Exo-Obs Course: Photometry and Transits

I will be sharing the slides I developed for a graduate level course on Exoplanets and Observational Astronomy. This is the third completed slide deck for this course. It covers topics on fundamentals of photometry and transit measurements for exoplanets.

The photometry slides were adapted from Jay Strader's graduate course elective on observational astronomy.

Transcript

1. PHOTOMETRY + TRANSITS 1 Created by: Adina Feinstein
2. RELEVANT READINGS 2 Seager & Mallén-Ornelas (2003)
3. 3 DETECTORS SLIDES ADAPTED FROM JAY STRADER
4. Photographic Plates 4 The first permanent images of the sky were made with photographic plates. Photographic plates dominated astronomical imaging from ~1900 - 1980.
5. Photographic Plates - Limitations 5 Size: Typically large (10+ inches), but had a wide field of view.
6. Photographic Plates - Limitations 6 Emulsion: The emulsion on the plates was silver bromide, which is very light sensitive. Electrons could easily obtain energies to excite them to the “conduction band.” In this state, electrons can move freely in the medium. This could result in small spatial inaccuracies that became visible when the plates were chemically developed.
7. Photographic Plates - Limitations 7 Efficiency: Low (~ few %) Uniformity: Every plate had a slightly different response Wavelength range: Really only worked in the blue
8. Digitizing Plates 8 Many observatories (Harvard, Yerkes Observatory, Lick Observatory) are digitizing their photographic plates. Science case: Search for transients/variables on 100+ year timescales https://dasch.cfa.harvard.edu/
9. Harvard “Computers” 9 1881: First women were hired by the Harvard Observatory to study and care for the photographic plates. Helped catalog stars for the Henry Draper (HD) Catalog. “The Glass Universe: How the Ladies of the Harvard Observatory Took the Measure of the Stars” by Dava Sobel
10. Photomultipliers 10 A photoemissive device in which the absorption of a photon results in the emission of an electron (electrical signal).
11. Photomultipliers 11 The majority of photometry measurements in the 1950s-1970s used these tubes. They are very sensitive and accurate and had a high time resolution, but were limited to observing one source at a time.
12. Charge-Coupled Devices (CCDs) 12 Solid state (semiconductor) devices that are now the basis for essentially all detectors in optical/NIR (photometry + spectroscopy)
13. Charge-Coupled Devices (CCDs) 13 CCDs were invented at Bell Labs in 1969 as a solid state storage device (e.g., modern day SSDs). Their potential for imaging was quickly realized. They were first used in space in 1976 on a spy satellite. Ed Loh (MSU Professor Emeritus) was the first person to use a CCD for astronomy for his Ph.D. thesis (Princeton, 1977). https://web.pa.msu.edu/astro/faculty/index.html
14. Charge-Coupled Devices (CCDs) 14 CCDs rely on semiconductors: a material with electrical conductivity between that of a conductor and an insulator.
15. Charge-Coupled Devices (CCDs) 15 Incoming photons of sufficient energy can knock an electron above the band gap into the conduction band. E E f Valence band Conduction band Unfilled bands Filled bands Band gap
16. 16 Finding the Appropriate Semiconductor The energy carried by a photon is given by E = (hc)/λ h is the Planck constant c is the speed of light λ is the wavelength of light If the Si band gap is 1.11 eV, what wavelength does this correspond to? [in nm]
17. Metal Oxide Semiconductor (MOS) Capacitor 17 The electrons generated by incoming photons are held in place until they are ready to be transferred / read out.
18. CCD Readout 18 The rows and columns of a CCD are fundamentally different because charge can only be transferred in one direction. Sometimes CCDs have additional electronics and amplifiers that can read out part of the chip.
20. CCD Illumination 20 Traditional CCDs are face “up,” meaning they are illuminated at the gates. This is bad for UV/blue sensitivity, since photons get absorbed near the gates. Incoming light Silicon Silicon Dioxide Gates
21. CCD Illumination 21 Traditional CCDs are face “up,” meaning they are illuminated at the gates. This is bad for UV/blue sensitivity, since photons get absorbed near the gates. Instead, you can make a thin CCD and illuminate it from the back. In this configuration, the red sensitivity is bad and devices are more expensive to create. Incoming light Silicon Silicon Dioxide Gates Incoming light Thinned Silicon
22. CCDs: Quantum Efficiency 22 The fraction of photons that transverse the gates and are successfully read out by the device.
23. CCDs: Quantum Efficiency 23 Red optical/near-infrared CCDs will be back-illuminated. Modern coatings can beat thin CCDs at wavelengths > 400 nm.
24. CCDs: Dark Current 24 Normal thermal motion jostles e- into the conduction band. This is mitigated by cooling the CCD to ~ 100oC with liquid nitrogen.
25. CCDs: Charge Transfer Efficiency 25 Pixels must have their charge transferred thousands of times during each readout, so the efficiency of transfer must be >99.999%. Radiation damage can be problematic for CCDs in space.
26. CCDs: Full Well 26 A single pixel has a maximum number of e- it can collect before they diffuse (leak) into surrounding pixels. Saturation typically happens at 100k-200k e-. As a pixel approaches this limit, the response deviates from linear. You want to stop your exposure in the linear regime.
27. CCDs: Amplifier and Readout 27 As each pixel is passed to the readout electronics, it senses the accumulated charge as a voltage. To avoid negative values, a pedestal level is added so that the mean unexposed value is > 0 (typically ~100 - 1000 ADU).
28. CCDs: Read Noise 28 Measuring the per pixel voltage (and the accumulated charge) is an analog electronic process. There is a trade-off between speed and accuracy. Fast read out = noisier read out. The read noise is represented in terms of the standard deviation in e-, not ADU.
29. CCDs: Analog to Digital Conversion 29 The voltage gets turned into a digital number (DN) with an ADC (“counts” = DN = ADU). The simplest ADCs are 16-bit, which means you can represent 216 -1 = 65535 unsigned integer numbers.
