Maurice Wilson's

Astronomy Research and Code

High Precision Photometry of Transiting Exoplanets

During the summer of 2015, I had the privilege of conducting exoplanetary research at the Harvard-Smithsonian Center for Astrophysics (CfA) in Cambridge, MA. This opportunity was due to the excellent graduate school preparatory program at the CfA known as the Banneker Institute. Here, I will discuss the key concepts and methodology of my research project. This research is still ongoing but I am happy to explain my progress thus far and continue updating this web site as I learn more about the intriguing exoplanets and stars that I am analyzing. Enjoy!

Objective: I seek to determine the physical properties, such as the mass and radius, of Earth-like exoplanets around Sun-like stars via high precision photometry and spectroscopy by utilizing the defocusing technique and the unique capabilities of MINERVA. Currently, the photometry is the primary focus of this research.


The MINiature Exoplanet Radial Velocity Array (MINERVA) is a four-telescope array recently built to detect, confirm, and characterize Earth-like exoplanets. It is located on Mount Hopkins in Arizona. Each telescope is a PlaneWave CDK-700 telescope with a diameter of 0.7 meters, a 20.8 x 20.8' field of view and an Andor iKON-L camera. MINERVA is unique because I can achieve a Doppler Shift precision of 0.8 meters per second or less AND a photometric precision of less than 1 millimagnitude. When both of these goals are achieved, I can obtain highly precise measurements on a star that has an exoplanet revolving around it, and such precision will allow me to ascertain the physical properties of the exoplanet with low uncertainties.

For more information on MINERVA, view the MINERVA website and the paper "Miniature Exoplanet Radial Velocity Array (MINERVA) I. Design, Commissioning, and First Science Results", Swift et. al., 2015, JATIS, 1, 2. Also, one astrophysicist took the time to explain in the simplest way possible why MINERVA is awesome in its unique own way. See this elucidation here. Furthermore, if you want a concise and comprehensible explanation of what a "Doppler Shift" is, I highly recommend you read these blog posts about Radial Velocity and Doppler Shift written by a handsome, young Harvard summer intern.


It is quite feasible to achieve the previously mentioned Dopper Shift precision of 0.8 m/s or less when I observe bright stars. For this reason, only bright stars with exoplanet companions are on my mind all day and every day. Let's consider the bright star seen below.

Figure 1: Bright star and exoplanet system.

The bright star is wobbling because of the gravitational pull of the exoplanet revolving around it. The wobble of the star is noticed by Earthlings like myself by looking at the Doppler Shift measurements of the star after examining the star's spectra over time. After analyzing these spectroscopic measurements, I obtain the lower limit (\( m_{p_{low}} \)) of the exoplanet's mass. This is good but I want the most precise evaluation of the exoplanet's mass. I can deduce this mass (\( m_{p} \)) via the following equation: \[ m_{p_{low}} = m_{p}sin(i) \] where \(i \) is the inclination angle. The angle of inclination describes the orientation of the exoplanet's orbit, with respect to the observer's line of sight. Considering this equation, I can solve for the mass of the exoplanet as long as I know the lower limit on the mass (which I do know) and the planet's angle of inclination. The problem here is that the spectroscopy alone does not reveal the planet's inclination angle. This is where photometry saves the day.

Physical Properties

Figure 2: Light curve of transiting exoplanet. (

Figure 2 is showing that a light curve reveals the change in a star's brightness of over time. In this case, the star gets slightly fainter because a planet crossed in front of the face of the star and blocked some of the stellar light from reaching our instruments on Earth. The planet's act of crossing in front of the face of the star is known as a transit. If a planet transits its host star, I can use the photometric data to create a light curve and subsequently determine the inclination angle! The simplified explanation for why this is true is illustrated by Figure 3.

Figure 3: Relationship between light curve and inclination angle.

Figure 3 shows that if the planet were located at different inclination angles, we get a completely different light curve each time. Knowing how this relationship works, I can use the light curve that I have produced from the photometry to get the angle of inclination. Therefore, I can now solve for the mass of the exoplanet! Furthermore, it's pretty cool that with the photometry and Doppler Shifts, I can determine the radius of the exoplanet as well. (For a more in-depth and mathematical explanation of this, I refer you to Joshua Winn's Transits and Occultations.)

Aperture Photometry

My analysis of the bright star and exoplanet begins with aperture photometry. In order to create the light curve, I must first extract the star's flux from the photometric data. However, in order to distinguish this stellar flux from the flux of other sources, I must perform aperture photometry. Figure 4 illustrates how the code I have written performs this method.

