Maurice Wilson's

Astronomy Research and Code

MINERVA Spectrograph's Instrumental Profile

The MINERVA telescopes commenced the RV survey in 2016. Since then, we have taken over 3,000 spectra of various targets in our 80-star target list. Now that I have completed photometry and robotic telescope work for MINERVA, I decided to work on the radial velocity side of things. Before I took this initiative, no one was analyzing the MINERVA spectra for potential planets. There are several targets on our target list that have well-known planetary companions. Those would be the easiest planets to find in our Doppler spectroscopy data, but MINERVA has been short-staffed for a long time now. So, it was up to me to dig through these spectra and find planets.

Before I could find planets in the spectra, I needed to first make sure that the starlight is properly being measured by our Doppler spectroscopy pipeline and properly converted into radial velocity (RV) estimates. This was originally not the case. Before I tackled the MINERVA RV data, our RV pipeline was incorrectly calculating the radial velocities of the stars. This is where I come in. My objective was to determine the primary culprit that was responsible for our previously inaccurate RV calculations and fix it.

Spectrograph Stability

Figure 1: Diagram of major components of MINERVA hardware. Telescope, photometry camera, fiber guide camera, (cartoon blue) optical fibers, and spectrograph (casing) are shown.

The MINERVA mission has a goal of achieving a precision of less than 1 m/s. It is helpful to have a stable spectrograph when trying to achieve this goal. MINERVA contains a lot of components that contribute to the overall stability of the spectrograph. When the telescopes are in RV mode (see Control Software for elaboration), the starlight bouncing off the telescope's mirrors will go to the fiber acquisition unit (FAU). The FAU has a (red) guide camera that keeps a target star centered on the fiber's aperture. Refraining the starlight from moving all over the fiber's aperture is how the FAU constributes to the stability. From there, the light goes into the optical fibers. We have octagonal-shaped fibers that have a circular butt-couple connecting them to the spectrograph. The octagonal shape helps scramble the light so that the result is, ideally, a uniformly well-scrambled image in the near field limit. The circular butt-couple contributes to the image being more circular. From there, the light enters the spectrograph. In our case, MINERVA's Kiwispec spectrograph contains an iodine absorption cell. This, of course, gives us a well-defined wavelength solution. Additionally, the sharpness of the iodine absorption lines helps us better characterize the spectrograph's instrumental profile (also known as the point spread function or in this case a line spread function).

Considering all of this, a stable spectrograph should mean that the spectrograph's instrumental profile is stable. Seeing as a stable spectrograph improves RV precision, a stable instrumental profile should affect the RV precision as well. However, in our case it is not clear if the characterization of the instrumental profile is the dominant variable affecting our RV precision. Perhaps the fibers don't scramble as well as we think they do. Perhaps our barycentric correction is inaccurate. Perhaps too much moonlight is contaminating our spectra. There are many things that could be of greater importance to MINERVA's RV precision than the instrumental profile characterization. However, we were most suspicious of the instrumental profile and thus we tackled issues with our instrumental profile instead of possible issues with the other potentially limiting factors of the RV precision.

We suspected that our characterization of the instrumental profile was predominantly affecting our precision. This is likely considering that the spectrograph's instrumental profile affects our raw measurements before any processing of the data is done and subsequently evaluated for an RV precision. This can be understood from the perspective of the forward modelling procedure used for modelling the observed spectra. This can be described mathematically as $$ F_{\rm obs} = [ F_{I_2}(\lambda(x)) \times F_{\star}(\lambda^\prime (x)) ]* \textrm{IP}(x), $$ where \(F_{\rm obs}\) is the observed flux, \(F_{I_2}\) is the normalized iodine absorption, \({\lambda(x)}\) is the non-Doppler-shifted wavelength solution, \(F_{\star}\) is the stellar flux, \(\lambda^\prime (x)\) is the Doppler-shifted wavelength solution, IP(\(x\)) is instrumental profile, and \(x\) is the pixel position in the dispersion direction.

