Orbital motion

In this lab you will study the orbital motion of the planets by their orbits as circular orbits. You will determine and confirm the necessary conditions for a circular orbit. You will compute the orbital speeds and periods of the planets and compare to simulated and measured values.

Learning outcomes:

Understand the motion of the planets.

Understand circular orbits and how they arise from Newton’s law of gravity.

Simulation

Instead of using a physical apparatus, this lab uses the “Gravity and Orbits” simulation provided by PhET. The simulation can be accessed at:

https://phet.colorado.edu/en/simulation/gravity-and-orbits

The simulation can be downloaded and run later without an internet connection.

Part 1: Orbital motion simulation

First select “Model” and try playing around with the simulation. In the upper right, you can change which objects you are simulating. You can turn gravity on and off. You can choose what information to display. You can adjust the mass of the objects. Try changing the mass of the Earth and the mass of the Sun. You can also move the Sun and the Earth around by clicking on them and dragging them around. The orange button in the bottom right corner will reset the simulation.

Question 1: Does Earth’s orbit change if you change the mass of the Earth? Does Earth’s orbit change if you change the mass of the Sun?

Notice that the orbits are not always circular, but more generally take the form of an ellipse. We will stick to modeling circular orbits.

Part 2: Newton’s law of gravity

Newton’s law of gravitation: Every mass exerts a gravitational force on every other mass. The force is always attractive, and the strength is given by

Newton’s law of gravity played a pivotal role in explaining the motion of the planets around the Sun.

Question 2: Compute the gravitational force on the Earth from the Moon and the gravitational force on the Earth from the Sun. Which exerts a stronger force on the Earth, the Moon or the Sun? Use the following standard values:

Masses M_Sun=2.0×10^30 kg M_Moon=7.3×10^22 kg M_Earth=6.0×10^24 kg

Distances d_Sun=1.5×10^11 m d_Moon=3.8×10^8 m

The distances are from the center of the Sun to the center of the Earth and from the center of the Moon to the center of the Earth.

Uniform circular motion: An object moving in a circle with a constant speed is said to be moving in uniform circular motion. Even though the speed of the object is constant, the velocity of the object is changing because the direction of motion is changing. There is a relationship between the (tangential) velocity and the (radial) acceleration of the object.

a=v^2/r

This acceleration is always caused by a force or combination of forces. Any force which causes circular motion is often called a centripetal force. For example, the electric force is responsible for causing electrons to orbit protons.

Question 3: What force plays the role of the centripetal force for planets orbiting the Sun?

Circular orbits: The simplest kind of orbital motion is when the orbiting object follows uniform circular motion. We will assume that a little mass m is orbiting a big mass M in a circular orbit with radius r. Applying Newton’s law to the small mass, we have:

F ⃗_total=m a ⃗

We use the fact that the total force is given by Newton’s law of gravity and that the acceleration must obey uniform circular motion. This allows us to re-write Newton’s law as

G (M m)/r^2 =m v^2/r

Note that the mass of the orbiting object drops out. We solve this equation to determine the speed of the orbiting object in terms of the large object’s mass M and the orbital radius r:

v=√((G M)/r) (1)

We will test this relationship in the simulation. Restart the simulation and select “to scale”. This will allow you to measure distances in the simulation using the “measuring tape” tool, located on the right.

First, we will study the Earth orbiting the Sun and compute the orbital speed theoretically using our above equation.

Computation 1: The mass of the Sun is given in table 1 below. Measure the orbital radius using the tape measure tool and record your result as “r (thousand miles)” in table 1. To do so, place one end of the measurement tool at the center of the Sun and the other at the center of the Earth. Note that the units the simulation gives are in thousands of miles. Convert your value to meters using the conversion given below and record your result as r (m). Finally use the mass of the Sun and your value for the orbital radius to compute the theoretical velocity using equation 1. Record your answer in table 1 as the theoretical velocity, v_th. Repeat the process for the Moon orbiting the Earth and the satellite orbiting the Earth and fill out table 1 below. The Moon orbiting the Earth is the third option in the upper right and the satellite orbiting the Earth is the fourth option. Turn in your computations with the lab report.