Figure 4: Raw image of a star, illustrating the aperture photometry performed by my code.

In Figure 4, the star is encompassed by a green circle. Within that green circle, there is flux due to the star and the sky background noise. Initially, these two are indistinguishable. This is why my code creates the annulus (colored in yellow). Within the annulus, there is only the sky background noise. Thus, my code calculates the average flux within that region in order to evaluate the sky background noise. Afterwards, my code subtracts this sky background from the region within the green circle and consequently I am left with only the flux of the star within that green circle. Now, I can make the light curve!

More Photons, More Problems

Ironically, the photometry of bright stars (or ~7 magnitude or less in the V filter) is more difficult to obtain than that of relatively faint stars. This is even more counterintuitive when you consider the fact that bright stars make my job of obtaining high precision spectroscopy easier, as I mentioned previously. Figure 4 pretty much sums up the plight that astronomers, who seek photometry of bright stars, must endure.

Figure 5: Image of the star Tau Bootes, demonstrating the struggle that comes with attempting to study bright star photometry. (DSS, STScI, and Palomer)

    Tau Bootes, in Figure 5, has an exoplanet companion. If I wanted to analyze this star and determine the exoplanet's physical properties, I would be hindered by two major problems that occur frequently when dealing with bright star photometry:
  1. Another method necessary to perform before I can create the light curve is relative photometry. Relative photometry requires the observation of comparison stars. Comparison stars are stars that are similar in brightness and color to the primary star of interest, i.e. the star with the transiting exoplanet. A major issue with relative photometry arises in the case of Tau Bootes because it is a bright star, and thus this bright primary star of interest needs to have nearby stars that are similar in brightness to it. Within that image, where can you see stars that are equally as bright at Tau Bootes? There are none. Therefore, I cannot conduct relative photometry on this star.
  2. The second major issue is evident in the extremely bright circular features and four spikes seen around the star. Stars do not have such features and should not look like this in an image. These features are due to the CCD camera reaching its diffraction limit and being saturated while imaging this bright star. The camera was exposed for too long on that particular star. Consequently, the photometry gathered on this bright star is no longer trustworthy. Thus, I cannot determine the true flux of the star and create the light curve.
  3. Despite these two immense obstacles, there is a way to analyze bright star photometry. That way is through the power of MINERVA's four telescopes.

Relative Photometry

Figure 6: Relative Photometry with MINERVA
(Click the image.)

  1. One of my Harvard research advisors is standing next to one of the four MINERVA telescopes.
  2. Let's just imagine that he is monitoring all four MINERVA telescopes simultaneously.
  3. After observing, my advisor managed to get these images of four bright stars. One of these stars is our primary star of interest with the transiting exoplanet. The other three stars serve as the comparison stars because all four stars are similar in brightness. This is great! Within one telescope's field of view, we could not find more than one bright star. However, with the use of MINERVA's four telescopes, we can use 1 telescope to observe the bright primary star while the other three telescopes look at nearby patches of the sky to find the bright comparison stars. Now, we can conduct relative photometry with a bright primary star.

Now that I have used the three other telescopes to measure the flux of the bright comparison stars, I can perform relative photometry in my analysis of the primary bright star.

Figure 7: Relative Photometry Analysis
(Click the image.)

  1. One MINERVA telescope has observed the star, WASP 52. The light curve for this star is shown. As can be seen, there is a lot of variation in the flux over the course of a few hours. Much of the variation is due to the exoplanet that transited the star during that time interval. However, I currently cannot tell where the exoplanet transit occurred in this light curve. Moreover, I cannot determine how much of the variation is due to the transit, as opposed to other mechanisms that also are responsible for some of the variation within the light curve. These "other mechanisms" are simply the effects of the Earth's atmosphere altering the star's light as this light travels through Earth's atmosphere to reach the telescope. Therefore, I need relative photometry to distinguish the flux variation that is due to the star-exoplanet system from the flux variation that is due to the Earth's atmospheric effects.
  2. Now the flux of the comparison stars come into play. The light curves of 3 comparison stars and the primary target (WASP 52) is presented. The most important criterion for a star to be a comparison star is that its flux does not vary (i.e. this star does not have a transiting exoplanet, eclipsing companion star, etc.). Thus, if flux variation is seen in the light curves of the comparison stars (which is conspicuously true in this case) then that variation must be due only to the Earth's atmospheric effects. Now that I have data on how the atmospheric effects altered the incoming stellar light, I can distinguish it from the flux variation, due to the exoplanet, in WASP 52's light curve. To do this, I divide the flux (at each data point) of WASP 52 by the average flux of the 3 comparison stars.
  3. Subsequently, I end up with a nice light curve that reveals the astrophysical information that I was looking for. There is a dip in the light curve that indicates where the exoplanet transit took place. After the dip, the flux remains pretty constant throughout the rest of the light curve.
    After all of this, I need to fit a model to this light curve. Once a satisfactory model is determined, I can go on to find the physical properties of the exoplanet.

