Lab 15 Exercise Question 1: (5 points) Describe the positions on the star’s orbi

Question

Lab 15 Exercise Question 1: (5 points) Describe the positions on the star’s orbit with the letters corresponding to the labeled positions of the radial velocity curve. Remember, the radial velocity is positive when the star is moving away from the Earth and negative when the star is moving towards the Earth. Note that the diagram won’t be included when you paste into the submit box.

Question 2: (5 points) Describe the positions on the planet’s orbit with the letters corresponding to the labeled positions of the radial velocity curve. Hint: the radial velocity in the plot is still that of the star, so for each of the planet positions, determine where the star would be and in which direction it would be moving. The next few questions refer to the Exoplanet Radial Velocity Simulator. Select the preset labeled Option A and click set. This will configure a system with the following parameters – inclination: 90º, longitude: 0º, star mass: 1.00 MSun, planet mass: 1.00 MJup, semimajor axis: 1.00 AU, eccentricity: 0 (effectively Jupiter in the Earth’s orbit). Question 3: (5 points) Describe the radial velocity curve. What is its shape? What is its amplitude? What is the orbital period? Increase the planet mass to 2.0 MJup and note the effect on the system. Now, increase the planet mass to 3.0 MJup and note the effect on the system. Question 4: (5 points) In general, how does the amplitude of the radial velocity curve change when the mass of the planet increases? Does the shape change? Explain. Return the simulator to the values of Option A. Increase the mass of the star to 1.2 MSun and note the effect on the system. Now, increase the star mass to 1.4 MSun and note the effect on the system. Question 5: (5 points) How is the amplitude of the radial velocity curve affected by increasing the star mass? Explain. Return the simulator to the values of Option A. Question 6: (5 points) How is the amplitude of the radial velocity curve affected by decreasing the semimajor axis of the planet’s orbit? How is the period of the system affected? Explain. Return the simulator to the values of Option A so you can explore the effects of system orientation. It is advantageous to check show multiple views. Note the appearance of the system in the earth view panel for an inclination of 90º. Decrease the inclination to 75º and note the effect on the system. Continue decreasing inclination to 60º and then to 45º. Question 7: (5 points) In general, how does decreasing the orbital inclination affect the amplitude and shape of the radial velocity curve? Explain. Question 8: (5 points) Assuming that systems with greater amplitude are easier to observe, are you more likely to observe a system with an inclination near 0° or 90°? Explain. Return the simulator to Option A. Note the value of the radial velocity curve amplitude. Increase the mass of the planet to 2 MJup and decrease the inclination to 30°. What is the value of the radial velocity curve amplitude? Can you find other values of inclination and planet mass that yield the same amplitude? Question 9: (5 points) Suppose the amplitude of the radial velocity curve is known, but the inclination of the system is not. Is there enough information to determine the mass of the planet? Question 10: (5 points) Typically, astronomers don’t know the inclination of an exoplanet system. What can astronomers say about a planet's mass even if the inclination is not known? Explain. Select the preset labeled Option B and click set. This will configure a system with the following parameters – inclination: 90º, longitude: 0º, star mass: 1.00 MSun, planet mass: 1.00 MJup, semimajor axis: 1.00 AU, eccentricity: 0.4. Thus, all parameters are identical to the system used earlier except eccentricity. Question 11: (5 points) Does changing the longitude affect the curve in the example above? Question 12: (5 points) Describe what the longitude parameter means. Does longitude matter if the orbit is circular? Select the preset labeled HD 39091 b and click set. Note that the radial velocity curve has a sharp peak. Question 13: (5 points) Determine the exact phase at which the maximum radial velocity occurs for HD 39091 b. Is this at perihelion? Does the minimum radial velocity occur at aphelion? Explain. (Hint: Using the show multiple views option may help you.) Select the preset labeled Option A and click set once again. Check show simulated measurements. Set the noise to 3 m/s and the number of observations to 50. Question 14: (5 points) The best ground-based radial velocity measurements have an uncertainty (noise) of about 3 m/s. Do you believe that the theoretical curve could be determined from the measurements in this case? (Advice: check and uncheck the show theoretical curve checkbox and ask yourself whether the curve could reasonably be inferred from the measurements.) Explain. Select the preset labeled Option C and click set. This preset effectively places the planet Neptune (0.05 MJup) in the Earth’s orbit. Question 15: (5 points) Do you believe the theoretical curve shown could be determined from the observations shown? Explain. Select the preset labeled Option D and click set. This preset effectively describes the Earth (0.00315 MJup at 1.0 AU). Set the noise to 1 m/s. Question 16: (5 points) Suppose the intrinsic noise in a star’s Doppler shift signal – the noise that you cannot control by building a better detector – is about 1 m/s. How likely are you to detect a planet like the Earth using the radial velocity technique? Explain. Question 17: (5 points) Use the table in Question 17 of the Student Guide to summarize the effectiveness of the radial velocity technique. What types of planets is it effective at finding? The following questions refer to the Exoplanet Transit Simulator. Select Option A and click set. This option configures the simulator for Jupiter in a circular orbit of 1 AU with an inclination of 90°. Question 18: (5 points) Determine how increasing each of the following variables would affect the depth and duration of the eclipse. (Note: the transit duration is shown underneath the flux plot.) Radius of the planet: Semimajor axis: Mass (and thus, temperature and radius) of the star: Inclination: The Kepler space probe photometrically detected extrasolar planets during transit. It’s estimated to have a photometric accuracy of 1 part in 50,000 (a noise of 0.00002). Question 19: (5 points) Select Option B and click set. This preset is very similar to the Earth in its orbit. Select show simulated measurements and set the noise to 0.00002. Do you think Kepler will be able to detect Earth-sized planets in transit? Question 20: (5 points) How long does the eclipse of an Earth-like planet take? How much time passes between eclipses? What obstacles would a ground-based mission to detect Earth-like planets face?

Explanation / Answer

Q3. Radial velocity curve: This is based on the fact that when a planet orbits around a star it is not completely stationary but moves slighty in the form of ellipses or small circles as a response to its gravitational pull. These motions can be detected in terms of wavelengths of the lights emitting from the star. The light of the star will appear redder if it moves away from the earth and bluer if moving towards the earth. These are expressed as radiation velocity on the y axis and the phase on the x axis.

On increasing the planet mass to 2 MJup the amplitude of the radial velocity increases, its inclination will decrease and the speed of the planet will also decrease. On further increasing the mass to 3 Jup the same results will further increase.

Q4. The radial velocity curve amplitude is given by K=Mp sini/Ms Vp where Vp= planetary speed, Mp= mass of planet,Ms= mass of star. SInce amplitude of radial velocity curve is directly proportional to mass of planet, therefore amplitude will increase if increase in mass of planet.

On increasing the mass of the star to 1.2 M sun and then further to 1.4 M sun the amplitude of radial velocity will decrease whereas speed of planet and inclination increases.

Q5. The amplitude of the radial velocity will increase because it is directly to the mass of the stars. Lighter the mass greater will be the amplitude signal and vice versa.

Q6. The semi amplitude of the radial velocity is given by

K=(2pi G/ Porb)1/3 Mpsini/(Ms+Mp)2/3 1/square root(1-e2)

where P orb+ orbital period, Ms=mass of star, Mp=mass of planet , i= inclination ,e= eccemntricity

The amplitude of radial velocity curve will increase on decreasing the distance of semi major axis wheras the orbital period will increase.

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