30. CCDs: Inverse Gain 30 If you have 65535 as the maximum pixel value but your full well is 200,000 e-, it doesn’t make sense to have 1 e- = 1 DN. If we introduce an inverse gain, such that each DN = some number of e-, we can fully sample the full well and get the highest possible dynamic range.
31. CCDs: Undersampling Read Noise 31 Let’s say the full well is 300,000 e- and you set the gain to be ~4.6 to sample the ADC fully. If your read noise was only 3 e- rms, you would be adding noise with every ready (since 1 DN = 4.6 e-). In general, this is not too important, except at very low S/N. Always remember the physical units are e-!
32. CCDs: Binning 32 Binning increases the effective full well and decreases the relative read noise of your observations.
33. CCDs: Size 33 A typical big CCD is ~4k x 4k pixels. Assuming a pixel scale of 0.15”/pixel, what would the field-of-view of this CCD be [in arcmin]?
34. CCDs: Size 34 A typical big CCD is ~4k x 4k pixels. Assuming a pixel scale of 0.15”/pixel, what would the field-of-view of this CCD be [in arcmin]? ~ 10 arcmin
35. CCDs: Size 35 It’s hard to achieve a pixel scale finer than ~0.3”/pixel if you want to ever sample the seeing. Hence, if you want to cover an area larger than ~20’, you need to use CCD arrays.
36. CCDs: Size 36 Due to electronics, it is very difficult to put CCDs right next to each other without gaps. Thus, for almost all CCD arrays, you will see gaps between the chips.
37. 37 SIGNAL-TO- NOISE
38. Case Study: Aperture Photometry 38
39. SNR Variables 39 Parameter Definition Units R star Count rate from the star e- / seconds
40. R star 40 The count rate of the star is set by the apparent magnitude of the star and by your telescope/instrument/filter/CCD. While R star could be calculated from first principles, in reality it is measured. For example, some instruments provide expected count rates for stars of varying magnitudes. These examples can also help you calculate your exposure time.
41. SNR Variables 41 Parameter Definition Units R star Count rate from the star e- / seconds R sky Count rate from the background e- / seconds
42. R sky 42 The count rate of the background depends on the atmosphere, the moon, and by your telescope/instrument/filter/CCD. The main source of R sky is the moon. The lunar phase makes R sky bright in the blue due to scattering. There is only a modest effect at red-optical wavelengths, and nearly no effect in the near-infrared. However, during a full moon, R sky may be x100 brighter than during a new moon.
43. R sky 43 Pure sky (i.e., if we blew up the moon) is brighter in the red than in the blue. This is because there are strong O and OH emission lines at redder wavelengths.
44. SNR Variables 44 Parameter Definition Units R star Count rate from the star e- / seconds R sky Count rate from the background e- / seconds t Exposure time seconds
45. SNR Variables 45 Parameter Definition Units R star Count rate from the star e- / seconds R sky Count rate from the background e- / seconds t Exposure time seconds r Radius of the aperture pixels
46. Selecting the “Ideal” Aperture 46 If we assume the star has a Gaussian point spread function (PSF): I(r) ~ e-r^2/2σ^2 Then the count rate within an aperture of R ap is: R star (r) = ∫ 0 R_ap I(r) 2 π r dr Ignoring constants, this roughly translates to R star ~ [1 - e^(-R ap 2)]
47. Selecting the “Ideal” Aperture 47 https://wise2.ipac.caltech.edu/staff/fmasci/GaussApRadius.p df
48. Selecting the “Ideal” Aperture 48 Sometimes, “ideal” is relative to your scientific objectives. We often try many different aperture shapes, sizes, and weights, especially with space-based photometry. It depends on what you’re trying to achieve. In transit searches, we are trying to minimize the scatter in the light curve to detect planets with small transit depths. This is a different objective than, say, measuring photometric variability.
49. SNR Variables 49 Parameter Definition Units R star Count rate from the star e- / seconds R sky Count rate from the background e- / seconds t Exposure time seconds r Radius of the aperture pixels n pix The number of pixels in the aperture π ྾ r2 (assuming circular)
50. Calculating SNR 50 When we set our exposure time, we expected to collect N star = R star ྾ t N star is the number of electrons. N star will change every time you expose. R star is the count rate from the star. t is the exposure time.
51. 51 We assume N star follows a poisson distribution f(x) = (eμμx) / x! μ is the mean and variance. It is the rate at which some process occurs if the events are independent. x is a discrete integer. Calculating SNR
52. 52 Repeat calculation of the number of electrons for the sky, N sky N sky = R sky ྾ t ྾ n pix This also obeys Poisson statistics. Calculating SNR
53. 53 For broadband imaging: SNR = (R star ྾ t) / [R star ྾ t + R sky ྾ t ྾ n pix ]½ The 1/R sky dependence justifies why you want the sky to be as faint as possible. The 1/n pix suggests that you want the aperture to be as small As possible. Calculating SNR
54. 54 In the bright object case, where R star >> R sky , which is most exoplanet cases: SNR = (R star ྾ t)½ ∝ t1/2 In the faint object cases, where R star << R sky , SNR = (R star ྾ t)/(R sky ྾ t ྾ n pix )½ ∝ t1/2 Calculating SNR
55. 55 This is probably the single-most important equation to know as an observer. If you double the exposure time, your SNR only increases by ~40%. So, in the limit of faint objects, if you have a several hour exposure, adding a few more hours isn’t going to increase your SNR all that much. Calculating SNR
56. SNR Variables 56 Parameter Definition Units R star Count rate from the star e- / seconds R sky Count rate from the background e- / seconds t Exposure time seconds r Radius of the aperture pixels n pix The number of pixels in the aperture π ྾ r2 (assuming circular) G Inverses-gain e- / DN
57. SNR Variables 57 Parameter Definition Units R star Count rate from the star e- / seconds R sky Count rate from the background e- / seconds t Exposure time seconds r Radius of the aperture pixels n pix The number of pixels in the aperture π ྾ r2 (assuming circular) G Inverses-gain e- / DN D Dark current e- / pixel / second
58. Calculating SNR 58 A more generalized SNR calculation is SNR = (R star ྾ t) / [R star t + R sky t n pix + RN2n pix + D t n pix ]½ This accounts for the scatter from the read noise, RN, and dark current, D. RN only matters for spectroscopy or narrowband imaging. D only affects (near)infrared observations.
