FINDING PLANCK FUNCTION

1. In this lab we will be looking at the speed with which planets orbit a star as well as how we can detect planets orbiting distant stars by measuring the back reaction
of the star due to the orbiting planet. We will use two different interactive elements to do this lab: the Orbital Velocity lab and the Extrasolar Planets lab. To start with, we will use the Orbital Velocity lab. Let us consider a planet orbiting at the distance that the Earth orbits the sun. Set the orbital radius to 1.00 AU and the mass of the star to 0.5 solar masses. Note the speed of the orbiting planet.

What must the mass of the star be in order to double the speed of the planet? (Points : 2)
1.0 solar mass
1.5 solar masses
2.0 solar masses
4.0 solar masses



Question 2. 2. Given your answer from the first question, answer the following items.
(a) How many times larger does the mass of the star have to be in order to double the orbital speed of the planet?
(b) Let's see if this value changes when you change the orbital distance. Set the mass of the star to either 0.5 or 1.0 solar masses. Set the orbital distance to some value other than 1.0 AU. What is the orbital speed for the parameters you chose?
(c) Identify the mass of the star that is needed in order to double the orbital speed at this orbital distance. How many times larger did the mass of the star need to be in order to make the orbital speed twice as fast? (Points : 4)




Question 3. 3. The results from the previous question confirm that the orbital speed scales with the square root of the mass (as indicated by the displayed formula). This is the same relationship that we studied last week. Now let's turn our attention to the orbital distance of the planet. This, too, affects how fast the planet orbits. Here the orbital distance is also under the square root sign (or radical), but now in the denominator.
(a) Choose any mass for the star and set the orbital distance to any number between 0.3 and 0.4 AU. Note the orbital speed of the planet. Keeping the mass of the star fixed, what is the orbital distance needed to reduce the speed of the orbit by half?
(b) How many times larger is the second orbital distance than the first?
(c) Select a different mass and again an orbital distance between 0.3 and 0.4. Note the orbital speed. What is the orbital distance needed to reduce this orbital speed by half?
(d) How many times larger is the second orbital distance than the first? (Points : 6)




Question 4. 4. We will now turn to the Extra Solar Planets interactive. A common method to find extrasolar planets (planets that orbit distant stars) is called the radial velocity technique. It involves measuring variations in the velocity of a star along the line of sight due to the presence of an orbiting planet. (The word radial
comes from the coordinate system that is used and refers to the radius of a sphere centered on the observer.) In a planet–star system, it turns out that both the planet and the star orbit their common center of mass. The period of the orbit is the same for both bodies, but the distance at which they orbit is different; the star has a much, much smaller orbit. As a consequence, the star also has a much smaller orbital velocity (less distance to travel in the same amount of time means a slower velocity). A good approximation for a planet–star system is that the velocity of the star is equal to the velocity of the planet times the ratio of the mass of the planet to the mass of the star:



So, if a planet is 1/1000 of the mass of its host star, then the star's orbital velocity will be 1/1000 of the orbital velocity of the planet; if the planet is 1/100,000 of the mass of the star, the star’s velocity will be 1/100,000 of the velocity of the planet. Note: This gives the velocity of the star in terms of the velocity of the planet, not in terms of gravity. As we have already seen, the ultimate cause of this velocity (gravity) has a different dependence on the mass of the objects. This relationship assumes that you already have the velocity of one of the two objects.

Using the Extra Solar Planets interactive, do the following.

i. Select a Hot Jupiter.
ii. Set the eccentricity to 0.0.
iii. Set the semimajor axis to 0.6AU.
iv. Set the mass of the star to 1 solar mass.
v. Set the inclination to 90 degrees.
vi. Set the Graph Scale to 125% (this is the easiest scale for this question).

(a) What is, approximately, the peak-to-peak amplitude for the velocity of the star in this case? That is, what is the difference (in m/s) between the peaks and the troughs in the radial velocity signal?
(b) Set the eccentricity to a value larger than 0.2. What is the peak-to-peak amplitude for this eccentricity?
(c) Given these two measurements, does the "size" of the radial velocity signal vary significantly with the eccentricity of the orbit? (Significant would be a change that is larger than about 20% on this graph.) (Points : 4)




Question 5. 5. One issue that arises from detecting planets with this technique is that there is a degeneracy between the inclination of the orbit and the mass of the planet. In the introduction to the previous question there is a relationship between the velocity of the planet and the velocity of the star. This relationship shows that if the mass of the planet were reduced by half, the velocity of the star would also be reduced by half. If the mass of the planet were reduced to 1/3 of its original mass, then the velocity of the star would also be reduced to 1/3 of its original value.
(a) Reset the parameters of this simulation back to those for the first part of the previous question (zero eccentricity, 90 degree inclination, 1 solar mass, and semimajor axis to 0.6 AU)); note the size of the signal in this scenario. Now adjust the inclination of the planet until the radial velocity signal is equal to 1/2 of the original value. What inclination angle gives you a signal that is 1/2 of the original signal when the inclination is set to 90 degrees? This shows that once a planet is discovered, there is still confusion regarding the mass of the planet and the inclination. We don't know either exactly, only some combination.
(b) Reset the inclination to 90 degrees and select any combination of parameters using the sliders. What is the size of the signal that this system produces?
(c) What inclination do you need in order to reduce this signal to 1/2 of the original?
(d) Did you get the same answer for parts (a) and (c)? (Points : 4)