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Kepler's Laws of Planetary Motion Lab Report - Fall 2021 (PHYS 101)

Course:
Intro Astr Lab: Solar System (ASTR 112)

66 documents

University:
George Mason University

Uploaded by

Student
Roger N
2023/2024
1
Kepler’s Laws of Planetary Motion
Name: Roger Newport
INTRODUCTION:
In this lab you investigate Kepler’s Laws of Planetary motion and their consequences.
Learning Goals:
Students will become familiar with Kepler’s laws of planetary motion and what they tell us about planets’
trajectories.
Learning tools:
Excel spreadsheet with planetary motion simulator (provided).
Background:
Johannes Kepler published three laws of planetary motion, the first
two in 1609 and the third in 1619. The laws were made possible by
planetary data of unprecedented accuracy collected by Tycho Brahe.
The laws were both a radical departure from the astronomical
prejudices of the time and profound tools for predicting planetary
motion with great accuracy. Kepler, however, was not able to describe in a significant way why the laws
worked.
1stLaw:LawofEllipses
Theorbitofaplanetisanellipsewhereone
focusoftheellipseistheSun.
How “elliptical” an orbit is described by
its eccentricity(see lab 2 for description of
eccentricity).
2ndLaw:LawofEqualAreas
A line joining a planet and the Sun sweeps out
equal areas in equal time intervals.
With elliptical orbits a planet is sometimes closer
to the sun than it is at other times. The point at
which it is closest is called perihelion. The point at
which a planet is farthest is called aphelion.
Kepler's second law basically says that the planets
Figure 1.
Figure 2.
Figure 3.
2
speed is not constant – moving slowest at aphelion and fastest
at perihelion. The law allows an astronomer to calculate the
orbital speed of a planet at any point.
3rdLaw:LawofHarmonies
Thesquareofplanet’sorbitalperiod
P
(expressedinyears)
equalstothecubeofitsaverageorbitaldistance
a
(orsemi-
majoraxis)expressedinastronomicalunits(AUs).
P2=a3
This equation is only good for objects that orbit our Sun. Later on Isaac Newton was able to derive a more
general form of the equation using his Law of Gravitation.
This law tells us that not only planets that are further from the Sun take longer to orbit it, something that
could be anticipated since the planet in a more distant orbit will have a longer trajectory to travel through as
its orbit has a greater circumference, but that planets that are further away actually also move more slowly
(see orbital speed in the worksheet for lab 2). Isaac Newton was able to explain that: the further a planet is
from the Sun, the weaker the force of gravity that the Sun exerts on it, which results in slower orbital motion
of a more distant object.
Kepler developed his three laws to describe the motion of the planets in our solar system. When the third law
is derived using Newton’s Laws of Motion and his Universal Law of Gravitation (see appendix A), which allows
to take the mass of the orbited body into account, Kepler’s three laws apply to other objects e.g. moons that
orbit a planet, planets that orbit other stars and etc.
Planetarymotionsimulator (in your Excel worksheet for this lab)
Open the Excel worksheet simulator. The simulator contains a spreadsheet calledInput and graphs—Orbit,
DistancetotheSun, and OrbitalSpeed.
Look at the Input spreadsheet contains
cells in a table titled “input quantities”
(see figure 5) where you input the
variables that you’ll adjust: the semi-
major axis,
a
, and eccentricity,
e
, for
two planets.
Note that the minimum and
maximum values for semi-major axis
a
and eccentricity
e
are given in parentheses. Do not exceed them, the
simulator is not designed to handle values beyond this range. Two other variables can also be adjusted (angle
between the orbits and inclination), but in this experiment we’ll leave them at their default values What you
will vary are semi-major axes and eccentricities of the two planets.
Figure 4.
Figure 5.
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Kepler’s Laws of Planetary Motion

Name: Roger Newport

INTRODUCTION:

In this lab you investigate Kepler’s Laws of Planetary motion and their consequences.