59. 59 APERTURE PHOTOMETRY
60. Aperture Photometry 60 “Ideal” aperture Sky annulus: measure and subtract off a per-pixel sky value
61. Aperture Photometry 61 To formalize this, the flux is: To convert to magnitude:
62. Calculating Magnitude 62 In order to actually calculate the magnitude of a source, you need to: 1. Find the center of your object 2. Select an aperture (shape + size) 3. Measure the sky background 4. Optional: Iterate on the selected aperture There are lots of pre-existing routines to do this, but you should still know the fundamentals of what’s going on behind the scenes.
63. Step 1: Identifying your Target 63 Sum along column Sum along row x ρ x
64. Step 1: Identifying your Target 64 x ρ x From here, you can either do a weighted average by intensity:
65. Step 1: Identifying your Target 65 x ρ x Or you can fit a model to the marginal sum: x ρ x
66. Step 1: Identifying your Target 66 Note that this technique to identify sources will not work will for faint sources because everything will be noisier. Instead, what routines like daofind do, is fit a PSF to every pixel or fraction thereof. This will enhance the stars while suppressing cosmic rays or resolved galaxies.
67. Step 2: Selecting an Aperture 67 The ideal answer is ~⅔ to 1 FWHM, but slightly large is generally better because of issues with mis-centering the PSF, variations, etc.
68. Step 3: Measuring the Sky 68 Sky annulus Let’s say that you have a sky annulus around your star. What is the best statistic to measure the sky?
69. Step 2: Selecting an Aperture 69 From the ground, we typically use a circular annulus. In space, you can explore using apertures of different sizes and weights. This is because the pixel scale of space-based facilities are (mostly) larger than ground-based images. Without being able to find the center of the target, we sometimes have to get creative.
70. Step 3: Measuring the Sky 70
71. Step 3: Measuring the Sky 71 Because all real deviations from the sky background are positive, the mode is the best estimate of the sky. mode = 3*median - 2*mean
72. Step 3: Measuring the Sky 72 Light from the star can extend to very large radii – never expect to get “all” of the starlight into your ideal aperture. Your inner sky radius needs to be large enough that the star is contributing basically nothing. The typical distance between the central aperture and the sky annulus is ~15-20-ish pixels. Your annulus should be wide enough that the sky is well measured. A typical annulus width is ~5-10-ish pixels.
73. (Optional) Step 4: Iterate on Aperture Selection 73 You only need to iterate if you want to get most of the light from our source. The standard approach is called “curve of grown” analysis to understand how to correct your aperture size. Technique: You measure the maximal S/N magnitude in a tiny aperture. Then, you derive a correction to a big aperture using a few bright, isolated stars.
74. (Optional) Step 4: Iterate on Aperture Selection 74 You will measure the differences between successively large apertures until things get too noisy.
75. (Optional) Step 4: Iterate on Aperture Selection 75 You can fit a function to these measurements to get the asymptotic correction that can be applied to all stars.
76. (Optional) Step 4: Iterate on Aperture Selection 76 This technique only works for objects with the same profile (e.g., stars and not galaxies)
77. (Optional) Step 4: Iterate on Aperture Selection 77 It may be worth doing this technique if you are calibrating your data using separate standard star fields obtained with the same instrument/filters and you have enough bright, unsaturated stars. This is especially true if your seeing is variable, such that the ideal aperture varies significantly between images.
78. Photometry Cookbook Recap 78 1. Detect your object using a pre-existing algorithm, like daofind, or by eye 2. Determine your aperture size 3. Pick your sky annulus and measure the sky counts 4. Calculate and apply aperture corrections using bright, isolated, unsaturated stars if you need “total” magnitudes for calibrations This call all be done in Python (e.g., photutils).
79. PSF Photometry 79 What is the best way to get all of the information out of a given image?
80. PSF Photometry 80 In cases where sources are overlapping (e.g. crowded fields) and aperture photometry is hopeless, we use PSF (point-spread function) photometry.
81. PSF Photometry 81 PSF photometry does iterative steps of defining the PSF empirically (rather than using a predefined model). This is followed by fitting the sources and subtracting them, allowing you to individual sources and fainter sources.
82. PSF Photometry 82 There are some pre-existing codes that do PSF photometry, but fewer than those that exist for aperture photometry.
83. 83 PROCESSING IMAGING DATA
84. Science Frame 84 Sky level (~900 ADU) Bias level (~500 ADU)
85. Bias (Zero) Frames) 85 Bias frames are additive to your image frames and hence should be subtracted. Bias frames should be taken with a 0-second readout. Most modern CCDs have ~nearly featureless bias frames. Take an odd number and then average (~15-25).
86. Bias (Zero) Frames) 86 Bias frames do not depend on the filter, so you can use one set for all observations taken with the same binning. If there are no features, you can take the median of the frames and subtract from all images.
87. Bias (Zero) Frames) 87 Sometimes, there are faint features (e.g. amplifier glow). These “bright” features have amplitudes of ~0.1%. You can use bias frames + flat fields to measure the gain/read noise of your CCD.
88. Overscan 88 More so in spectroscopy, but sometimes in imaging, you will see ~32 columns at the edge of the image. These are fake columns made by continuing to cycle the electronics after the CCD is readout.
89. Overscan 89 This “overscan” region can be used to subtract the bias level, rather than using the full bias frames. The most common thing to do is average across the overscan region on a per-row basis, then fit a function to this in case there is structure.
90. Darks 90 Darks are rarely used in optical CCDs, but are used in near-IR CCDs. Darks are an additive correction, so you subtract them from your image. Note that darks are not always filter-independent.