As the equation shows, the IP(\(x\)) is being convolved with the product of the well-known, non-Doppler-shifted iodine spectrum and the Doppler-shifted stellar spectrum. This convolution is the model we use to describe how the true starlight is being distorted by our optics. After collecting the data \(F_{\rm obs}\), we can deconvolve it with a model IP(\(x\)) that I deduced and then we can solve for the stellar continuum, wavelength solution, and the Doppler shift.


Before we can calculate the Doppler shift (and thus the RVs), we must first characterize our spectrograph's instrumental profile with a reasonable model IP(\(x\)). To do this, I must first understand the specific kind of spectra I am dealing with--in this case, spectra from MINERVA.

Figure 2: Exemplary full frame MINERVA echellogram from spectra taken of the daytime sky.

Our Kiwispec spectrograph has an echelle grating that spreads the light for the primary dispersion (horizontal) and a grism is used to disperse the light in the cross-dispersion (vertical) direction. The orders of our spectrum are dispersed vertically. In each order you can see 4 traces which pertain to the 4 MINERVA telescopes.

Figure 3: Close-up of echellogram, exemplary "chunk" of spectra, and cross-dispersion curves of the chunk.

As Figure 3 illustrates, we divide each trace into "chunks" of spectra and we define an IP(\(x\)) for each chunk. In fact, each chunk acts as the \(F_{\rm obs}(x)\) data and thus we find a new model for each chunk distinctly. The convention for determining an IP(\(x\)) is to assume it follows some function comprised of gaussian structures and to characterize the instrumental profile from the dispersion direction. I however took a unique approach. I modelled the instrumental profile from the cross-dispersion direction. In Figure 3, you can see how some of the columns, or "crosscuts", within a chunk look. Each chunk spans ~100 pixels wide and thus I have ~100 crosscuts to analyze per chunk.

One reason I took a unique approach is because it seemed like conventional methods were not working for us. Before I came along, the RV pipeline could choose between two conventional IPs to use in the forward modelling procedure. Despite the many complex components that constitute an RV pipeline, we believed that the characterization of the instrumental profile was the primary factor contributing to the imprecise and inaccurate radial velocity calculations. However, it is possible that our model IP was fine while the stability of the intrinsic instrumental profile was not fine. As I previously mentioned, a stable instrumental profile should help in producing high precision RVs. The potentially poor characterization and stability of the instrumental profile are the two reasons why I set out to define a new IP in an unconventional way.

Fixed IP

To clearly see the structure of the spectrograph's instrumental profile, we needed data with a high signal-to-noise ratio. Instead of observing a B-type star, we observe the sky in the daytime. The daytime sky provides Rayleigh-scattered light from the Sun that we assume is uniformly distributed across the optical fiber's aperture. This would yield a well-scrambled image in the near field limit. The fiber scrambling is further improved by coupling the octagonal fiber from the telescope to a circular fiber into the spectrograph. This provides a circular image as opposed to an octagonal one. An added bonus to these sky spectra is that they come from a light source that follows the same optical path that our RV target starlight follows. Ideally, the daytime sky spectra results in a uniformly scrambled, circular image; and as we neglect the distortion from the optics, we derive our fixed IP from the cross-dispersion direction of raw 2D spectra like that of Figure 3.

Within one frame of daytime sky spectra, I can gather all of the chunk's crosscuts and normalize them. I then fit them simultaneously with a cubic spline function. The result can be seen in Figure 4.

Figure 4: Fixed IP constructed from daytime sky spectra taken on October 8, 2017.
Top: The blue points represent the normalized crosscuts and the orange line is the fixed IP.
Bottom: Residuals of normalized data and fixed IP. The reduced \(\chi^2\) of the resiudals is shown.

The residuals of Figure 4 look as we expected them to. The discrepancy gets worse towards the center simply because of the shot noise is worse as the signal increases. Fortunately, there is no systematic structure in the residuals. This means that the IP is a good fit to the data. It is typical for observatories to characterize their instrumental profile every night or even every frame. We however have a stable spectrograph. To test the stability of the spectrograph's instrumental profile, I do not change my IP in any way for months. This is why I call this the "fixed IP." It is fixed over time. I use the same fixed IP to model data taken on June 20, 2018. Figure 5 presents the result of this test.

