Description
You have to complete the hw on the file .Please don’t accept the work if you will not be sure of your answersit contains physics n math. Accepting this hw means you will help getting more than 85%, if higher I’m gonna tip. I won’t accept lower.Thank you…
Unformatted Attachment Preview
Introduction to Space Sciences
SPS 1020-01 Fall 2023: Homework #3
Assignment is due Wed Nov 8 2023, 11:59pm.
No late submissions will be accepted.
Instructions: The majority of the instructions for this assignment are in the “Instructions for Submission
of Problem Sets” handout that is available on Canvas. Please review these instructions. Also review the
solutions for the previous homeworks and use those as guidelines for how your submission should look like
and the level of details and calculations that you should include.
Questions:
1. A satellite in geostationary orbit appears to remain stationary in the sky as seen from any particular
location on Earth. To do this, the satellite has to orbit at an altitude of about 35,785 km (almost
the circumference of the planet!).
(a) Briefly explain why a satellite in geostationary orbit must orbit Earth in 1 sidereal day, not 1
solar day.
(b) Why must that satellite be in an equatorial orbit, rather than in some other orbit (e.g., inclined
or around the poles)?
2. In 2009, NASA launched a telescope called the Kepler Space Telescope that has discovered many
planets around other stars. One of the many new discoveries of Kepler (the satellite, not the 15th
century scientist!) was planets that orbit around two stars at the same time. Let’s figure out the orbit
of the two stars of Kepler-34, called A and B. The two stars together are called a binary. Kepler-34
is shown in the figure. This figure shows a top-down view of the orbits and the sizes of the objects
in the top portion aren’t to scale. The cross shows the center of mass of this system. (From Welsh
et al. 2012)
(a) Assume that star A has a mass of 1 solar mass and star B also has a mass of 1 solar mass. The
semi-major axis is 0.23 and the eccentricity is 0.53. What is the orbital period of the stellar
A-B binary in days? Ignore the (much less massive) planet and focus on the orbit of the binary.
(b) Now let’s consider the orbit of the planet, called “b” (shown at the bottom of the figure). As
you can see, the planet orbits some distance away from the stars. Therefore, it is an okay
approximation to pretend like the stellar binary is a like a single star with a mass that is the
sum of the masses of stars A and B and that “b” is moving on a regular orbit. Using this
approximation and that the mass of “b” is very small compared to the stars’ mass, calculate
the semi-major axis in AU of the planet’s orbit given a period of 269 days. (It should match
pretty close what is shown in the figure.)
3. The element Helium is named after “Helios” or Sun because it was first discovered through an
unknown spectral line in the solar atmosphere. (Only decades later did they find it on Earth! How
cool is that! Go science!)
(a) Notice the bright yellow emission line from Helium in the image of the spectrum above (from
Wikipedia, so it must be correct). Approximately what is its wavelength? What is its frequency
(in Hz) and energy (in eV)? Note that 1 eV = 1.6 × 10−19 Joules.
(b) If the excited Helium electron that emits a yellow photon in this line starts with a potential
energy of 8 eV, what is the potential energy of the electron afterwards? Assume that the
emission of a yellow photon is allowed by the laws of quantum mechanics. Also don’t worry
about the other electron.
(c) Consider a Helium ion with 1 electron in the ground state. It turns out that it would take about
28 eV of energy to remove this electron through ionization. Blue photons have an energy of
about 4 eV. So, I think this electron can emit about 7 blue photons. Do you agree? Justify
your response with 1-2 sentences.
4. Online, there are many physics applets that help teach principle through interactive investigation.
Many of these are hosted on the PhET website. Let’s figure out why no stars look green. Go to:
https://phet.colorado.edu/en/simulations/blackbody-spectrum Stars are (nearly perfect) blackbodies and their color is represented in the upper right hand portion of this applet.
(a) Write down a random number between 0 and 20, let’s call it n. Add 540 to this number, this
will be the wavelength that you will work with. Use Wien’s Law to find the temperature of a
blackbody that peaks in the green at that wavelength (i.e., 540 + n nm).
(b) Now, in the applet dial up the temperature that you obtained in part (a) by 100K. At this
temperature, what color does the star look like? Roughly how much of red, green, and blue are
there?