91. Darks 91 Ideally, you have a sampling of darks that match your calculated science exposure times.
92. Cosmic Rays 92 Similar to spectroscopy, cosmic rays can be removed using the L.A. Cosmic function.
93. Flat Fields 93 You will want to correct for this. Flat fields highlight the pixel-level quantum efficiency variations (~few %). Large scale variations are due to CCD properties (e.g., thinning), nonuniform illumination, or dust/hairs? on the filter.
94. Flat Fields 94 Flat-fielding is multiplicative. You will want to make a normalized flat and then divide your science spectrum by your flat field.
95. High Precision Photometry 95 In order to achieve ~1% photometry (a common goal), you will want any controllable errors caused by flat fielding to be << 1%. For a goal of 0.1% Poisson errors in flat-field, we need (1/1000)2 = a million e- per pixel. For a typical gain of 3.8 e-/ADU, this is ~260k counts, which can be accumulated over ~10-15 frames. You will have to repeat this for every filter.
96. Dome Flats 96 Most telescopes have a flat white screen suspended inside of the dome. During your afternoon calibrations, you can turn on a quartz lamp and take images of this screen.
97. Dome Flats 97 Pros Cons Can do in the afternoon and take as many as you want The lamp is much colder than the Sun, so the color doesn’t match the sky Easy to combine multiple frames for high-N flat Screen is not at infinity, so the focus is different than the sky
98. Dome Flats 98 Pros Cons Can do in the afternoon and take as many as you want The lamp is much colder than the Sun, so the color doesn’t match the sky Easy to combine multiple frames for high-N flat Screen is not at infinity, so the focus is different than the sky
99. Dome Flats 99 Pros Cons Can do in the afternoon and take as many as you want The lamp is much colder than the Sun, so the color doesn’t match the sky Easy to combine multiple frames for high-N flat Screen is not at infinity, so the focus is different than the sky
100. Dome Flats 100 Pros Cons Can do in the afternoon and take as many as you want The lamp is much colder than the Sun, so the color doesn’t match the sky Easy to combine multiple frames for high-N flat Screen is not at infinity, so the focus is different than the sky
101. Dome Flats 101 Pros Cons Can do in the afternoon and take as many as you want The lamp is much colder than the Sun, so the color doesn’t match the sky Easy to combine multiple frames for high-N flat Screen is not at infinity, so the focus is different than the sky
102. Twilight Flats 102 Right after sunset, or right before sunrise, you can take images of the darkening, or brightening, sky and use this as your flat instead.
103. Twilight Flats 103 Pros Cons Focus is an excellent match for the dark sky Sky brightness changes quickly and it’s hard to get enough counts in multiple filters Color is an okay match for the dark sky Frame combination must be done carefully Sky gradient can be an issue for a wide field of view Sometimes you have to do it in the morning when you’re tired
104. Twilight Flats 104 Pros Cons Focus is an excellent match for the dark sky Sky brightness changes quickly and it’s hard to get enough counts in multiple filters Color is an okay match for the dark sky Frame combination must be done carefully Sky gradient can be an issue for a wide field of view Sometimes you have to do it in the morning when you’re tired
105. Twilight Flats 105 Pros Cons Focus is an excellent match for the dark sky Sky brightness changes quickly and it’s hard to get enough counts in multiple filters Color is an okay match for the dark sky Frame combination must be done carefully Sky gradient can be an issue for a wide field of view Sometimes you have to do it in the morning when you’re tired
106. Twilight Flats 106 Pros Cons Focus is an excellent match for the dark sky Sky brightness changes quickly and it’s hard to get enough counts in multiple filters Color is an okay match for the dark sky Frame combination must be done carefully Sky gradient can be an issue for a wide field of view Sometimes you have to do it in the morning when you’re tired
107. Twilight Flats 107 Pros Cons Focus is an excellent match for the dark sky Sky brightness changes quickly and it’s hard to get enough counts in multiple filters Color is an okay match for the dark sky Frame combination must be done carefully Sky gradient can be an issue for a wide field of view Sometimes you have to do it in the morning when you’re tired
108. Twilight Flats 108 Pros Cons Focus is an excellent match for the dark sky Sky brightness changes quickly and it’s hard to get enough counts in multiple filters Color is an okay match for the dark sky Frame combination must be done carefully Sky gradient can be an issue for a wide field of view Sometimes you have to do it in the morning when you’re tired
109. Twilight Flats 109 Pros Cons Focus is an excellent match for the dark sky Sky brightness changes quickly and it’s hard to get enough counts in multiple filters Color is an okay match for the dark sky Frame combination must be done carefully Sky gradient can be an issue for a wide field of view Sometimes you have to do it in the morning when you’re tired
110. Combining Twilight Flats 110 To combine twilight flats, subtract the bias, then scale (normalize) to a common value. From there, you can average the normalized frames, throwing out the lowest pixel and the highest 50-70% of the pixels.
111. Combining Dome + Twilight Flats 111 To take advantage of the pros of each method, you can combine dome and twilight flats via the following: 1. Median smooth your combined dome flat 2. Median smooth your combined sky flat 3. Divide your smoothed combined dome flat by the unsmoothed dome flat (preserves pixel variation info) 4. Multiply the output of (2) x (3)
112. Dark Sky Flats 112 If you have many observations in the same filter that are not too crowded, with a lot of moving around, you could use minmax rejection to get an amazing flat field. This is rarely an option, but worth investigating.
113. Validating Your Algorithm 113 Look at some statistics: variations across your image should be < 1% (ideally less than that). If your variations are larger, this will directly translate to large photometric uncertainties. Don’t trust your eyes (in this case). A badly flat-fielded image may look fine.
114. When Should I Take my Cals? 114 Bias frames are typically stable with time. They are cheap to do, so you might as well do them every time.
