Understanding Radial Velocity in Exoplanet Characterization

Averie Kammerdiener 00 3 Characterizing Extrasolar Planets 1: Reading the graphs Part 1: Radial Velocity A star with a planet will orbit the common center of mass, causing the star to wiggle. If Earth is in the same plane as the other planet, the star will move toward and away from us, which we can detect using the doppler shift. 1. If Earth is perpendicular to the plane of the planet’s orbit (that is, if we are looking straight down on the system, like “view from above” in the introduction), will the star move toward and away from us? Why or why not? (you may want to use a sketch in your explanation). N o, the star will not be moving away or towards us because the Doppler shift only works to measure face on not side-to-side. Thus, we only see forward and backward motion when we look at it face-on. Taking a single stellar spectrum, astronomers can determine one instantaneous measurement of the star’s velocity. If they make many observations, they can make a graph of the star’s velocity, like this one: M = 1 M Jup , M star = 1 M Sun , a = 1 AU, P = 1 year, e = 0 2. What is the period of the planet’s orbit? 1 year_ ______ Page 1 of 12

3. What is the amplitude?__ Amplitude= (ymax-ymin)/2= (30+30)/2= 30 m/s_ __ 4. Describe the shape of the velocity curve: The shape of the velocity curve is much like a wave graph because the planet’s motion is going back and forth. It is very sinosidal. Part 2: Effects of the orbit and planet properties on the RV graph The next section will explore some of the things that have the biggest effect on the shape of the RV graph. Be sure to pay attention to both the shape of the curve and the amount of scatter of the data. Effects of eccentricity Here are 3 graphs of similar systems (1 M sun star and 1 M jupiter planet orbiting at 1 AU), but with eccentricities of 0.2, 0.5 and 0.8. All graphs are from a longitude of 0º, which means Earth is aligned with the planet’s minor axis (see sketches, to the right of the graphs.) e=0.2 (nearly circular) e=0.5 (similar to the sketch above) e=0.8 (fairly flat) Page 2 of 12

How does eccentricity affect the velocity curve? The higher the eccentricity, the higher the relative velocity amplitudes is which is due to Keplers 3rd law. Additionally, the higher eccentricity allows the planet to be nearer to the star for a shorter period of time, and we see a slingshot effect as the planet quickly speeds up for a short period of time and then slows down. Effects of semi-major axis Here are 3 more graphs of similar systems (1 Msun star and 1 M jup planet), with an eccentricity of 0.5 and semi-major axes of 0.5AU, 2 AU, and 10 AU. a = 0.5AU; Period = 129 days a = 2AU; Period = 1030 days a = 10AU; Period = 11500 days Page 3 of 12

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How does the semi-major axis affect the velocity curve? As the semimajor axis grows larger, the period will also grow larger as the data becomes less consistent. As the distance grows, wider orbitted planets cause the star to wobble less, making it harder to determine the stars movement. Newton’s laws of gravitation explain this as the distance between two objects increases, their gravitational impact on each other decreases. Effects of planet mass Here are 3 more graphs of similar systems (1 M sun star with a planet orbiting at 1 AU), with an eccentricity of 0.4 but this time, the mass of the planet is 0.1 MJup, 2 MJup and 10 MJup. M = 0.1 MJup M = 2 MJup Page 4 of 12

M = 10 MJup How does the planet mass affect the velocity curve? The large the semi-axis, the longer the period is as well as the more scattered and inconsistent the data. With more distance, a wider orbitted planet causes the star to wobble less, making it harder to determine the star’s movement. Newton’s laws of gravitation explains this as the distance between two objects increases, their gravitational impact on each other decreases. Part 3: Transits Below is the light curve of the same system we started with above: M = 1 M Jup , M star = 1 M Sun , a = 1 AU, P = 1 year, e = 0. Page 5 of 12

