## Satellite Orbit

### Activity Information

#### Learning Goals

- Use mathematical representations of Newton's Law of Gravitation and Coulomb's Law to describe and predict the gravitational and electrostatic forces between objects HS-PS2-4
- Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy and associated with the relative positions HS-PS3-2

#### Prior Knowledge Required

- Kinematics
- Conservation of Energy
- Newton's 2nd Law
- Net Force

- Momentum
- Newton's Law of Gravitation
- $F_{g}=\dfrac{Gm_{1}m_{2}}{r^2}$

#### Code Manipulation

- Interpret code and relate to physics concepts
- Copy and paste code
- Replicate motion maps

- Add variables to premade graph

### Activity

#### Handout

##### Satellite Orbit

**Part 1**

Copy and paste the following GlowScript code into your own GlowScript account. Read through the code and predict what might happen during the simulation

- Run the program, observe and describe what happened and how it differed from your predictions
- How long does the simulation run for?
- What is the initial momentum of the satellite? Which direction?
- Which line of code indicates Newton's Law of Gravitation?
- How far way is the satellite from the Earth initially?
- What is the mass of the satellite? What is the mass of the Earth?
- What does the following line of code mean in physics terms? What does it do?
- “pSatellite = pSatellite + Fnet*dt” (line 48)

- How is velocity represented in the code?
- Modify the code in order to make the satellite orbit the Earth. What changes did you make?
- Qualitatively and quantitatively described the orbit of your satellite in terms of the following, and provide a sketch:
- Shape:
- Net Force:
- Velocity:
- Kinetic Energy:
- Potential Energy:
- Total Energy:

- Add a net force vector to your satellite
- Add a bar graph for the total mechanical energy
- Challenge: modify the code to make the orbit of the satellite circular. What evidence do you have that it is circular (besides visual inspection)?

**Part 2**

Conceptual Physics Claim, Evidence, and Reasoning (C.E.R.):

- Do you support or refute the following claims? Provide evidence and reasoning:
- Energy is not conserved in the Earth-satellite system.
- The magnitude of the satellite's velocity is constant.
- Changing the mass of the Earth and/or the satellite changes the force of gravity involved in this system.
- If the distance between the Earth and the satellite is doubled, the force between the Earth and the satellite is halved.
- Changing the initial momentum of the satellite changes the shape of the satellite's orbit around the Earth.

- For the following concepts, create a claim, cite your evidence, and provide your reasoning for each prompt:
- The kinetic energy of the satellite.
- The potential energy of the satellite.
- The total mechanical energy of the system.
- Relationship between the net force on the Earth and net force on satellite.
- Relationship between the mass of the satellite and the shape of the orbit.
- Relationship between radius and velocity of the satellite.

- Respond to the following statements/questions using your prior knowledge and physics concepts
- Discuss how changing the parameters and initial conditions (initial momentum, mass, G) affect orbit shape, net force, kinetic energy, potential energy, total mechanical energy, and momentum.
- Using this simulation and your own experience, how can you explain how the Earth-Moon system could have originated?
- Why does the satellite remain in orbit without crashing to Earth?

#### Code

GlowScript 2.7 VPython get_library('https://rawgit.com/perlatmsu/physutil/master/js/physutil.js') rom __future__ import division from visual import * from visual.graph import * #Window setup scene.range = 7e7 scene.width = 1024 scene.height = 760 #Objects Earth = sphere(pos=vector(0,0,0), radius=6.4e6, texture=textures.earth) Satellite = sphere(pos=vector(6.6*Earth.radius, 0,0), radius=1e6, color=color.orange, make_trail=True) #Parameters and Initial Conditions mSatellite = 15000 pSatellite = vector(30000000,0,0) G = 6.67e-11 mEarth = 5.98e24 r = (Earth.pos - Satellite.pos) g1 = gcurve(color=color.cyan,label="kinetic energy") g2 = gcurve(color=color.red,label="gravitational energy") #Time and time step t = 0 tf = 60*60*24*10 dt = 1 graphv = gdisplay(xmin=-0.25, xmax=1.25, ymin=-12e10, ymax=12e10, ytitle="Energy") g3 = gvbars(gdisplay = graphv, color = color.red, delta = 0.2, label = "Kinetic Energy") g4 = gvbars(gdisplay = graphv, color = color.blue, delta = 0.2, label = "Potential Energy") #MotionMap/Graph pSatelliteMotionMap = MotionMap(Satellite, tf, 200, markerScale=0.2, markerColor=color.blue, labelMarkerOrder=False) #Calculation Loop ev = scene.waitfor('click') while t < tf: rate(6000) g3.delete() g4.delete() Fnet = vector(0,0,0) r = (Earth.pos - Satellite.pos) Fnet = vector(G*mEarth*mSatellite/(mag(r)**2)*(r/mag(r))) pSatellite = pSatellite + Fnet*dt Satellite.pos = Satellite.pos + (pSatellite/mSatellite)*dt if mag(Satellite.pos) < Earth.radius: text(text='You Crashed!!', pos=vec(0, 4e7, 0), color = color.red, depth=1, height= 7e6) break pSatelliteMotionMap.update(t, pSatellite) t = t + dt KE = 1/2*mSatellite*mag(pSatellite/mSatellite)**2 PE = -G*mSatellite*mEarth/mag(r) g1.plot(t, KE) g2.plot(t, PE) g3.plot(0, KE) g4.plot(0.5, PE) #Earth Rotation (IGNORE) theta = 7.29e-5*dt Earth.rotate(angle=theta, axis=vector(0,0,1), origin=Earth.pos)