115. When Should I Take my Cals? 115 Flat fields are rarely stable. New flat fields should be taken every night.
116. Calculating the Gain and Read Noise of the CCD 116 You can use the bias and flat field frames to directly calculate the gain and read noise of your CCD. For flat frames F 1 , F 2 and bias frames B 1 , B 2
117. 117 PHOTOMETRIC CALIBRATION
118. Calibration 118 Assuming you do everything correctly, you now have a carefully measured integrated flux over some time and want to convert that to a magnitude:
119. Calibration 119 Assuming you do everything correctly, you now have a carefully measured integrated flux over some time and want to convert that to a magnitude: What is c?
120. What is c (not the speed of light)? 120 c is a magically number that puts your magnitude on a standard system to compare to other magnitudes.
121. What is c (not the speed of light)? 121 An example: photometry of a star in g filter -2.5 log(I/t) This is the instrumental magnitude as discussed previously. g inst should include aperture correction (if necessary).
122. What is c (not the speed of light)? 122 An example: photometry of a star in g filter zeropoint Once measured, the zeropoint could be ~constant for a given telescope/instrument/CCD/filter.
123. What is c (not the speed of light)? 123 An example: photometry of a star in g filter color term This term arises due to differences between your filter and that in a comparison system.
124. Color term 124 The choice of color is nearly arbitrary. Don’t extrapolate to your filter. This is ideally measured using stars with a wide range of colors. Again, we try to avoid extrapolating at all stages.
125. What is c (not the speed of light)? 125 An example: photometry of a star in g filter Extinction coefficient
126. What is c (not the speed of light)? 126 An example: photometry of a star in g filter Extinction coefficient The extinction coefficient depends on wavelength. It is generally steeper in the blue. You can measure it directly from your data, standard star fields, or assume typical values for a given site. However, this does vary in time. Why?
127. Calibration Issues 127 To check for problems in your calibration: - Plot (measured-standard) magnitude versus g-r, airmass, etc. and look for any trends If you want really high precision, you can image adding more terms to this calibration equation.
128. Standardizing Magnitudes 128 The old-timey way to put your magnitudes on a standard system is to observe standard fields over a range of airmasses during the night. You should look for fields that contain stars with a range of colors (often near a WD) and have very accurate photometry.
129. Standardizing Magnitudes 129 Problems: - This only works on photometric nights (obeying the previous equation). - It is very time-consuming to observe standard fields multiple times throughout the night. - The seeing can change between your target and standard field, so you have to do very careful aperture corrections.
130. Standardizing Magnitudes 130 In modern times if you are working in an SDSS-like photometric system, things are much simpler. There are many stars with ugriz magnitudes in your field of view. You can use these for photometric calibration without needing to observe additional fields.
131. Standardizing Magnitudes (in space) 131 Since there is no atmosphere in space, the calibration can be done carefully and independently of your observations. A lot of zeropoints already exist for space-based telescopes.
132. 132 EXTINCTION
133. DUST (interstellar extinction) 133 Interstellar dust (is the worst) scatters and absorbs light in a wavelength dependent manner. In nearly every situation, you need to think about the effects of dust on photometric observations. Light from star Blue reflection Red transmission Dust cloud
134. Interstellar Extinction 134 Dust associated with cold molecular clouds that sit in the Galactic disk are mostly carbon and silicon-based. The light absorption and scattering properties of the dust depends on the grain size and composition. That is why it varies among galaxies.
135. Interstellar Extinction 135 Variations are largest in the ultraviolet and can generally be ignored in the optical. This is debilitating when trying to measure UV fluxes of stars at large distances.
136. Interstellar Extinction 136
137. Interstellar Extinction 137 Ratio of total to selective extinction
138. Interstellar Extinction 138 Extinction in V band
139. Interstellar Extinction 139 (B-V) reddening
140. Interstellar Extinction 140 The typical Galactic extinction curve assumes R V = 3.1. Under this assumption, each line of sight has a single reddening represented as E(B-V). Given a mean reddening law, this determines extinction at all wavelengths.
141. “Classic” Map 141 The “classic” reddening map is from IRAS+COBE far-IR maps of the sky. This map has ~few arcmin resolution. https://irsa.ipac.caltech.edu/applications/DUST/
142. Interstellar Extinction 142 In regions of low to moderate extinction and at optical/near-IR wavelengths, using these maps to estimate extinction works well, but remain skeptical for: - High extinction regions (i.e. plane/bulge) - “Nearby” sources (won’t see full extinction) - Extinction in UV (stupid dust)
143. Interstellar Extinction 143 Note: you don’t have to correct your photometry for extinction/reddening. It depends on your scientific goals. But you always have to make an explicit decision about how to handle dust. Anytime you see photometry, your first thought should be, “what about dust?” (but don’t be that person at every conference who asks about it, like that one person who asks about magnetic fields.)
144. 144 EXOPLANET TRANSITS
145. The Transit Method 145 A transit occurs when a planet crosses between us and its host star, causing an apparent dimming of the stars overall flux.
146. Transit Geometry 146 Assuming an idealized transit light curve: Y X t I t II t III t IV T dur δ τ Flux Time
147. Direct Measurables 147 T dur - the duration of the transit Y X t I t II t III t IV T dur δ τ Flux Time
148. Direct Measurables 148 t I, II, III, IV - points of contact Y X t I t II t III t IV T dur δ τ Flux Time
149. Transit Contact Points 149 t I - The start of the transit, when the edge of the planet first touches the limb of the star. This is the beginning of ingress. t II - The planet is fully within the stellar disk. This is the end of ingress. t III - The last time the planet is fully within the stellar disk. This is the beginning on egress. t IV - The end of the transit, when the planet is fully off the stellar disk. This is the end of egress.
150. Direct Measurables 150 τ - Duration of transit ingress/egress Y X t I t II t III t IV T dur δ τ Flux Time
151. Transit Geometry 151 δ - Transit depth Y X t I t II t III t IV T dur δ τ Flux Time
152. 152 Deriving the Transit Depth Assuming both the star and the planet are spherical, the transit depth can be derived from: δ = πR P 2 / πR S 2 δ = R P 2 / R S 2 So to get the planet’s true radius, you must have an accurate measurement of the star’s radius.