Learning Goals:

Students will become familiar with Kepler’s laws of planetary motion and what they tell us about planets’

trajectories.

Learning tools:

 Excel spreadsheet with planetary motion simulator (provided).

Background:

Johannes Kepler published three laws of planetary motion, the first

two in 1609 and the third in 1619. The laws were made possible by

planetary data of unprecedented accuracy collected by Tycho Brahe.

The laws were both a radical departure from the astronomical

prejudices of the time and profound tools for predicting planetary

motion with great accuracy. Kepler, however, was not able to describe in a significant way why the laws

worked.

1 st Law: Law of Ellipses

The orbit of a planet is an ellipse where one

focus of the ellipse is the Sun.

How “elliptical” an orbit is described by

its eccentricity (see lab 2 for description of

eccentricity).

2nd Law: Law of Equal Areas

A line joining a planet and the Sun sweeps out

equal areas in equal time intervals.

With elliptical orbits a planet is sometimes closer

to the sun than it is at other times. The point at

which it is closest is called perihelion. The point at

which a planet is farthest is called aphelion.

Kepler's second law basically says that the planets

Figure 1.

Figure 2.

Figure 3.

speed is not constant – moving slowest at aphelion and fastest

at perihelion. The law allows an astronomer to calculate the

orbital speed of a planet at any point.

3 rd Law: Law of Harmonies

The square of planet’s orbital period P (expressed in years)

equals to the cube of its average orbital distance a (or semi-

major axis) expressed in astronomical units (AUs).

P

####### 2

=a

####### 3

This equation is only good for objects that orbit our Sun. Later on Isaac Newton was able to derive a more

general form of the equation using his Law of Gravitation.

This law tells us that not only planets that are further from the Sun take longer to orbit it, something that

could be anticipated since the planet in a more distant orbit will have a longer trajectory to travel through as

its orbit has a greater circumference, but that planets that are further away actually also move more slowly

(see orbital speed in the worksheet for lab 2). Isaac Newton was able to explain that: the further a planet is

from the Sun, the weaker the force of gravity that the Sun exerts on it, which results in slower orbital motion

of a more distant object.

Kepler developed his three laws to describe the motion of the planets in our solar system. When the third law

is derived using Newton’s Laws of Motion and his Universal Law of Gravitation (see appendix A), which allows

to take the mass of the orbited body into account, Kepler’s three laws apply to other objects e. moons that

orbit a planet, planets that orbit other stars and etc.

Planetary motion simulator (in your Excel worksheet for this lab)

Open the Excel worksheet simulator. The simulator contains a spreadsheet called Input and graphs—Orbit,

Distance to the Sun, and Orbital Speed.

Look at the Input spreadsheet contains

cells in a table titled “input quantities”

(see figure 5) where you input the

variables that you’ll adjust: the semi-

major axis, a, and eccentricity, e, for

two planets.

Note that the minimum and

maximum values for semi-major axis a and eccentricity e are given in parentheses. Do not exceed them, the

simulator is not designed to handle values beyond this range. Two other variables can also be adjusted (angle

between the orbits and inclination), but in this experiment we’ll leave them at their default values What you

will vary are semi-major axes and eccentricities of the two planets.

Figure 4.
Figure 5.

This time step between two consecutive data points will not change if the eccentricity changes. For e= 0 the calcu-

lated time steps amount to 9-degree intervals between the data points. You will investigate what happens when the
eccentricity changes: does the time step change? Does the angle change?

Procedure:

1. Adjust the input quantities settings to the following: for Planet 1: a= 1 AU and e= 0 and for Planet 2: a=

1 AU and e= 0. What are the values of the period P and the mean annual motion N? Record the results in

table 1 below.

Table 1.

Object Orbital period P (years) Mean annual motion N (degrees)

Planet 1 1 360.

Planet 2 1 229.

2. Adjust the values of the semi-major axis for Planet 2 to values listed in table 2 below. For each value of

semi-major axis, record the period P and the mean annual motion N. From Orbital speed graph, read the

orbital speed of Planet 2 for each value of semi-major axis. Record it in table 2 below.