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### Answer Key

#### Handout

**Part 1**

- The satellite begins with a velocity pointing away from the Earth. Eventually, the satellite comes to a complete stop, then accelerates towards the Earth until it crashes.
- Approximately 6 seconds
- 3,0000,000 kg·m/s in the positive x-direction (line 18)
- Line 50: "Fnet = vector(G*mEarth*mSatellite/(mag(r)**2)*(r/mag(r)))"
- Line 14 indicates that the satellite is a distance of 6.6 times the Earth's radius away. The Earth's radius (line 13) is $6.4*10^6$ meters, therefore the satellite is initially $6.6*6.4*10^6=4.2*10^7$ meters away
- The mass of the satellite is 15000 kg (line 17) and the mass of the Earth is $5.98*10^24$ kg (line 20)
- The line of code that says “pSatellite = pSatellite + Fnet*dt” is located within the while loop and it updates the momentum of the satellite. In physics terms, this reads as “The updated momentum of the satellite is equal to the original momentum plus the impulse (force multiplied by time) acting on the satellite due to the force of gravity”
- In line 49, the velocity is represented as “(pSatellite/mSatellite).” This comes from the fact that momentum is mass times velocity ($p=mv$), so dividing the momentum of the satellite by its mass, the velocity can be calculated
- See updated code below
- Satellite orbit description:
- Shape: The orbit is elliptical about the Earth
- Net Force: The strength of the gravitational force is strongest when the satellite is closest to Earth (perigee) and weakest when when it is furthest (apogee)
- Velocity: Similarly to net force, the velocity of the satellite is highest at its perigee and lowest at its apogee
- Kinetic Energy: Kinetic Energy is defined as $KE=\dfrac{1}{2}mv^2$ (line 56). Since kinetic energy is proportional to velocity squared, it makes sense that the kinetic energy is highest when the velocity is at a maximum (when the satellite is closest to Earth) and lowest when the velocity is at a minimum (when the satellite is furthest from Earth)
- Potential Energy: Gravitational potential energy is defined as $PE=-\dfrac{GMm}{r}$ (line 58). Since the gravitational constant and masses of the Earth and satellite remain fixed, the potential energy only varies as the distance between the Earth and satellite change. Given an inversely proportional relationship between potential energy and distance, the magnitude of the potential energy is greatest when the satellite is closest to Earth and smallest when the satellite is furthest from Earth
- Total Energy: The total mechanical energy in the Earth-satellite system is simply the sum of the kinetic and gravitational potential energies. This value is constant throughout the entire orbit, and always negative, which means that the satellite is bounded to Earth

