ASTR 101 Lab 5 - Kepler
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101
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Astronomy
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Jan 9, 2024
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5
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March 23, 2023
ASTR 101 Lab 5 - Kepler (Mars)
Objective
The objective of this lab is to familiarize myself with the orbits of the planets in
the Solar System and help to understand the seasonal changes in the Sun's positions
and the planets in the sky as seen from Earth. During this lab, I will be introduced to two
essential coordinate systems, the heliocentric and the geocentric systems that were
used by astronomers to locate the positions of solar system objects. This lab will be
based on Kepler’s laws of planetary motion.
Introduction
Around the year 1609, Johannes Kepler was an astronomer who came up with
the three laws of planetary motion that are still being used today. He based these laws
on the analysis of another astronomer named Tycho Brahe who observed Mars and the
other planets. And this was before the telescope was invented. Because of Kepler,
Copernicus and Galileo’s observations and efforts, they were able to change the way
humanity views the solar system. Before, it was believed that everything orbited the
Earth (geocentric view). But now, it’s the correct heliocentric model. The Earth orbits
around the Sun every 365.24 earth days and with a period of one year. The Sun
appears to be stationary. However, when astronomers were observing the Sun’s
position in the sky throughout the year, it appeared to move because it was against a
fixed background of distant stars. And this was the case with the other bright planets.
Their orbits are in the shape of elliptical. Kepler used triangulation to determine the
position of Mars. From this technique, he learned the distance of Mars from Earth as
well as the orbital period which is 687 days. For this lab, I will be using a geometrical
method based on triangulation to determine the orbit of Mars.
Procedure
For this lab, I will be following what Kepler did to determine the distance of Mars
based on the angles of the Earth and the Sun. First, I drew a circle with a radius of 5cm
(diameter of 10cm). This represents the orbit of the Earth. I drew a dot in the middle of
the circle which represents the Sun. Then I drew a straight line from the Sun towards
the first point of Aries which is an angle of 0 degrees. Second, following the table of
angles (Table 1), I used heliocentric and geocentric measurements to determine where
the Earth is relative to the Sun and the position of Mars relative to the Earth. If the angle
was greater than 180, I would subtract 180 deg from the angle, flip my image (Figure 1)
and use that result to draw the angle. I repeated this six times to find the 6 positions of
Mars. Once I found all six positions, I traced an almost circular path that represents the
orbit of Mars.
Observations, Tables, Graphs and Figures
Figure 1 - Position of Mars Image
Table 1 - Orbital Positions of Earth and Mars
Answers
1.
Three Laws of Planetary Motion
a.
planets move in elliptical orbits with the Sun as a focus (The Law of
Orbits).
b.
a planet covers the same area of space in the same amount of time no
matter where it is in its orbit (The Law of Areas).
c.
a planet’s orbital period is proportional to the size of its orbit (its
semi-major axis) (The Law of Periods).
2.
Ratio and Average (5cm / 1 AU)
a.
Mars A = (8.3cm x 1 AU) / 5cm =
1.66 AU
b.
Mars B = (8.2cm x 1 AU) / 5cm =
1.64 AU
c.
Mars A = (8.0cm x 1 AU) / 5cm =
1.60 AU
d.
Mars A = (8.9cm x 1 AU) / 5cm =
1.78 AU
e.
Mars A = (9.5cm x 1 AU) / 5cm =
1.90 AU
f.
Mars A = (7.7cm x 1 AU) / 5cm =
1.69 AU
Average is the sum of all six Mars AUs divided by 6. Therefore the
average is
1.69 AU
3.
Eccentricity
a.
e = focal length (f) / semi-major axis (a)
b.
f = 0.8cm, a = 6.9cm
c.
e = f / a = 0.8cm / 6.9cm = 0.1159 =
0.116
4.
Perihelion and Aphelion
a.
Perihelion (closest) = (7.7cm x 1 AU) / 5cm =
1.54 AU
b.
Aphelion (furthest) = (8.1cm x 1 AU) / 5cm =
1.62 AU
5.
The orbital period of Earth is 365.24 days and for Mars is 687 days. Earth and
Mars are closest to one another every 26 months which is just over 2 years. This
is due to their different orbit periods and that they orbit in an ellipse in the same
direction.
6.
When Earth and Mars are on the same side of the Sun, they will be at the closest
approach to each other. Based on the data from the table and the position of
Mars C and Earth
8
, the Opposition will occur in the month of December.
7.
When Earth and Mars are on opposite sides of the Sun, they will be furthest
away from each other. Based on the data from the table and the position of Mars
E and Earth
9
, the Conjunction will occur in the month of December.
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8.
Kepler’s Third Law (P^2 = a^3)
a.
P = orbital period, a = semi-major axis
b.
First, I need to convert the orbital period from days to years and the
semi-major axis from centimetres to AU.
c.
P(earth) = 365.26 days / 365.26 days = 1 year
d.
a(earth) = 1.0 AU
e.
P(Mars) = 686.98 days / 365.25 days = 1.88 years
f.
a(Mars) = (6.9cm x 1 AU) / 5cm = 1.38 AU
g.
Ratio = 1 = (P^2 / a^3)
h.
Ratio of Earth = (1^2 / 1^3) = 1
i.
Kepler’s formula holds true for Earth since
1
= 1.
i.
Ratio of Mars = (1.88^2 / 1.38^3) =
1.34
i.
Kepler’s formula does not hold true for Mars because the ratio is
not equal to 1. But it is very close though.
Discussion
This lab required me to trace, draw and measure multiple things. My results
could have been better. There were some factors that might of lead to some inaccurate
data. For example, I used an online protractor. Even though I used an actual ruler to
measure the distances, using an online calculator was a bit tricky since I had to trace it
from the screen of my laptop on a piece of paper. Had I used an actual protractor I
would have gotten much better and more accurate results. I am also not the strongest
artist so when I was tracing the angles, I could have made some small mistakes that
affected the results. Another thing that could have made my results less accurate is the
fact that I assumed Earth’s orbit was a perfect circle instead of an ellipse. The average
distance between the Sun and Mars in Astronomical Units that I calculated was 1.69
AU. Compared to the actual distance of Mars from the Sun which is 1.52 AU. For the
eccentricity of Mars, I got 0.1159 which is pretty large for an orbit. But considering the
orbit path in Figure 1 is almost circular, that could be the reason why. When I looked up
the eccentricity of Mars online, it was 0.093 which is significantly smaller than what I
got.
Conclusion
From this lab, I learned how to use the triangulation technique to determine
aspects of Mars like its distance and trace its orbital path. While doing the lab, I did
some of my own research that the days on Earth and Mars are very close. The Earth
makes a full revolution every 24 hours while on Mars, it’s 23.9 hours. I also learned that
the Sun is not stationary, but also that it and almost every other celestial body are in an
elliptical orbit. This lab has expanded my knowledge of how Kepler found the orbital
path of Mars and its distance.
References
J. Tate, “Kepler’s Laws”, in Universe Today. [Online]. Available:
https://www.universetoday.com/55423/keplers-law/
[Accessed: February 11, 2010]
M. Williams, “When Will Mars Be Close to Earth”, in Phys.org. [Online]. Available:
https://phys.org/news/2017-04-mars-earth.html
[Accessed: April 10, 2017]