GS107KeplersLawLab_12AUG23
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Rogue Community College *
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107
Subject
Astronomy
Date
Dec 6, 2023
Type
docx
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9
Uploaded by SuperHumanOxideBison32
Name: Derek Greene
GS 107
Kepler’s Laws Used for Analysis on Exoplanets and Stellar Orbits
Kepler’s Laws can be applied, with some slight modifications, to any object orbiting another. It is particularly straightforward when one object is much more massive (the primary) and one much less massive (the secondary).
Consider the following table of some of the earliest exoplanets discovered: 1.
If you are at lab table #1, use the data for Kepler 3b. Likewise, use the data for Kepler 4b if you are at lab table #2. Use the data for Kepler 6b if you are
at lab table #3. Use the Kepler 8b data if you are at lab table #4. If you are at lab table #5 or #5, you get to choose your planet. 2.
Clearly identify here which planet you are using. Note that each exoplanet listed here orbits a different star.
Keppler 3b
3.
Use Kepler’s Laws to find the orbital distance in AU for your planet if each of these planets orbited a star that was the same mass as the Sun.
P^2=a^3
0.013^2=a^3
Cube root of 0.000169= 0.05528774 AU
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4.
Consider the following table. What would the actual orbital distance be using the proper star mass?
Exoplanet
Star’s Name
Star’s Mass (solar masses)
Kepler 3b
HAT-P-11
0.81
Kepler 4b
Kepler 4
1.117
Kepler 6b
Kepler 6
1.209
Kepler 8b
Kepler 8
1.213
Most of this data taken from Wikipedia.
0.051543 AU
5.
Is your assigned planet closer or farther from its star than Earth is to the Sun? [Hint: What distance does an AU represent?]
My assigned planet is closer to its star than earth is to the sun.
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6.
What do you think the temperature would be like on your assigned planet compared to Earth based on this information? Would it be warmer, cooler or
about the same? Why? [To know for sure, we’d need to have information about its atmosphere (density, composition, etc.) and many other things. Make this guess based on the distance from the star.]
Soley based on the distance, I would expect my assigned planet to be much hotter than earth and due to solar radiation, unable to build an atmosphere to trap heat in so also very cold when not in direct sunlight. Page 3
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7.
Weight is the force of the Earth’s gravitational pull on you. What is your weight in pounds? [Feel free to use your target weight rather than your COVID weight!] Weight is the force of the Earth’s gravitational pull on you.
145 lbs
8.
Your weight on Earth corresponds to a given mass. A mass of 0.454 kg corresponds to a force of 1.00 pounds on the surface of the Earth. What is your mass?
65.83 kg
9.
Newton built his work with gravity based on Kepler’s work. As part of Newton’s work is his universal force of gravity (see section 3.3 in our text):
F = (GM
p
M
y
)/R
2
p
where, in this case, F is the size (magnitude) of the force of gravity, G is the universal gravitational constant (see textbook appendices), M
p
is the mass of the planet, M
y
is your mass and R
p
is the radius of the planet (i.e., the distance between the center of gravity of the planet and the surface of the planet). What is the force of gravity (i.e., weight) in newtons you would have on your planet?
The force of gravity in newtons on my planet would be 3.41055419x10^-24 newtons
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10.There are 0.225 pounds of force for every newton. What would your weight
be on the surface of your planet in pounds?
7.67374692x10^-25
11.
These same sorts of calculations can be used for finding the mass of the object at our galactic center. For a 3D visualization of the data we will consider, see https://www.youtube.com/watch?v=SGJY0aeWBNc&t=66s
You can see some data collected by UCLA that we will use at https://galacticcenter.astro.ucla.edu/black-hole-science.html
While the video file showing the 2D orbits that will be easier for us to use doesn’t seem to be on the site where it is easily found, you can find a recording of it in this week’s lab folder.
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Let’s consider the star S0-2. You can see it starting out towards the top and near center:
Its orbit is traced out in yellow in the animation. A screen shot is shown below but consider the animated graphic linked above for the time of the orbit. Page 6
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The year data is displayed in the upper right corner. Observe the time for one period of S0-2 and record it here. 12.This image has a scale of 0.05 arc seconds shown as the length of the white double-headed arrow. At the distance of our galactic center, this corresponds to about 400 AU. Use a ruler to measure the length of the bar in millimeters AND the length of S0-2’s major axis. Use a ratio of the length of the scale bar in millimeters to its actual size in AU and equate that to a ratio of the length of the star’s orbit in millimeters and its actual size in AU. Watch your
units carefully here. Have each member of your lab measure these distances as carefully as can be as being a bit off can give results that are pretty far off.
Scale bar [mm]/Scale bar [AU] = Major Axis [mm]/Major Axis [AU]
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13.Calculate the length of the major axis of S0-2 using the ratio above. Divide your answer by two to get the semimajor axis, a. ~12.2 mm
14.Newton’s Law of Gravity can be combined with Kepler’s Third Law which gives a form shown in Section 3.3 with the heading “Orbital Motion and Mass”. Using that equation considering M
1
is much greater than M
2
, which is reasonable given the graphic referenced above, how large is the mass that these stars are orbiting around based on your calculations in solar masses? Show your work.
They are orbiting around a star with the mass of ~2.63 solar masses
15. The measurement for the star S0-2’s semimajor axis assumed that it is not tilted from our point of view. (From the 3D video, you can see not all of the stars are in the same plane.) If it turns out that the star’s orbit is tilted relative to us, will that make the actual semimajor axis larger or smaller than
what you measured? Will that make the mass of the central object larger or smaller than your estimate?
This would make the actual semimajor axis smaller than measured and would make the mass of the central object larger than measured.
16.Astronomers are strongly convinced that there is a black hole at the center of
our galaxy. Explain, with reference to your calculations and the animated graphic, why this view is justified.
Some reasons for why this theory is justified is that scientists have been able to observe how stars react to the forces of gravity, which a black hole is
essentially comprised of. Thus, it can be argued that there is a black hole at the center of our galaxy. 17. The text associated with this graphic gives a value of the central mass taking all of the stellar orbits and tilts into consideration from our point of view of 4.6 +
0.7 x 10
6
solar masses. Does your value fall into this range? If not, how close is it? (It may not be as close as you’d like but it should be between 1 and 10 million solar masses.)
Yes my value falls into this range. Page 8
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18. Write a summary of the usefulness of Kepler’s Laws (1610-1619) and Newton’s Modification (1687) in astronomy today.
Keppler’s Laws and Newton’s Modification have had an extraordinary impact on astronomy and how we are able to study space and the stars around us. Keppler’s Laws have helped us to predict the orbits of many stars and various other objects. Additionally, Kepler’s law was pivotal in the discovery of dark matter in the milky way. Newtons Modification helps us to determine the masses of objects from the orbital motion as well as in various other areas of astronomy such as understanding
gravity and how it works. Page 9
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