P112L08_Extrasolar_Planets_v20150831 2

Extrasolar Planets Introduction The abundance and characteristics of planets around other stars has long been a topic of great interest in astronomy. It has direct implications on our understanding of our own solar system and planet. It is also directly linked with the broader philosophical question “are we alone in the universe?”. If other stars commonly have planets, this would greatly increase the likelihood of us discovering life elsewhere in the universe. DETECTING EXTRASOLAR PLANETS Direct Detection Planets around other stars are difficult to observe directly. Even with the 2.4 m Hubble Space Telescope and many ground-based telescopes with apertures over 8 m, extrasolar planets are simply to close to their stars. These distances are minuscule compared to the distances between stars, and a star can be billions times as bright as the reflected light from any extrasolar planet orbiting it. Indirect Detection Astronomers have had much better success at indirectly detecting extrasolar planets. Instead of detecting the planet, they try to infer its existence by observing the effects that it has on its parent star. These efforts take three major forms: astrometric, radial velocity and transit methods. We will be concerning ourselves with the first two methods in this lab. 1) Astrometric Methods In a system planet-star, both the star and the planet actually orbit a fixed point in space called the center of mass. By carefully observing the motion of the star (as it wobbles around the center of mass) astronomers can in principle obtain information about the companion planet. The center of mass is clearly a very important concept when discussing extrasolar planets. In general, the center of mass of two objects can be thought of as a “balancing point” between them: • It is always found on the line between the two objects. • If the two masses are equal the center of mass is exactly halfway between them. • If they have different masses, the center of mass is closer to the more massive object. (If one is considerably more massive, the center of mass may even be inside of it.) Phys 112 -- Extrasolar Planets 1/9 Your Name: Your Lab Section:

Open the Center of Mass Simulator , shown in Figure 1 . Experiment with the masses of the two objects and notice the location of the center of mass. Repeat after unchecking “ keep CM fixed ”. The location of the center of mass along the line between the objects (the distances r 1 and r 2 ) clearly depend on the masses m 1 ( object 1 mass ) and m 2 ( object 2 mass ) of the two objects and on their separation. Procedure Question 1: Make sure “ keep CM fixed ” is checked and maintain “ separation ” at 10.0 for the time being. Using the masses in the first two rows of Table 1 determine a simple relationship between the masses m 1 and m 2 and the distances r 1 and r 2. Use it to complete the rest of the entries in the table. Question 2: What is the relationship between the masses m 1 and m 2 and the distances r 1 and r 2 ? As important as the Center of Mass is, its location depends on the orbital distance a of the two orbiting masses. As Figure 2 shows, the orbital distance a and distances r 1 and r 2 of each mass to the Center of Mass satisfy a = r 1 + r 2 . This identity together with the relationship that you just derived easily leads to To try to understand the consequences of this expression, let’s apply it to Jupiter and the Sun: Phys 112 -- Extrasolar Planets 2/9 m 1 m 2 m 1 / m 2 r 1 r 2 r 2 / r 1 7.0 3.0 9.0 3.0 5.0 5.0 8.0 4 1.5 4.0 Table 1 Masses and distances to CM (Question 1) Figure 1 Center of Mass simulator m 1 m 2 × CM r 1 r 2 a Figure 2 Orbital distance and distances to the Center of Mass r 1 = a 1 + m 1 / m 2 . 713 3 74/3 3 2.5 7. 5 3 3.O 3.0 I I 2.0 4 2. O 8.0 6.0 4.8 6. O 1.5 = En

Question 3: What do you notice about the position of the Sun? Does this agree with what you expect based on the center of mass? Question 4: Now open the Hammer Thrower Comparison animation. Does this agree with your findings? Is our Sun stationary in space? Explain. Astrometric techniques have been successfully used to identify binary stars for many years. Considerable effort has been devoted to trying to detect exoplanetary systems with the same technique, but no extrasolar planets have been discovered to date. The wobbles are just too small to be detectable at the great distances of stars. Question 5: Finally, open the Effect of Planets on the Sun simulation shown in Figure 4 . The animation allows you to see the effect on the sun of the motion of the planets in our solar system. Experiment with a) Jupiter alone, b) Jupiter & Saturn, and c) Earth alone before trying other combinations. What kind of planet is more likely to be detected with this technique? Explain. Phys 112 -- Extrasolar Planets 4/9 Figure 4 Effect of Planets on the Sun (Question 5) If Mars orbits around center of masses, the the Sun also orbits around the center of masses. However, since the mass of the Sun is much bigger, the radius of the Sun's orbital is much smaller Yes, it agrees with my findings. No, our Sun is not stationary in space. Its motion depends on every planet that orbits around "the Sun" a) With Jupiter, we can see the sun orbits the cil of the Sun and Jupiter. b) With Jupiter and Saturn, we can see that Saturn alters the motion of the Sun compared to Jupiter's. a) With Earth alone, we can hardly see the Sun's motion because the Earth is two light. => Detect heavy planets.