Table 2.

Semi-major

axis (AU)

Orbital period P (years)

Mean annual motion N

(degrees)

Orbital speed (AU/year)

0 .13 2880 12.

0 .35 1018 8.

0 .65 554 7.

1 1 360 6.

1 1 257 5.

1 1 195 5.

1 2 155 4.

2 2 127 4.

Based on this data, complete the following statements:

As the value of the semi-major axis increases, the period Increases (enter: “increases”, “decreases” or “stays

the same”)

As the value of the semi-major axis increases, the mean annual motion Decreses (enter: “increases”,

“decreases” or “stays the same”)

As the value of the semi-major axis increases, the orbital speed Decreases (enter: “increases”, “decreases” or

“stays the same”)

3. Now start with Planet 2 at a = 1 AU and e= 0. Then adjust the value of eccentricity e for Planet 2 to the

values listed in table 3. For each value of eccentricity, record the period and the period P and the mean

annual motion N. Also, for each value of eccentricity, look at the graph showing the orbits. Record the data in

table 3 below.

Table 3.

eccentricity Orbital period P (years) Mean annual motion N (degrees)

0 1 360

0.

1 360

0 1 360

0.

1 360

0.

1 360

0 1 360

Based on this data, complete the following statements:

As the value of the eccentricity increases, the period Stays the same (enter: “increases”, “decreases” or “stays

the same”)

As the value of the eccentricity increases, the mean annual motion Stays the same (enter: “increases”,

“decreases” or “stays the same”)

Describe how the shape of the orbit changes as eccentricity increases? The orbit becomes more oval.

As you increase the eccentricity for Planet 2, does the Sun stay at the center of the orbit? No

4. Adjust the input quantities settings to the following: for Planet 1: a= 1 AU and e= 0 and for Planet 2: a= 1.

AU and e= 0.

Look at the Orbits graph. Are the fourty data points still evenly spaced around the orbit? Yes

Look at the Distance to the Sun graph. Is the distance of each planet to the Sun constant? Yes

The distance from the Sun to Planet 1 is 1 AU

The distance from the Sun to Planet 2 is 1 AU

Do not forget to include units!

Look at the Orbital speed graph. Are the orbital speeds of the planets constant? Yes

The orbital speed of Planet 1 is 5 AU/yr

The orbital speed of Planet 2 is 6 AU/yr

Do not forget to include units!

In the Input spreadsheet, look at the data table for Planet 1 (columns B, C and I; rows 23-63) and Planet 2

(columns O, P and V; rows 23-63).

Object Eccentricity Semi-major axis (AU) Orbital period P (years)

Least eccentric

major planet

Venus .72 AU.

Most eccentric

major planet

Mercury .39 AU.

Set Planet 1 parameters in the simulator to semi-major axis and eccentricity of least eccentric of the real major

planets and those for Planet 2 to those for the most eccentric. Look at the Orbits graph. Describe how the

most and least eccentric of the real major planets’ orbits appear on the simulation. Click or tap here to enter

text.

Copy the Orbits graph (made for two major planets listed in table 4) and paste it here:

Submission details:

Submit into this lab’s drobox on Blackboard:

MS Word report (this document with your entries) only,

Appendix A

Newton’s three Laws of motion:

1. An object at rest will remain at rest and an object in motion will continue to move at a constant speed

along a straight line, unless it experiences a net external force

2. Object’s acceleration equals to the net external force divided by its mass.

3. When a body A exerts a force on body B, body B will exert an equal amount of force on body A in opposite

direction.

Newton’s Law of Gravitation:

The force of gravity between two masses (M 1 and M 2 ) is directly proportional to the product of the two

masses and inversely proportional to the square of the distance d between them:

F=G

M 1 × M 2

d

####### 2

Here G is the universal gravitational constant G = 6× 10 -11 N×m 2 /kg