- See lines 38 and 56 below
- See lines 34, 37, and 69 below
- If the orbit of the satellite is circular, then it can be modeled using centripetal motion. Specifically, if the centripetal “force” is equal to the gravitational force, then we can find the initial velocity required to keep the satellite in circular orbit:
- $\dfrac{mv^2}{r}=\dfrac{GMm}{r^2}$
- Solving for velocity: $v=\sqrt{\dfrac{GM}{r}}$
- Plugging in known values: $v=\sqrt{\dfrac{6.67*10^{-11}\text{m³·kg⁻¹·s⁻²}*5.98*10^{24}\text{kg}}{6.6*6.4*10^6\text{m}}}=3073\text{m/s}$
- The velocity of the satellite must lie tangent to the path of the orbit in order for the above to remain true. Therefore, in order to obtain a circular orbit, the momentum of the satellite must be entirely in the y-direction (line 18). In the code, this would look like “pSatellite = vector(0,3073*mSatellite,0)”
- Circular motion means that the satellite will remain the same distance away from the Earth throughout the entire orbit. Additionally, the magnitude of the satellite's velocity will also remain constant. Therefore, if the orbit of the satellite is circular, then its potential and kinetic energy will remain constant throughout, and checking the graphs will prove if you have achieved circular orbit

**Part 2**

- Claim Support/Refutation:
- Claim: Energy is not conserved in the Earth-satellite system.
- Refute; no work is being done on the system. Instead there are internal energy transfers. Although the energy may transform from kinetic to potential, it will always sum to a constant amount, and the total mechanical energy is conserved

- Claim: The magnitude of the satellite's velocity is constant.
- Refute; this is only true for circular orbits (Part 1 Problem 13). As answered in Part 1 Problem 10c, the velocity increases towards the perigee and decreases towards the apogee

- Claim: Changing the mass of the Earth and/or the satellite changes the force of gravity involved in this system.
- Support; provided Newton's equation of gravitation ($F_{g}=\dfrac{Gm_{1}m_{2}}{r^2}$), it becomes clear that changing either of the masses will affect the magnitude of the force (although the direction will still point from one mass to the other)

- Claim: If the distance between the Earth and the satellite is doubled, the force between the Earth and the satellite is halved
- Refute; because the force is inversely proportional to the distance squared, a two-fold increase in distance results in the force being quartered
- $\dfrac{Gm_{1}m_{2}}{(2r)^2}=\dfrac{Gm_{1}m_{2}}{4r^2}=\dfrac{1}{4}F_{g}$

- Claim: Changing the initial momentum of the satellite changes the shape of the satellite's orbit around the Earth
- Support; mass of the satellite does not affect the orbit, while velocity does affect the shape. Therefore, altering the initial momentum, which is mass times velocity, will alter the size and shape of the orbit
- At velocities less than circular speed (Part 1 Problem 13b), the orbit is elliptical
- At velocities exactly at circular speed, the orbit is circular
- At velocities greater than circular speed, but less than escape speed (defined as $v=\sqrt{\dfrac{2GM}{r}}$) the orbit is elliptical
- At velocities exactly at escape speed the orbit is parabolic
- At velocities greater than escape speed, the orbit is hyperbolic
- Larger satellite mass results in smaller orbit size

- Claims + Evidence
- Claim: The kinetic energy of the satellite increases as its distance to the Earth decreases
- Evidence: Based on Newton's Law of Gravitation, as the satellite gets closer to the Earth, the gravitational force acting on the satellite increases. Based on Newton's 2nd Law, force is directly proportional to acceleration, so the satellite will begin to accelerate, or in other words, increase its velocity. A larger velocity corresponds to a greater amount of kinetic energy, as proven by the equation for kinetic energy (Part 1 Problem 10d)

- Claim: The potential energy of the satellite increases as its distance from the Earth increases
- Evidence: When a satellite orbits a planet, it has negative potential energy and is bounded to the planet. Moving the satellite closer to Earth makes its potential energy become more negative. As the satellite drifts farther away from the Earth, its potential energy will approach zero (see Part 1 Problem 10d)

- Claim: The total mechanical energy of the satellite remains constant throughout the orbit of the satellite
- Evidence: If you were to calculate the kinetic and potential energy of the satellite at any point during its orbit, you will find that the sum will always be the same negative value (negative because it is bounded, as explained above). The total energy does not change because no work is being done on the system: only a transfer of internal energies. You can also see this in the graphs displayed below your code; the line or bar representing total mechanical energy will remain fixed

- Claim: The net force acting on the Earth and the satellite are equal and opposite
- Evidence: According to Newton's 3rd Law of Motion, every action has an equal and opposite reaction. That means that the gravitational force from the Earth acting on the satellite is equal in magnitude and opposite in direction to the force from the satellite acting on the Earth

- Claim: The mass of the satellite does not affect the orbital shape
- Evidence: We discovered earlier that the shape of the orbit is dependent on the speed of the satellite, not the mass (Part 2 Problem 1eI). You can test this by changing the values of “mSatellite” in your code (line 17), and you will find the shape of the orbit stays the same regardless

