as follows:
The Earth's Orbit Around the Sun
The BASIC LOOP code can be modified to calculate orbits around the sun by replacing the mass of the earth (the constant mEarth in the code) with the mass of the sun (mSun). The mass of the sun is 1.98892 ◊ 1030kg, the earthís orbit around the sun is an ellipse with a semi-major axis of 150,000,000km, the sunís radius is 695500km. The period of the earthís orbit is one year. Using these numbers produces an approximation to the earthís orbit around the sun as shown in the figure. The code for the simulation is as follows:
The code for the simulation is as follows:
Atmospheric Drag and Terminal Velocity
An object falling close to the earth is also acted on by the force of atmospheric drag. One model of the drag force is given by the equation
Amospheric density is 1.3 kg/m3 at sea level to falls to 0 at altitude 35 km and above. We implement a simple linear model of atmospheric density shown in the graph.
Atmospheric density is computed by the following code:
The drag force is computed and resolved into its x and y components with the following code:
The code can be inserted into the BASIC LOOP, and the drag acceleration added to the gravitational acceleration, to
The Juno Space Probe
The US launched a probe to Jupiter on Aug. 5, 2011. The trip to Jupiter will take five years, and utilizes a
slingshot, or gravity assist, trajectory. The boost rocket put a Centaur upper stage and the Jupiter probe into a low
orbit around the earth. The Centaur then launched the probe into a deep space orbit around the sun that will return to the earth in two years! The gravitational field of the earth will then accelerate the probe and send it on its way to Jupiter. The flight of the probe is un-powered except for two deep space maneuvers a year into the flight. A diagram of the actual planned trajectory is shown in the figure.
Once the probe detaches from the Centaur booster it is acted on only by the gravitational fields of the earth and the sun (except for the two deep space maneuvers which we will not model). In an example above we computed the earth's orbit around the sun. We will also need to compute the probe's trajectory. The forces acting on the probe are the gravitational force of the sun on the probe and the gravitational force of the earth on the probe.
The variables x, y, vx, and vy will store the position and velocity of the earth relative to the sun. We'll add variables xp, yp, vxp, and vyp, to store the position and velocity of the probe relative to the sun.
The gravitational force of the sun on the probe and the resulting acceleration is calculated by:
Once the probe is launched from the Centaur, and minus the mid-course maneuvers of the actual probe, the trajectory of
the probe is completely determined by the gravitational forces of the sun and earth acting on the probe, and the code
above accounts for those forces, so with it added to the BASIC LOOP we are good to go. The only thing required is to tweak the initial conditions so that when the probe returns to its approximate launch point, two years into the mission, it comes in behind the earth and gets close enough to the earth so that the earth's gravitational force accelerates it
into the Jupiter bound orbit. We did that, and the results are shown:
The earth and probe orbits can also be displayed in 3-d using the plot3 command, as shown in the figure. The code required is