03-A Satellite Orbiting Earth

The code below has almost everything required in order to show the motion of a Satellite orbiting the Earth. The satellite is given an initial position of 10 Earth radii and an initial velocity of 2000 m/s that is perpendicular to the line between the Satellite and the Earth.

• Complete the code by adding one line for for each of the comments labeled A thru F inside the while loop.
• Try different initial speeds of 2500 m/s, 3000 m/s, 3400 m/s, and 3600 m/s.
• The flags showVelocity and showForce can be changed to True or False depending upon whether or not you want to show velocity and force vectors.  The force vector is also shown with its components perpendicular and parallel to the velocity direction.

#
# Starter code for modeling the motion of a satellite around the Earth.
# 
# We will start the satellite at an initial position of 10 Earth radii and 
# with an initial velocity vector that is perpendicular to the line between
# the satellite and Earth.
#
# This isn't a typical orbital radius for satellites.  It is MUCH larger than 
# low Earth orbit and is nearly twice as large as the geosynchronous orbit
# associated with communications satellites.  Instead, consider this a demo
# of the mechanics involved with the gravitational force.

# Define the size of the scene
scene = canvas(width=600, height=600)

# The following boolean flags control whether a green velocity vector and 
# red force vectors (and its components) are shown.
showVelocity   = True  # Set to True or False
showForce      = True  # Set to True or False
showComponents = False # Set to True or False

# Define the Earth as a sphere and assign its attributes
Earth = sphere(pos=vector(0,0,0), texture=textures.earth)
Earth.radius = 6.4e6    # Set the radius of the Earth, in m
Earth.m      = 6.0e24   # Set the mass of the Earth, in kg

# Define and object for the Satellite
Satellite = sphere(make_trail=True, trail_type="points", trail_radius=0.5e6, interval=10, retain=100)
Satellite.radius = 1.0e6                        # Not realistic of course, helps us to see it                        
Satellite.color = color.yellow 
Satellite.m   = 15e3                            # Mass of Satellite, in kg
Satellite.pos = vector(-10*Earth.radius, 0,0)   # Initial Position of Satellite
Satellite.v   = vector(0, 2.0e3, 0)             # Initial Velocity of Satellite (also try speeds of 2.5e3, 3.0e3, 3.4e3 and 3.6e3)
Satellite.p   = Satellite.m * Satellite.v       # Initial Momentum of Satellite

# Create some arrow objects to represent Velocity and Force vectors
if showForce:
    F_scale = 7500  # Adjustment to allow us to see the vector
    ForceVector = arrow(pos=Satellite.pos, axis=vector(0,0,0)*F_scale, color=color.red, shaftwidth=8e5)

if showComponents:
    ForcePerpVector = arrow(pos=Satellite.pos, axis=vector(0,0,0)*F_scale, color=color.red, shaftwidth=5e5)
    ForceParaVector = arrow(pos=Satellite.pos, axis=vector(0,0,0)*F_scale, color=color.red, shaftwidth=5e5)

if showVelocity:
    v_scale = 7500  # Adjustment to allow us to see the vector
    VelocityVector = arrow(pos=Satellite.pos, axis=Satellite.v*v_scale, color=color.green, shaftwidth=7e5, opacity=0.5)

# Define necessary constants
t  = 0          # Initialize the time counter to 0 seconds
dt = 60         # Time step (60 s = 1 minute)
G  = 6.7e-11    # Gravitational Constant, N m**2 / kg**2

while True:
    rate(100)   # Since dt = 1 minute, we see 100 minutes of animation each real second.
    
    # A - Compute the position (r) of the Satellite relative to the Earth
    
    
    # B - Compute the unit vector direction pointing from Earth to Satellite (rhat)
    
    
    # C - Compute the magnitude of the force (Fmag) on the Satellite by the Earth
    
    
    # D - Compute the vector force (F) on the Satellite by the Earth
    F = vector(0,0,0)
    
    # E - Use Momentum Principle to Update the Satellite's Momentum
    
    
    # F - Update the Position of the Satellite
    
    
    # Update the Satellite's Force Vector
    if showForce:
        ForceVector.pos = Satellite.pos
        ForceVector.axis = F*F_scale
    
    # Calculate the parallel and perpendicular components of the Force
    if showComponents:
        p_hat = norm(Satellite.p)
        ForcePara = dot(F,p_hat)*p_hat
        ForcePerp = F - ForcePara
    
        # Show the Force Vector Components
        ForceParaVector.pos = Satellite.pos
        ForceParaVector.axis = ForcePara*F_scale
    
        ForcePerpVector.pos = Satellite.pos
        ForcePerpVector.axis = ForcePerp*F_scale

    # Update the Satellite's Velocity Vector
    if showVelocity:
        VelocityVector.pos = Satellite.pos
        VelocityVector.axis = (Satellite.p/Satellite.m)*v_scale

Questions

Start out using initial speed of 2.0e3 m/s

1. Can you use the dots (trail markers) to tell where the satellite is moving the fastest? the slowest?
2. What phrase describes the direction of the force on the satellite at all times?
3. What phrase describes the direction of the velocity (and momentum) of the satellite at all times?
4. What phrase describes the direction of the CHANGE in MOMENTUM of the satellite at all times?
5. At what positions is the force on the satellite perpendicular to the momentum of the satellite?
6. Set the ShowComponents flag to True.  Describe the relationship between the force on the satellite and the direction of the velocity when the satellite is speeding up?
7. Describe the relationship between the force on the satellite and the direction of the velocity when the satellite is slowing down?
8. Set the initial velocity of the satellite to 2.5e3 m/s.  Describe the orbit?  At what positions is the force on the satellite perpendicular to the momentum of the satellite?
9. Set the initial velocity of the satellite to 3.4e3 m/s and increase the rate to 1000 to speed the animation up.  Observe the motion.  Now try an initial velocity of 3.6e3 m/s and observe the motion.  Do you notice a fundamental difference in the motion for these two initial conditions?