AST 101 - HW 4 - Jesse Brown - keplers-laws
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Syracuse University *
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101
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Astronomy
Date
Jan 9, 2024
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Tutorial-Exercise – Kepler’s Laws
In this exercise, you’ll explore Kepler’s laws of orbital motion.
Remember that these exercises are not meant for you to do alone; you should work with others
near you on them, and should raise your hand and ask questions as you have them.
Your fourth homework assignment is included on the back of this handout. You should complete
it by class time on October 12 (or 7) and put it in your TA’s mailbox.
1
Kepler’s First Law
Kepler’s first law says that planets orbit the Sun in an ellipse with the Sun at one
focus
. An ellipse
has two foci; they lie along the long axis of an ellipse, are nearer the center for ellipses that are
less eccentric (stretched out), and are nearer the edge for ellipses that are more eccentric (more
stretched out).
Here are four orbits.
1. Two of them have
two
possible locations for the Sun. Draw them in.
2. One of them has only
one
possible location for the Sun. Draw it in.
3. One of them is not a possible shape for a planet’s orbit. Cross it out.
Of the three possible orbits, which one has zero eccentricity?
Which one has the highest eccentricity?
The Second ellipse has the highest eccentricity
The fourth Ellipse has zero eccentricity.
2
Kepler’s Second Law
As we saw demonstrated a bit ago, Kepler’s second law says that an imaginary line between the
Sun and a planet orbiting it sweeps out an equal amount of area in an equal amount of time.
First, let’s apply this to Earth’s orbit, which is very close to a perfect circle.
Pretending for now that Earth’s year is divided into twelve equal months and that its orbit is a
perfect circle, here is a cartoon of Earth going around the Sun.
1. Using your pencil, shade the region that the imaginary line between Earth and the Sun
sweeps out during January, and again during September.
2. Does Earth’s orbit follow Kepler’s second law? How do you know?
3. Kepler described his observations of planetary motion in terms of the “area swept out” by
the line, but we can also think about the
speed
of Earth’s motion.
Does the Earth move faster during January or during September? How do you know?
Yes Earth's orbit follows Kepler's second law because the shaded areas between each month cover the
same amount of area.
Based on the example above the earth moves the same speed during January
and September.
4. Now, let’s consider a different planet whose orbit is more eccentric.
Here is a cartoon of its orbit, showing its position at the beginning of January and February,
along with several other positions
Shade in the area that the line connecting the Sun to the planet would sweep out during
January, as you did before.
5. Then, find
two other
points in the orbit during which the line connecting the planet to the
Sun would sweep out approximately that same area. You would like to fill in the blanks in
the statement:
“As it moves from point
to point
, the line connecting this planet
to the Sun sweeps out the same area as it does during the month of January.”
Remember what “area” means: this means that if you were to color both areas in with ink,
you would use the same amount of ink on both.
6. Does the motion on the left through the region you’ve highlighted take
more than
,
less than
,
or
exactly
one month? How do you know? (Look back at the text of Kepler’s second law on
the front page.)
7. Does the planet cover
more distance
during January or during the highlighted portion you’ve
shown on the left?
A
D
If the area is actually the same, then it will take exactly one month.
It covers more distance during the highlighted potion on the left.
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8. Does the planet
travel faster
during January or during the motion shown on the left? How
do you know?
9. Now, let’s think about the orbits of the planets known in Kepler’s time. The eccentricity of
Earth’s orbit is 0.016. This is very small, so its orbit is very nearly circular. The orbits of
Earth and the other inner planets look like this:
Planet
Perihelion (closest to Sun)
Aphelion (furthest from Sun)
Eccentricity
Mercury
0.47 AU
0.31 AU
0.206
Venus
0.72 AU
0.73 AU
0.006
Earth
0.98 AU
1.02 AU
0.016
Mars
1.38 AU
1.67 AU
0.093
Which of these planets
changes speed by the largest fraction
over the course of one orbit?
How do you know?
10. Which of these planets
changes speed by the smallest fraction
over the course of one orbit?
How do you know?
During the motion shown on the left because they are the same amount of time but the left has a
grouter distance to cover.
Mercury because it is the most eccentric.
This would be the least eccentric planet.
3
Kepler’s Third Law
Kepler’s third law says that:
“The square of the time that it takes a planet to go around the Sun is proportional
to the cube of the long axis of its orbit.”
Translated into modern mathematics, we could also say:
“The time it takes for a planet to go around the Sun is proportional to the 3/2
power of the long axis of its orbit.
We will do some more precise calculations based on this in lab. For now, you can just know:
“If a planet’s distance from the Sun increases by some factor, its orbital period
increases by
more
than that factor.”
Earth is in a circular orbit around the Sun with a radius of about 1 AU, while Saturn is in a
circular orbit with a radius of about 10 AU. Will Saturn take less than one year, between one and
ten years, about ten years, or more than ten years to go around the Sun? How do you know?
Saturn will take more than 10 years because the proportion is increased m by 10 fold and the equation is
squared.
Here are some data on the planets:
Orbital Distance
Orbital Period
Mass
(AU)
(years)
(Earths)
Mercury
0.38
0.24
0.06
Venus
0.72
0.61
0.82
Earth
1
1
1
Mars
1.52
1.88
0.11
Jupiter
5.20
11.86
318
Saturn
9.54
29.46
95.2
Notice that the planets have wildly varying masses: Jupiter is five thousand times more massive
than Mercury.
Based on these data and what you know about Kepler’s third law, what is the relationship between
the mass of a planet and how long it takes to orbit the Sun?
If there
is
a relationship, describe where that relationship comes from. If there is
not
a relationship
between mass and orbital period, discuss how the data from the table tell you that.
What is the dominant factor in determining how long it takes for a planet to go around the Sun?
There is no relationship.
The data tells us that there is no relationship because there is no measurable proportionality between the
masses of the planets and the planet distance as well as orbital period.
The dominant factor is orbital distance, because the orbital time seems to have a direct correlation with the orbital
distance of a planet.
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Homework 4 – Kepler’s Second Law
Due Tuesday, October 12
Suppose that the fictional planet that divides its years into twelve equal months has a pretty
elliptical orbit. Astronomers on this planet are trying to figure out how it moves around its star
during a year. They propose two options:
Choice A
Choice B
Three astronomers – Alex, Bethany, and Chen – are debating which one is correct.
Alex:
I think Choice A is correct. The planet is supposed to go fastest when it is near the Sun,
right? So the months are spaced closer there since it is going faster.
Bethany:
Yes it is. But Choice B shows an orbit where the planet is moving slowly far away
from the Sun, since it covers less distance each month.
Chen:
I think neither one can be right. The months are supposed to be equally-spaced and last
the same amount of time, right? So the dots should be equally far apart, since the orbit is one
year, and the months are equal portions of that. So you should draw another diagram like that.
1. Who is correct? Write a few sentences explaining why and how you know.
Bethany is correct because when the planet is a closer to the Sun, it moves faster so it needs to cover
more distance to still take up the same amount of time. When the planet is farther, it needs to cover
less distance, so it moves slower to accomodate taking up the same amount of time.
2. Is the planet moving faster in July or January? How do you know?
3. Is the planet speeding up or slowing down in April? How do you know?
The planet is moving faster in July because it is closer to the Sun at this time and the gravitational force put on
it makes it go faster. In addition, if you think about it another way, it has more distance to cover to continue
its orbit in equal time if it will be moving slowly when it is farther away from the Sun.
The planet is speeding up in April because it is getting closer to the Sun instead of
on the other side of its orbit where it is getting farther from the Sun.