Figure 5: The fixed IP from October 8, 2017 (orange line) is used to model data taken on June 20, 2018 (blue points). The reduced \(\chi^2\) of the residuals is shown. The same telescope and chunk from Figure 4 are used here.

The residuals in Figure 5 have the same noise pattern as in Figure 4 but remains absent of any systematic structure. Because the old fixed IP is still providing a good fit to the data, this suggests that our instrumental profile was stable for about 9 months from October 8, 2017 to June 20, 2018. Back when I was performing this stability test, the last daytime sky spectra we had at the time was taken on June 20, 2018. Thus, I could not extend the test to see if the exact same fixed IP would be fitted well to data of a later date. So back then, I decided to go backwards in time and just extend the test by creating a different fixed IP used on the same data.

Lesson Learned

Figure 6: The fixed IP from April 7, 2017 (orange line) is used to model data from June 20, 2018 (blue points). The same telescope and chunk from Figure 4 are used here.

Figure 6 shows what happens when I push the fixed IP too far. The fixed IP I constructed from daytime sky spectra taken on April 7, 2017 is not a good fit to the data collected on June 20, 2018. This is evident based on the clear systematic structure in the residuals. (Based on the shape, it looks like some Gauss-Hermite polynomials would do me some good, but that is not true for reasons I won't bother elucidating here.)

The two fixed IPs used to model data taken on June 20, 2018 were constructed from spectra taken 6 months apart. Although I would have liked to create fixed IPs that were closer to another, I could not do this because of the lack of daytime sky spectra we took during those 6 months. This happens whenever Mt. Hopkins in Arizona receives a long monsoon season. Every year, during this time we do not open the observatory for fear of the rain. So, I played the cards I was dealt and just used the data that was nearest to October 8. This happened to be daytime sky spectra taken on April 7.

Figures 5 and 6 prove that our instrumental profile evidently evolved substantially between April 7 and October 8. This can be explained by Figure 7.

Figure 7: The pressure within our spectrograph spanning from March 2017 to December 2018. After ~50 days (or in mid-April), a power outage occurred at the MINERVA facility and the pressure swiftly rose. After ~220 days (or early October), the pressure was sufficiently stable for us to collect good quality spectra again. The two aforementioned fixed IPs were derived from spectra taken near both of these epochs. After October 2017, the pressure has some fluctuations but returns to a state of stability after ~470 days (or early June 2018).

While our spectrograph is normally continuously pumped, the power outage caused the valve from the pump to the spectrograph to close for an extended period of time, and the spectrograph slowly leaked up toward atmosphere. Figure 5 illustrates the abrupt disturbance in the pressure and the spectrograph's return to a stable state. This event permanently perturbed the spectrograph in such a way that the environment could not naturally return to its original instrumental profile as the pressure returned to its operating specification. When characterizing the instrumental profile with a fixed IP, a new fixed IP must be used whenever an event such as this occurs. If this is not done, a situation like that of Figure 6 is more likely to occur.

Although the power outage significantly disrupted my stability test, it provided me insight about how catastrophic things need to be in order for our stable spectrograph to become unstable and permanently change its instrumental profile. That's a valuable lesson that I was content in learning, especially since I managed to prove that our instrumental profile can be stable for 9 months--maybe even a year if the power outage didn't ruin the test. The ultimate test has yet to come though. I will test the stability of and my unconventional characterization of the instrumental profile by characterizing an exoplanet with my modifications to the RV pipeline.

The new IP is the most significant contribution I have made to MINERVA's RV pipeline. The other improvements to the RV pipeline I have made involve the rejection rates used in deciding which chunks are of insufficient quality to be included in the RV results, the choice among 3 iodine cell scans we can use in the RV pipeline, the new choice among 3 IPs we can use, and the cosmic ray detection code I wrote for raw 2D spectra. In tackling all of these very different problems, I learned a lot about how complicated and unstable an RV pipeline can be.

I will explain in detail the results of the ultimate test on another day. I will show how the fixed IP actually characterizes our instrumental profile greater than or equal to other conventional IPs. I will also show you how I confirmed MINERVA's first exoplanet detection with my version of the RV pipeline! is developed and managed by Maurice Wilson.