2) Radial Velocity Methods As the star moves about the Center of Mass, its velocity changes as seen from Earth. This produces small changes to the wavelength of the light that we receive from the star. This is called the Doppler effect. If you have ever heard the changing pitch of a siren as it passed by, you have experienced the Doppler Effect first-hand. It is the change in frequency (and wavelength) due to relative motion of the source and observer. It occurs (to both sound and light) when either the source, observer, or both are moving. In an exoplanetary system, like the one pictured in Figure 5 , an observer on Earth measuring the spectra from the star would see: • A shorter wavelength (blueshift) when the star is moving toward Earth. • A longer wavelength (redshift) when the star is moving away from Earth. • No shift when the star is moving perpendicularly to the line of sight. The magnitude of the shifts can be used to determine the velocity of the star as seen from Earth thanks to Doppler’s equation where v radial is the velocity of the star along the line of sight (how fast is the star moving toward us or away from us), c is the speed of light, λ shift is the observed wavelength from the star and λ rest is the wavelength of the light if there’s no relative motion between emitter and receiver. Figure 6 shows the end result of many velocity measurements using Doppler’s equation: a velocity curve for the star. The figure makes clear that by using the Doppler effect • A blueshift ( λ shift < λ rest ) yields negative radial velocities . • A redshift ( λ shift > λ rest ) yields positive radial velocities. • No shift means the star is moving perpendicularly (no radial velocity) to the line of sight. Phys 112 -- Extrasolar Planets 5/9 v radial c = l shift - l rest l rest Figure 5 Extrasolar system seen from Earth Top: blue shift. Bottom: red shift To Earth × Star Light from Star To Earth × Exoplanet Star Light from Star Figure 6 Radial velocity curve of a star from Doppler shifts Radial velocity To Earth × Light from Star

• Close your windows. Repeat the procedure with the spectrum of each of the next 10 days. • Calculate the corresponding shifts Δλ 1 = λ 1 − λ 1 rest and Δλ 2 = λ 2 − λ 2 rest and complete the corresponding columns in Table 3 . Remember that λ 1 rest = 588.995 nm and λ 2 rest = 589.592 nm. • For each wavelength, use Doppler's equation to calculate the radial velocity of the star on each day, For the speed of light you will use c = 3 × 10 5 Km/s. Add your results to Table 3 . • Calculate the average of both velocities for each day and complete the last column. You will use this average velocity as the radial velocity of the star in the next questions, to improve accuracy. Question 7: Open the Velocity Curve program. Enter the radial velocities (the averages) for each day in the input fields and notice the pattern that your data (red dots) takes. Question 8: You will now find the velocity curve of the star. Play with the four sliders in the Velocity Curve simulation until you find the best curve that fits your data. The parameters that determine this curve represent important information about the motion of the star. Phys 112 -- Extrasolar Planets 7/9 Day λ 1 (nm) Δλ 1 (nm) v radial 1 (in Km/s) λ 2 (nm) Δλ 2 (nm) v radial 2 (in Km/s) v radial (in Km/s) 0 1 2 3 4 5 6 7 8 9 10 Table 3 Star’s radial velocity from wavelength shifts (Question 6) Enter the radial velocities H in Km ê s L Day 1 0 Day 2 0 Day 3 0 Day 4 0 Day 5 0 Day 6 0 Day 7 0 Day 8 0 Day 9 0 Day 10 0 Day 11 0 Find the Velocity Curve of the star V orbit Period V CM offset Reset 0 1 2 3 4 5 6 7 8 9 10 - 30 - 20 - 10 0 10 20 30 days Km ê s V orbit = 0.00 Km ê s Period = 5.00 days V CM = 0.00 Km ê s Figure 9 Finding the velocity curve from your data (Questions 7 and 8) v radial c = l shift - l rest l rest . 589.040.045 22.92589.640.04824.4223.67 589.050.05528.01 589.65 0.058 29.51 28.76 589056.05528.01589.65 0.058 29.5128.76 589.030.03517.83589.63 0.038 19.33 18.58 589.000.0052.55589.900.0084.07 3.31 588.98 - 0.015-7.54589.58-0.012 - 6.1 - 6.87 588.96 - 8.035 - 17.83589.56 - 0.032- 16.28 - 17.85 588.96-0.038 - 17.83589.53-0.032 16.28- 17.05 588.98 - 0.025 - 7.64589.58 0.012- 6.11 - 6.87 589.888.0052.55359.680.8084.073.31 589.030.03577.83 589.630.038 19.34 18.58

The period of the orbit (how much time it takes for the star and companion to complete one orbit) corresponds in the graph to the time interval after which the graph repeats itself, as shown in the first panel in Figure 10. The orbital velocity of the star (how fast it orbits the center of mass 1 ) in the graph can be obtained from the maximum and minimum values of the curve, as in Figure 10 . Those are the radial velocities observed from Earth when the star moves exactly along the line of sight, away and toward Earth. Finally, if the system as a whole moves with respect to Earth (that is, if the Center of Mass itself has a velocity v CM describing how fast is the system moving away or toward us), your data will always have that extra velocity on top of the orbital velocities. This has the effect of making the velocity curve a bit asymmetric along the horizontal axis, shifting the entire graph, up or down, by exactly v CM . This is depicted in Figure 11. Mathematically, we are describing the velocity curve by a function of the form where each term has the following meaning • T is the period of the orbit • v CM is the vertical offset of our periodic data. It represents the part of the velocity of the star not related to its rotation about the Center of Mass. Instead it represents the velocity of the system as a whole (or of its Center of Mass) with respect to Earth. Phys 112 -- Extrasolar Planets 8/9 1 We will assume that the orbit is circular which means the star maintains a constant orbiting speed around the Center of Mass . In general these orbits are elliptical, with the star’s speed changing throughout the orbit. v radial = v CM + v orbit sin ⇣ 2 p t T + f ⌘ Figure 10 Extracting the period and orbital velocity from the velocity curve v orbit -v orbit Radial velocity Period Period Figure 11 Extracting the velocity of the system from the velocity curve To Earth × Light from Star v CM v CM {