153. 153 Guesstimating Radii Planetary Radius [R ⊕ ] Transit Depth around the Sun [%] 1 0.08 4 0.1 10 1
154. 154 Guesstimating Radii Planetary Radius [R ⊕ ] Transit Depth around the Sun [%] 1 0.08 4 0.1 10 1
155. 155 Guesstimating Radii Planetary Radius [R ⊕ ] Transit Depth around the Sun [%] 1 0.08 4 0.1 10 1
156. Direct Measurables 156 Some exoplanets properties are directly measured from a transit, while other properties are derived. Measured Properties: • Transit duration • Ingress/egress time • Transit depth • Time of mid-transit • Time between transits Derived Properties: • Orbital period/ Semi-major axis • Impact parameter, b • Orbital ephemerides (where the planet will be in its orbit at a given time) • Planet-to-star radius ratio, R p /R S
157. Impact Parameter and Inclination 157 Y X R S a b i Assuming e = 0, b R S = a cos i Not assuming e = 0, b R S = a cos i ( ) –- 1 - e2 1 + e sin ω
158. 158 Grazing Versus Full Transit Star P Grazing Grazing Full A transit is considered grazing when b + (R p /R s )2 ≥ 1.
159. Biases 159 • Large planet-to-star ratios • Short orbital periods • Favors inactive stars (little to no photometric variability)
160. 160 Probability of a Transit Star P Grazing Grazing Full p transit = R S / a ≈ 0.005 (R S /R ྾ ) (a / 1 AU)-1 (assumes R P << R S and e = 0)
161. Limb Darkening 161 Limb darkening is a phenomenon where the edges of a star are observed to be darker at the edge (limb) than the center. We can see this on the Sun, but we can’t resolve other stars. Venus
162. 162 Where Does Limb Darkening Come From? Stars have a known temperature gradient from the core to the surface. The photosphere has a ~1000K gradient. Going deeper, the core is T ⊙, core ~40MK. Note that the light we see from the Sun originates at the photosphere where τ ~ 1.
163. 163 Where Does Limb Darkening Come From? Light from the center of the disk are traveling radially outwards and originate deep in the photosphere, where the temperature is hotter. Light from the limbs skim through the photosphere at a shallow angle and originate in the upper photosphere, where temperatures are cooler. Core
164. Core 164 Where Does Limb Darkening Come From? So, there are two differences: temperature and angle. Looking at the radiative transfer equation, you can solve for I 𝜈 as a function of θ and find that I 𝜈 ≈ F(cosθ) θ = π/2 θ = 0
165. DISCUSSION QUESTION How would you expect limb darkening to change as a function of wavelength you observe at? 165
166. The Limbs of the Sun 166 ultraviolet optical infrared
167. Effects of Limb Darkening on Transits 167 The best way to measure limb darkening would be to resolve other stars, but we can’t do that (for most stars). However, we can still see the effect of limb darkening on exoplanet transits. Venus
168. Effects of Limb Darkening on Transits 168 Example: Transits of HD 209458 b from the UV through the IR as observed with HST.
169. 169 Limb Darkening x Inclination - Draw the transit
170. 170 Limb Darkening x Inclination - Draw the transit
171. 171 Limb Darkening x Inclination - Draw the transit
172. 172 Limb Darkening x Inclination - Draw the transit
173. 173 Limb Darkening x Inclination - Draw the transit
174. 174 Limb Darkening x Inclination - Draw the transit
175. Eccentricity 175 You cannot reliably measure the eccentricity, e, from a transit. However, there may be some observables that can tell you if a planet may be eccentric. What do you think those could be? Top down view
176. Eccentricity 176 The duration of the transit will be shorter at e > 0 than if e = 0.
177. 177 TRANSIT SURVEYS
178. Why do we need transit surveys? 178
179. Transit Probabilities 179 What is the transit probability of a “hot Jupiter” (P = 4 days; R ⛤ = 1 R ⊙ ) [in %]?
180. Transit Probabilities 180 What is the transit probability of a “hot Jupiter” (P = 4 days; R ⛤ = 1 R ⊙ ) [in %]? period = 0.0109 years a = 0.0493 AU p transit = ~10%
181. Why do we need transit surveys? 181 • Transit probability is relatively low, even for the “easiest” targets (short period, favorable R p /R star )
182. Transit Detection 182 We know that “hot Jupiters” have an occurrence rate of η ≈ 1%. How many stars do you have to look at to find 1?
183. Transit Detection 183 We know that “hot Jupiters” have an occurrence rate of η ≈ 1%. How many stars do you have to look at to find 1? P detect = η ྾ p transit 1000 stars
184. Why do we need transit surveys? 184 • Transit probability is relatively low, even for the “easiest” targets (short period, favorable R p /R star ) • Detection probability is low, meaning you have to stare at a bunch of stars
185. Transit Duration (assuming e = 0) 185 t I t IV T tot
186. 186 t I t IV T tot Transit Duration (assuming e = 0)
187. Chord Distance 187 R star b
188. Chord Distance 188 R star b Accounting for partial transits, R star - R p < b < R star + R p
189. Chord Distance 189 R star b Simplifies to
190. Chord Distance 190 R star b k = R p /R star
191. 191 Projected Velocity
192. 192 Projected Velocity
193. 193 Projected Velocity
194. 194 Projected Velocity
195. Transit Duration 195 k = R p /R star Calculate T tot , assuming: • P = 4 days • R ⛤ = 1 R ⊙ • R p = 1 R Jupiter • i = 90° • b = 0
196. Transit Duration 196 k = R p /R star Calculate T tot , assuming: • P = 4 days • R ⛤ = 1 R ⊙ • R p = 1 R Jupiter • i = 90° • b = 0 ~ 3 hours This is reasonable to observe from the ground.