(c) Can you find a temperature where the star looks green? Why or why not? (2 sentences)
5. The figure below shows a spacecraft (in the middle) and five planets (A-E). The motion of the
spacecraft is indicated by the arrow. The spacecraft is continuously broadcasting a radio signal in
all directions.
(a) Which planets will receive a radio signal that is redshifted, assuming that all the planets are
fixed in their location. Explain your reasoning in a sentence or two.
(b) Now, let all these planets have their own orbital motion with velocities that are all equal and all
smaller than the spacecraft’s velocity. Remember that the space probe is broadcasting equally
in all directions and its velocity is shown by the arrow. Let A be moving downward, B upward,
C to the left, D down and to the left, and E up and to the right, again, all with equal velocities.
Answer the question from part (a) again with these variations. [Remember that Doppler shift
depends only on the relative radial velocity.]
6. Assume that we have a distant star-planet system with a single planet in it. The star has properties
that are identical to those of our Sun and the planet has properties that are identical to those of
Earth (this is called an “Earth analog” or “Earth twin”). The masses and orbits are exactly the
same as Earth’s and the Sun’s, except you can assume a perfectly circular orbit for the planet.
(a) Calculate the equation for the orbital velocity of a planet on a circular orbit. To do this use
the equation for average speed: distance = rate × time. In a circular orbit, the speed (not the
velocity!) is always the same, so can use a time that is one full orbital period (i.e., 1 year).
What is the distance that a planet on a circular orbit traverses in this time? Calculate the speed
of the Earth twin in meters/second.
(b) Although we say that the Earth orbits the Sun, in reality they are both orbiting the center of
mass. It turns out that the equation for the size of the Sun’s orbit around the center of mass is
to take the full semi-major axis (1 AU) and multiply it by MEarth /(MSun + MEarth ), sometimes
known as the fulcrum equation. What is the semi-major axis of the Sun’s orbit around the
center of mass in km?
(c) What is the orbital speed of the Sun’s orbit in centimeters/second? Use the same method as in
part A, but now for the Sun, which traverses a much smaller distance. What animal moves at
this speed? Sailfish, Cheetah, Person, Sloth, Snail, something else?
(d) As we’ll discuss in more detail in class, one of the most prominent methods for detecting exoplanets uses the Doppler shift of light due to the motion of stars in their orbits. The Doppler
shift equation for light is v/c = ∆λ/λ, where v is the radial velocity, c is the speed of light, ∆λ
is the shift in wavelength, and λ is the wavelength of light. Use a wavelength of 550 nm and
assume you are looked at the system at a time when the radial velocity is equal to the speed
you calculated above. What is ∆λ in nm? (Use scientific notation.)
(e) If the star was bright enough, could you see this wavelength shift as a color change with your
naked eye?
7. Let’s look at light-collecting area and see if there is another factor that can figure in to how deep an
observation can go.
(a) How much greater is the light-collecting area of one of the 10-meter Keck telescopes than our
0.82m telescope on the roof?
(b) Do the same comparison for the 5m Hale telescope on Mt. Palomar.
(c) The Hubble Space Telescope is only a 2.4m diameter telescope, but its images can be fantastically
deep – far deeper, in fact, than any Earth-based telescope, including the Keck telescopes that
have a dimeter of about four times as big. Writing 1-2 sentences, justify why is that?
8. Telescopes often have small fields of view. In particular, the wide-field camera of the Hubble Space
Telescope has a field of view that is roughly square and is about 0.1 degrees on a side.
(a) What is the angular area of the camera’s field of view in square degrees?
(b) The angular area of the entire sky is about 41,200 square degrees. How many pictures would
HST have to take to obtain a complete picture of the entire sky?
9. Suppose you were looking at our own solar system from a distance of 8 parsecs, at a wavelength of
510 nm.
(a) What angular resolution (in arcseconds) would you need to see the Sun and Jupiter as distinct
points of light?
(b) What about Sun and Earth?
(c) Can this be done by the Hubble Space Telescope?
(d) Technological advances are now making it possible to link visible-light telescopes so that they
can achieve the same angular resolution as a single telescope over 300m in size. What is the
angular resolution of such a system of telescopes?
Purchase answer to see full
attachment