12. What was the period from part 1? ___ 1 year _____ 13. How long does the transit take from beginning to end? __ 14.3 hours as said on the graph ___________ 14. what fraction of the total orbit is this? __ 14.3 hrs = 0.6 days, so .6 days / 365 days, or .1643% of a total year_________ 15. Does it go from maximum to minimum brightness quickly or slowly compared to the entire transit? __quickly________ 16. Does it take the same amount of time to go from minimum to maximum as it did to go from maximum to minimum brightness? ___YES !_______ Part 4: Effects of the orbit and planet properties on the transit graph The next section will explore some of the things that have the biggest effect on the shape of the Transit graph. Be sure to pay attention to both the shape of the curve and the amount of scatter of the data. When most groups are done, you GSI may call for a group discussion of your observations of both the RV and transit graphs. Page 6 of 12

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Effects of Inclination Below are 4 light curves for the same system but with inclinations of 89.9º, 89.8º, and 89.75º. i = 89.9º i = 89.8º i = 89.75º 17. how does the inclination affect the light curve? There is a direct relationship between angle of inclination and eclipse time. As the inclination angle increases, the time of eclipse also increases. The light curve’s graph shape becomes less or more fluid because of this. Page 7 of 12

Effects of planet mass Below are 3 curves for a planet orbiting a 1 solar mass star at 1 AU and an inclination of 0º. The planets’ masses are 0.5, 5, and 25 Jupiter masses but the radius is held to 1 jupiter radius. M = 0.5 M_Jup M = 5 M_Jup M = 25 M_Jup Page 8 of 12

18. How does the planet’s mass affect the light curve Since the depth of the transit dip is the same between the greatly varied masses, it can be said that a planet’s mass does not affect the light curve. The depth of the dip in a transit conversation determines the mass of a planet, but there was no change in this dip as mass changed. Effects of planet radius Below are 3 curves for a 1 Jupiter mass planet orbiting a 1 solar mass star at 1 AU and an inclination of 0º. The planets’ radii are 2, 0.5, and 0.2 Jupiter radii. radius = 2R_Jupiter radius = 0.5R_Jupiter radius = 0.2R_Jupiter 19. How does the planet’s radius affect the light curve? There is a direct relationship between a planet’s radius to the normal flux, length of the transit Page 9 of 12

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eclipse, and the consistency of the data. A planet has more time for the eclipse to transmit the face of the planet with stars with different radii. Page 10 of 12

Concluding Questions: 1. What happens to the transit graph if the inclination is more than 1º away from 90º? If the inclination is more than 1º away from 90º, the transit graph would probably not be able to detect a dip that is noticeable enough because a dip on the transit graph represents the amount of light blocked by the planet that is orbiting. Thus if the inclination is greater than 90 degrees, the eclipse time is much shorter in length. 2. Describe a planet that is easy to detect in terms of the quantities in this lab (mass & radius of the planet, orbital inclination, eccentricity, period, and semi-major axis.) For this lab, a planet with a large eccentric orbit, large radius, orbital inclination near 90 degrees, large mass, a short orbit period, and a smaller semi-major axis, or close to the star would be easier to detect. 3. If there is an alien civilization on a planet around one of the nearby stars, and they have the same level of technology we have: a. Would Jupiter be easy or difficult to detect? Use your answer to the previous question to support your answer. Jupiter would be difficult to detect because its far away from the sun (so a long orbit and large semi-major axis), causes lots of wobble of the sun, and there is a low eccentricity. The planet does have a large mass but these other factors would cancel out. b. Would Earth be easy or difficult to detect? Justify your answer. Use your answer to the previous question to support your answer. Earth is closer to the Sun and has an appropriate orbital period. However, it does not have enough mass and there is also low eccentricity in its orbit, thus Earth would still be difficult to detect because its mass and eccentricity are not quite right. Page 11 of 12

Last updated by SAM. Original: SAM, Bell Copyright Regents of the University of Michigan. Page 12 of 12

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