197. Why do we need space-based transit surveys? 197 Assuming: • R p = [0.5 - 20] R ⊕ • P = [0.1 - 100] days Calculate T tot for R ⛤ = 0.5, 1, 2 R ⊙ . Plot this as a 2D grid with Period on the x-axis and planet radius on the y-axis.
198. Why do we need space-based transit surveys? 198
199. Why do we need space-based transit surveys? 199 Go through notebook
200. Why do we need transit surveys? 200 • Transit probability is relatively low, even for the “easiest” targets (short period, favorable R p /R star ) • Detection probability is low, meaning you have to stare at a bunch of stars • Need very long stares to observe a full transit ◦ Some system configurations are not feasible to do from the ground ◦ Need to do this x 3 (3 transits to confirm a planet)
201. History of Ground-Based Transit Surveys 201 HATNet (Hungarian-made Automated Telescope Network) • Used off-the-shelf Canon 200mm f/1.8 lenses with CCD detectors • Total of 4 facilities: 2 in Arizona and 2 in Hawaii • Discovered ~150 transiting planets (mostly Jupiters with some Neptunes)
202. History of Ground-Based Transit Surveys 202 WASP (Wide Angle Search for Planets) • Also used off-the-shelf Canon 200mm f/1.8 lenses with CCD detectors • Total of 2 facilities: Canary Islands (WASP-North) and South African Astronomical Observatory (WASP-South) • Discovered ~200 transiting planets (mostly Jupiters)
203. History of Ground-Based Transit Surveys 203 TRAPPIST (TRAnsiting Planets and PlanetesImals Small Telescope) • 60-cm telescope • Total of 2 facilities: Chile (TRAPPIST-South) and Morocco (TRAPPIST-North) • Discovered 1 transiting planet system
204. History of Ground-Based Transit Surveys 204 TRAPPIST - all it takes is 1
205. Why so few transits from the ground? 205 Scintillation from the atmosphere • D is the diameter of the telescope. • Δt is the exposure time. • h is the altitude of the turbulent layer. Osborn, Föhring, & Wilson (2015)
206. Why so few transits from the ground? 206 Ground-based light curves will have some combination (bottom) of white noise (top) and red noise (middle). white red White + red
207. Colors of Noise 207 White noise equally contains all frequencies across the spectrum. Red noise follows a Brownian motion spectrum. It is sometimes referred to as random walk noise. However, it is stronger at longer wavelengths. white red White + red
208. Transits in Context 208
209. NASA’s Kepler Mission 209 Kepler was a custom-built, 0.95m diameter space telescope dedicated solely to finding transiting exoplanets. Cost: $600M Launch Date: March 7, 2009
210. NASA’s Kepler Mission 210 Observing strategy: stare at the same patch of sky for 3+ years to find Earth-sized planets orbiting Sun-like stars in the habitable zone (year-long periods)
211. NASA’s Kepler Mission 211 Discovered 100s of exoplanets! https://www.youtube.com/watch?v=Td_YeAdygJE One of the four reaction wheels (used to point the telescope) failed in July, 2012. A second wheel failed in 2013. This ended the primary Kepler mission.
212. NASA’s Kepler Mission 212 Discovered 100s of exoplanets!
213. NASA’s K2 Mission - Kepler’s Second Light 213
214. NASA’s K2 Mission - Kepler’s Second Light 214 Pointing the spacecraft was now possible, but still limited. K2 could only look at regions along the ecliptic plane.
215. NASA’s K2 Mission - Kepler’s Second Light 215 The new pointing strategy provided an opportunity to look at lots of different fields and environments (e.g., had the opportunity to look at stellar clusters).
216. NASA’s K2 Mission - Kepler’s Second Light 216 The extended mission wasn’t perfect and it required a lot of new analysis techniques to be developed to mitigate spacecraft noise. Luger et al. (2016)
218. NASA’s Transiting Exoplanet Survey Satellite (TESS) Mission 218 TESS is NASA’s latest exoplanet hunting mission. The objective of TESS is to find small planets on short orbits around bright stars - ideal for follow-up with JWST. Cost: $200M Launch date: April 18, 2018
219. NASA’s Transiting Exoplanet Survey Satellite (TESS) Mission 219 4 x 10cm lenses
220. NASA’s Transiting Exoplanet Survey Satellite (TESS) Mission 220 Guerrero et al. (2021)
221. ESA’s CHaracterising ExOPlanet Satellite (CHEOPS) Mission 221 CHEOPS is ESA’s first space mission dedicated to studying bright, nearby stars that already host exoplanets. It makes very high-precision transit observations. Cost: €50M Launch date: December 2019
222. ESA’s CHaracterising ExOPlanet Satellite (CHEOPS) Mission 222
223. Why so few transits from the ground? 223 There is a high rate of false positives. a) Low-mass eclipsing binaries
224. Why so few transits from the ground? 224 There is a high rate of false positives. a) Low-mass eclipsing binaries b) Low-mass eclipsing binaries + background star
225. Why so few transits from the ground? 225 There is a high rate of false positives. a) Low-mass eclipsing binaries b) Low-mass eclipsing binaries + background star c) Partial transit of an eclipsing binary
226. Why so few transits from the ground? 226 There is a high rate of false positives. a) Low-mass eclipsing binaries b) Low-mass eclipsing binaries + background star c) Partial transit of an eclipsing binary d) Background eclipsing binary
227. Why so few transits from the ground? 227 There is a high rate of false positives. a) Low-mass eclipsing binaries b) Low-mass eclipsing binaries + background star c) Partial transit of an eclipsing binary d) Background eclipsing binary
228. 228 TRANSIT TIMING VARIATIONS
229. Transit Timing Variations (TTVs) 229 While we cannot measure a planet’s mass from a single transit, we can measure a planet’s mass by looking for transit timing variations. TTVs occur when the gravitational pull of the planets causes one planet to accelerate or decelerate along its orbit. TTVs can only be measured in multi-planet systems, where you have these gravitational effects.