The High Precision

The physical properties can now be evaluated, but achieving satisfactory precision for the properties of Earth-like exoplanets can be done by using the defocusing technique. Rather than taking images of the stars while in focus, the stars are intentionally set out of focus in order to maximize the amount of CCD pixels that the star's photons land on.

Figure 8: Exemplary Defocused Star (

Because the defocusing technique is not a new concept for the astronomy community, I will not go into serious detail about it. Although, I will emphasize that this technique is beneficial primarily because it allows me to expose the camera for a long period of time without saturating the CCD--despite the brilliance of the stars that I am interested in observing. This is due to the fact that saturation occurs when one CCD pixel reaches the maximum limit of photons that it can detect within one exposure time. Thanks to the defocusing technique effectively dispersing the star's photons across many more pixels, there are now many more pixels that share that load of stellar photons--thereby significantly reducting the chance that saturation will occur quickly. When observing a bright star, the CCD will more than likely saturate within a 1-second exposure time. The defocusing technique allows me to increase the exposure time, for example, to 2 minutes while observing the same bright star and avoiding saturation. Such a drastic increase in exposure time substantially reduces flat fielding errors, decreases the fractional overheads, and thus results in highly precise photometry.
For a more extensive explanation of this technique, I found Dr. John Johnson's paper, A Smaller Radius for the Transiting Exoplanet WASP-10b, to be quite helpful.

I will now describe how a photometric precision of less than 1 millimagnitude was achieved with observations from one MINERVA telescope.

Figure 9: Light Curves of star system 16 Cyg. All data was gathered by one MINERVA telescope.

In Figure 9, photometry of two stars, 16 Cyg A and 16 Cyg B, was collected by one of the MINERVA telescopes and the resultant light curves are shown. There was no transit or eclipse occurring during the observations, so it makes sense that dips in the light curves are not present. By using the defocusing technique on the telescope, an observational scatter of 2.7 millimagnitude (mmag) was achieved. Considering how bright those stars are, such a precision is quite fortunate. However, a photometric precision of less than 1 mmag was proven to be achieved with this same data. (Strange? Yes, I know. I thought this was weird too when I first learned about it.)
Allow me to elaborate on why this is true.

Figure 10: Allan Variance plot of the photometric time series of 16 Cygnus A. Root-Mean-Square (i.e. precision) is seen as a function of timescale. Dashed line reaches below the 1.0 mmag mark at about the 200 second timescale.

In Figure 10, you can think of timescale as a proxy for the binning of the light curve data. For example, if I were to bin the light curves of Figure 9 in such a way that 1 data point was equivalent to 200 seconds of time interval, then my timescale would be 200 seconds and the newly binned light curve would show a precision of less than 1.0 mmag. The sub-millimagnitude precision was only discerned after the light curve was binned.


    It has been proven that relative photometry of bright stars can be conducted with MINERVA. It is also confirmed that sub-millimagnitude precision can be achieved with a MINERVA telescope. Soon, my advisors and I will combine the multi-telescope aspect of MINERVA with the defocusing technique. Utilizing this defocusing technique on all four of the MINERVA telescopes is a task that has yet to be done. Thus, this research is ongoing for me. I am excited for the day that we can conduct such an observation, as it will be a precursor to the lengthy bright star survey that is planned for MINERVA.
    Due to the substantial amount of data that will result from this bright star survey, there is an obvious need for software that can automate the analysis as the data flows in. This is where my code saves the day. Here are the most significant capabilities of my code:
  • User can query for satisfactory comparison stars.
  • Code creates observation schedules which can be read by the autonomous MINERVA telescopes.
  • Post-observation
  • Perform aperture and relative photometry, even if the stars are defocused.
  • Produce light curve of primary target star. is developed and managed by Maurice Wilson.