Conversion: 1000 miles=1.609×10^6 m

Table 1: theoretical orbital velocity

M m M (kg) r (thousand miles) r (m) v_th (m/s)

Sun Earth 2.0×10^30

Earth Moon 6.0×10^24

Earth Satellite 6.0×10^24

Next, we will directly measure the orbital velocity using the simulation. Select the Earth-Sun system and display the path of the orbit using the “Path” tool on the right hand side. (You can also display the grid to help.) Start the simulation and let it run for only five to ten simulated days. The path should be a very short line. Use the measuring tool to measure the distance along the path that the Earth traveled during that time. Record the distance and time in table 2 below. Convert your distance to meters and time to seconds. Finally use your distance and time to compute the orbital velocity. Record your result as the simulated velocity, v_sim. Reset the simulation and repeat for the Earth-Moon and Earth-Satellite systems.

Table 2: simulated orbital velocity

Distance (thousand miles) Time Distance (m) Time (s) v_sim (m/s)

Question 4: Comment on how well your simulated values in table 2 and the theoretical values in table 1 agree for the orbital velocity.

Finding circular orbits: Orbits are not always circular but are more generally elliptical. Only for special values of the orbital speed and orbital radius will the orbits be circular. We will attempt to generate circular orbits in the simulation after changing the mass of the star.

The simulation starts with the same speed every time. Reset the simulation and select the Earth-Sun system. Record your value for the orbital speed for the Earth from table 1 into table 3 below. We will assume that the Earth always starts with this value for its orbital velocity.

Computation 2: The Star Mass slider can be set to 0.5, 1.5 and 2.0. The corresponding masses are given in table 3 as M. We can solve equation 1 to give the radius in terms of the speed and mass.

r=(G M)/v^2

Compute the orbital radius for each of the masses listed in table 3. Record your results in table 3 below as r (m) and then convert your answer to “thousand miles”.

Now we will check if your values for the radius correspond to circular orbits. Set the Star Mass slider to 0.5 and move the Earth so that its distance from the Sun is the value you computed below. Run the simulation and check if the Earth’s orbit is roughly circular. You can enable the “Path” option to see a drawing of the orbit. Reset the experiment and repeat for the other two mass values.

Table 3: circular orbits

Orbital speed (m/s) Star Mass M (kg) r (m) r (thousand miles)

0.5 1.0×10^30

1.5 3.0×10^30

2.0 4.0×10^30

Question 5: Comment on how circular the orbits were for the various Star Mass values.

Part 3: Orbital periods of the planets

Period: The period of an orbit is defined as the time it takes for the object to make one full orbit.

Notation: T

SI units: seconds

Since an object following a circular orbit travels at a constant speed, we can relate the period of the orbit to the circumference

x=v t → 2πr=v T → T=2πr/v

Using our expression in equation 1 for the orbital velocity to eliminate v in terms of r, we get an expression which gives us the period of a circular orbit in terms of the orbital radius

T=√((4π^2 r^3)/(G M)) (2)

We will test this expression using the simulation for the Earth-Moon and Earth-satellite systems. Copy your values for the distances from table 1 into table 4 below. Compute the theoretical value for the period using equation 2. Record your result as T_th in table 4. Next run the simulation and record the time it takes for the Moon to orbit once as T_sim. Repeat for the satellite orbiting the Earth. You can use the path to help you trace the orbit.

Table 4: period of Moon and satellite

M (kg) r (m) T_th (s) T_sim (s)

Moon M_Earth=6.0×10^24 kg

satellite M_Earth=6.0×10^24 kg

Next, we turn to determining the orbital period of the planets. In table 5, record the observed distance from the Sun, the orbital period and the orbit eccentricity for each planet using data from NASA, which is available at:

https://solarsystem.nasa.gov/planets/overview/

At the bottom of the page, you can select a planet. You will then be presented with an overview of the planet. Click “By The Numbers” to get a summary of the relevant information. Make sure the units are set to metric.

The orbital eccentricity gives a measurement of how circular the orbit is. For circular orbits, the eccentricity is zero. Note that the Earth has a very small orbital eccentricity of 0.02, indicating that its orbit is almost a perfect circle.

Computation 3: Use the observed radius and equation 2 to determine the theoretical value for the orbital periods of the planets. Record your answers as T_th in table 5. Turn in your computations with your lab report. To make your life easier, compute the constants appearing in equation 2 and then reuse your result for all the computations. That is first compute:

√((4π^2)/(G M))=

and then write equation 2 as

T=(√((4π^2)/(G M))) r^(3/2)

Table 5: orbital period of the planets

Planet r_obs (m) T_obs (s) Orbit eccentricity T_th (s)

Mercury

Venus

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

Question 6: Which planet has the most eccentric orbit (closest to one)? Which planet has the least eccentric orbit (closet to zero)?

Question 7: Which planet has the greatest disagreement between T_obs and T_th? Which planet has the best agreement between T_obs and T_th?