230. Transit Timing Variations (TTVs) 230 What you need: • A multi-planet system • High-quality data where transit times can be measured to a few seconds
231. 231 Agol et al. (2021) TRAPPIST-1 TTVs
232. River Plots 232 Agol et al. (2021)
233. Mass Comparisons 233 Agol & Fabrycky (2018) There is generally good agreement between RV- and TTV-measured masses.
234. The Benefits of TTVs 234 • Can get masses for planets that are hard to target with radial velocities (e.g., around active stars, faint stars)
235. The Benefits of TTVs 235 • Can get masses for planets that are hard to target with radial velocities (e.g., around active stars, faint stars) • Can get planet densities from a single method
236. The Downsides of TTVs 236 • Requires a significant baseline of observations (~years)
237. The Downsides of TTVs 237 • Requires a significant baseline of observations (~years) • Requires sampling transits frequently (may be hard to do from the ground)
238. The Downsides of TTVs 238 • Requires a significant baseline of observations (~years) • Requires sampling transits frequently (may be hard to do from the ground) • Requires high timing precision (needs high SNR observations)
239. 239 STARS
241. Stars! 241 The majority of exoplanet astronomers find stars annoying. When people complain to you about stars, your response should be something along the lines of, “Well, what can we learn about the star from these exoplanet observations?” E.g. You talk to someone who has a catalog of giant planets and eclipsing binaries, and they just toss the binaries. Instead, think about what science could be done with those systems.
242. Stellar Variability 242 Similarly to spectroscopy, stellar variability can contaminate, mimic, or mask exoplanet transit signals.
243. Stellar Variability 243 Similarly to spectroscopy, stellar variability can contaminate, mimic, or mask exoplanet transit signals. Different sources of variability can cause different changes in a star’s brightness over time.
244. Stellar Variability 244 Similarly to spectroscopy, stellar variability can contaminate, mimic, or mask exoplanet transit signals. Different sources of variability can cause different changes in a star’s brightness over time. Sources: Starspots, faculae, rotation, flares (to name a few)
245. Starspots 245 Starspots are cooler, darker magnetic regions on the star’s photosphere. On other stars, they can be larger and more persistent than on the Sun (see Rachael’s interferometry). Relevant for: M dwarfs and young stars.
246. Starspots in Photometry: Starspot Crossing Events 246 When a planet crosses a starspot, there is a temporary increase in flux. Why?
247. Starspots in Photometry: Starspot Crossing Events 247 When a planet crosses a starspot, there is a temporary increase in flux. Why? The stellar flux is a combination of the photosphere + spot. When the planet is passing over a darker region, the overall flux of the system appears to increase.
248. Starspots in Photometry: Starspot Crossing Events 248 Starspot crossing events are chromatic. This will become important when we talk about atmospheres. Mori et al. (2025)
249. Starspots in Photometry: Rotation 249 Made with starry
250. Starspots in Photometry: Rotation 250 Rotation can “smear” out transits, making them very hard to detect. Barber et al. (2025)
251. Faculae 251 Faculae are hotter, bright regions that, on the Sun, are co-located with spots. Unlike spots, faculae impede very high-precision photometry.
252. Stellar Flares 252 Stellar flares are magnetic reconnection events that result in a rapid release of energy, and sometimes a release of stellar plasma.
253. Stellar Flares 253 Flares last on the order of minutes to hours (yes, hours!). They are common on very active stars (e.g., young stars and M dwarfs).
254. Stellar Flares - Components 254 Jackman et al. (2019)
255. Stellar Flares - Components 255 Jackman et al. (2019) Impulse phase: caused by the magnetic reconnection process
256. Stellar Flares - Impulse Phase 256 Your average detectable stellar flare releases energy on average E ~ 1029-34 erg. This causes the rapid heating of local plasma to 10s MK.
257. Stellar Flares - Impulse Phase 257 During the impulse phase, one could also expect (based on solar flares):
258. Stellar Flares - Impulse Phase 258 During the impulse phase, one could also expect (based on solar flares): • Acceleration of electrons and ions to relativistic speeds.
259. Stellar Flares - Impulse Phase 259 During the impulse phase, one could also expect (based on solar flares): • Acceleration of electrons and ions to relativistic speeds. • Chromospheric evaporation - occurs when non-thermal electrons stream along magnetic field lines and impact the chromosphere, continuously heating it.
260. Stellar Flares - Components 260 Jackman et al. (2019) Decay phase: Caused by the thermal cooling of the stellar plasma
261. Stellar Flares - Decay Phase 261 The timing and morphology of the decay phase depends on a lot of different physical properties:
262. Stellar Flares - Decay Phase 262 The timing and morphology of the decay phase depends on a lot of different physical properties: • Flare size, loop length, plasma density, temperature
263. Stellar Flares - Decay Phase 263 The timing and morphology of the decay phase depends on a lot of different physical properties: • Flare size, loop length, plasma density, temperature • Post-flare loops may continue to reconfigure or shrink
264. Stellar Flares - Decay Phase 264 The timing and morphology of the decay phase depends on a lot of different physical properties: • Flare size, loop length, plasma density, temperature • Post-flare loops may continue to reconfigure or shrink • Secondary reconnection events can cause low-level heating and prolong the decay
265. Stellar Flares - No Two are the Same 265 Davenport et al. (2014)
266. Stellar Flares - Contamination 266 The presence of flares in photometry, and around transits, can: • Change the overall shape and depth of a transit
267. 267 The presence of flares in photometry, and around transits, can: • Change the overall shape and depth of a transit • Cause irregularities in transit shapes Stellar Flares - Contamination
268. 268 The presence of flares in photometry, and around transits, can: • Change the overall shape and depth of a transit • Cause irregularities in transit shapes • Cause irregularities that can be mistaken for transit timing variations Stellar Flares - Contamination
269. 269 The presence of flares in photometry, and around transits, can: • Change the overall shape and depth of a transit • Cause irregularities in transit shapes • Cause irregularities that can be mistaken for transit timing variations Stellar flares have the strongest effect at bluer wavelengths, but can still affect transits at red wavelengths. Stellar Flares - Contamination