- Claim: The larger the radius between the satellite and the Earth, the smaller the velocity
- Evidence: This has already be thoroughly discussed in Part a of this problem

- Connecting code and real-life physics concepts
- Initial parameters/conditions and their effects
- The relationships between momentum, mass, G, orbital shape, net force, kinetic energy, potential energy, and total mechanical energy have been thoroughly explored and discussed in parts 1 and 2

- Origin of Earth-Moon system
- There are three main theories that explain how the Earth-Moon system originated:
- Early in the life of the Solar System as the planets were just beginning to form, the Earth and Moon could have formed together in a process known as accretion
- At some point in the history of the Earth, the moon could have originated elsewhere and it was captured by the gravity of the Earth. In order for this to have occured, the moon would have had to slow down to less than escape speed (Part 2 Problem 1eIC)
- By far the most popular theory, sometime long ago, a Mars-sized object (commonly called Theia) collided with the Earth, ejecting a large amount of material from Earth. This material then accreted to form what is now the Moon. This accounts for the axial tilt of the Earth.

- Satellite-Earth orbit
- The satellite is always falling towards the Earth; however, so long as it is travelling fast enough, the satellite essentially moves out of the way of the Earth as fast as it is falling. Basically, the effects of the satellite's velocity and the gravitational force from the Earth are balanced.

#### Code

GlowScript 2.7 VPython get_library('https://rawgit.com/perlatmsu/physutil/master/js/physutil.js') rom __future__ import division from visual import * from visual.graph import * #Window setup scene.range = 7e7 scene.width = 1024 scene.height = 760 #Objects Earth = sphere(pos=vector(0,0,0), radius=6.4e6, texture=textures.earth) Satellite = sphere(pos=vector(6.6*Earth.radius, 0,0), radius=1e6, color=color.orange, make_trail=True) #Parameters and Initial Conditions mSatellite = 1000 pSatellite = vector(-1500*mSatellite,2598*mSatellite,0) G = 6.67e-11 mEarth = 5.98e24 r = (Earth.pos - Satellite.pos) g1 = gcurve(color=color.cyan,label="kinetic energy") g2 = gcurve(color=color.red,label="gravitational energy") g3 = gcurve(color=color.green,label="total mechanical energy") #Time and time step t = 0 tf = 60*60*24*10 dt = 1 graphv = gdisplay(xmin=-0.25, xmax=1.25, ymin=-12e10, ymax=12e10, ytitle="Energy") g4 = gvbars(gdisplay = graphv, color = color.red, delta = 0.2, label = "Kinetic Energy") g5 = gvbars(gdisplay = graphv, color = color.blue, delta = 0.2, label = "Potential Energy") g6 = gvbars(gdisplay = graphv, color = color.green, delta = 0.2, label = "Total Energy") #MotionMap/Graph FSatelliteMotionMap = MotionMap(Satellite, tf, 200, markerScale=4000, labelMarkerOrder=False) pSatelliteMotionMap = MotionMap(Satellite, tf, 200, markerScale=0.2, markerColor=color.blue, labelMarkerOrder=False) #Calculation Loop ev = scene.waitfor('click') while t < tf: rate(6000) g4.delete() g5.delete() g6.delete() Fnet = vector(0,0,0) r = (Earth.pos - Satellite.pos) Fnet = vector(G*mEarth*mSatellite/(mag(r)**2)*(r/mag(r))) pSatellite = pSatellite + Fnet*dt Satellite.pos = Satellite.pos + (pSatellite/mSatellite)*dt if mag(Satellite.pos) < Earth.radius: text(text='You Crashed!!', pos=vec(0, 4e7, 0), color = color.red, depth=1, height= 7e6) break FSatelliteMotionMap.update(t, Fnet) pSatelliteMotionMap.update(t, pSatellite) t = t + dt KE = 1/2*mSatellite*mag(pSatellite/mSatellite)**2 PE = G*mSatellite*mEarth/mag(r) g1.plot(t, KE) g2.plot(t, PE) g3.plot(t, KE+PE) g4.plot(0, KE) g5.plot(0.5, PE) g6.plot(1.0, KE+PE) #Earth Rotation (IGNORE) theta = 7.29e-5*dt Earth.rotate(angle=theta, axis=vector(0,0,1), origin=Earth.pos)

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