Homework week 2
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PHY 21430 Frontiers in Astronomy
Homework - Week 2
Instructions: Replace the Text in Orange with your answers. You can take up as much space as needed to answer, but please keep your answer colored orange to make it easier to grade. Upload your solutions to the Assignment in Blackboard. Kepler’s Laws
Based on meticulous data compiled by Tycho Brahe, Kepler proposed three general laws that summarized planetary orbits. These laws ultimately were explained to come from Newton’s Law of Gravitation, a triumph of Newtonian Mechanics. Kepler’s laws can be summarized as follows: Kepler’s first law:
Each planet moves around the Sun in an
orbit that is an ellipse, with the Sun at one focus of the ellipse.
The semi-major axis (a) is half the longest diameter. Eccentricity (e) indicates how elongated the ellipses are. It
ranges from e = 0 (circle) to e =1 (line). Kepler’s second law:
The straight line joining a planet and the
Sun sweeps out equal areas in space in equal intervals of time.
The second law indicates the speed of a planet along its orbit.
Kepler’s third law
: The square of a planet’s orbital period is
directly proportional to the cube of the semimajor axis of its
orbit.
If the period (T) of a planet is measured in years
, and the semi-major axis (a) is measured in A. U.
(mean distance from
Earth to the Sun) then T
2
=
a
3
PHY 21430 Frontiers in Astronomy
1. Orbits of the planets
Open https://javalab.org/en/solar_system_en/
and choose 3D to see the orbits of the planets in the solar system. You can click and drag to change the viewing angle and use the slider to zoom in and out. Click and drag the angle so that the motion of the planets is parallel to the screen (and you are looking directly down on the planets’ orbits. Zoom in to see the four inner planets clearly.
a.
(4 pts) Notice the shapes of the orbits. List the inner planets Mercury, Venus, Earth, Mars in order of increasing eccentricity. (Circular orbits have the least (e=0), and the more elongated the orbit, the higher the eccentricity.)
Write planets in increasing order of eccentricity. b.
(1 pt) Which of Kepler’s laws is question 1a related to?
Write the number of law here Now zoom out to see more of the planets and view from a convenient angle. c.
(3 pts) How does the time for a planet to make a complete orbit relate to the orbit size? Write your answer here. Be qualitative. d.
(1 pt) Which of Kepler’s laws is this question 1c related to?
Write the number of law here. 2. Kepler’s first law
Now, we get a closer look at the orbit of the Earth around the Sun. Open the simulation https://phet.colorado.edu/sims/html/gravity-and-orbits/latest/gravity-and-
orbits_en.html
This program simulates the orbit of a planet around the Sun.
PHY 21430 Frontiers in Astronomy
Click on the “To Scale” menu. Choose the
following options: Sun and Earth in the
upper right (default). Click on Velocity,
Path, and Grid. An image of what the screen should look
like is to the right. The green arrow shows the direction of
the velocity and the length of the arrow is
the planet’s speed. To start/stop simulation press the play/pause button at bottom center. You can reset the simulation with the button on the lower right in the likely case you need to start the simulation over. a.
(6 pts) Start the simulation and stop it right when one orbit. Click on the tape measure. Measure the horizontal distance from the Sun to the Earth on the left, r1, and on the right, r2. The Semi-major axis is a
E
= (r1+r2)/2. a
E
is called 1 AU (
astronomical unit)
. Sun-Earth distance to the left, r1 in km
Sun-Earth distance to the right, r2 in km 1 AU in km
Write r1
Write r2
Write a
E
= (r1 + r2)/2
Express answers in scientific notation with two decimal places in km (e.g., 1000000000 km is 1.00 x 10
9
km). If you don’t have a scientific calculator handy, you can search “calculator” on google and use 1 Exp 9 for 1.00 x 10
9
. b.
(1 pt) Is the Earth’s orbit really circular? Write yes or no here. c.
(1pt) How does your measured value of a
E
compare to the accepted value, 1 AU = 1.5 x 10
8
km?
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PHY 21430 Frontiers in Astronomy
Write answer here. 3. Kepler’s second law
According to Kepler’s second law
, the Earth changes speed as it orbits the Sun as the distance to the Sun changes. Since Earth’s orbit is nearly circular, it is hard to see the difference of the speed along its orbit. That is, for a nearly circular orbit, the speed is nearly constant. Let’s change the orbit to one with higher ellipticity. Restart the simulation and change the Earth’s initial velocity by stretching or shrinking the green arrow before the simulation starts. Explore different orbits by increasing and decreasing the length of the initial velocity vector so you get an orbit that is easier to see the speed changing during the orbit.
a.
(1 pt) Where is the planet relative to the Sun when it is moving fastest along its orbit?
Write answer. 4. Blackbody Radiation
The temperature of an object is a measure of the random motion associated with heat energy. When electrons and nuclei in atoms of molecules vibrate, they radiate light containing all different wavelengths with amounts characteristic of this temperature called the Blackbody Spectrum. By looking at the shape of the spectrum of light emitted by a star, we can tell something about its average surface temperature.
In this illustration, https://phet.colorado.edu/sims/html/blackbody-spectrum/latest/blackbody-
spectrum_en.html
you will explore how the temperature of an object affects the spectrum of radiated light. The curve indicates the intensity of each wavelength of light for an object at temperature T. Click on Labels
which show the ultraviolet, visible, and infrared regions of the electromagnetic spectrum. Also, click on graph values
to see the values of intensity and wavelength for points on the curve. a)
(3 pts) The temperature of the Sun is 5800 K. Describe the distribution of electromagnetic radiation emitted from the Sun (Is it all visible light? What parts of the electromagnetic spectrum is light emitted? How does the intensity at the violet end of the
PHY 21430 Frontiers in Astronomy
visible spectrum compare to the intensity of the red end of the visible spectrum? What color has the maximum intensity?). Write answer here. You can change the temperature using the slider on the right by clicking and dragging the blue arrow on the right of the thermometer.
b)
(3 pts) Describe the change to the spectrum as the temperature of the object increases or decreases. How does the wavelength of the peak intensity change? How does the intensity change?
Write answer here. c)
(3 pt) Estimate the surface temperature (in degrees K) of a star with a blackbody spectrum with a maximum intensity at the border between red and infrared light (wavelength of 0.75 μm).
Write answer here
d)
(3 pt) As an iron bar is heated, it starts to glow red when it reaches a temperature of about 900 K. Why do you think you don’t see a change in color as it is heating up?
Write answer here.