Efficient Follow-up of Exoplanet Transits Using Small Telescopes

To predict whether an exoplanet transit can be satisfactorily captured, one first needs to be able to predict the errors on an observation from all sources. Many of the sources contributing errors, such as target noise, are common to all observatories/telescope/camera combinations. Some embedded terms such as the observatory's altitude are specific to a given observatory. Each telescope/camera combination will have specific design characteristics, and consequently will have unique calibration values. The calibration values should be updated over time, as calibrations such as flat field will change. The precision also depends on the observations being made, in particular the chosen target and reference stars; the selected exposure time; and the values of air mass during the observations. For example, long exposures needed for satisfactorily capturing faint targets means fewer measurements and less well-defined light curves. On the other hand, bright targets can result in a very high number of data points, with great care needed to not exceed the pixel linearity limit, especially if the value of air mass changes significantly during an observation.

The total standard deviation (SD) for the target star and reference star is needed, and we present a condensed version of Beck (2018) and in the FAQ of the online tool. The predicted precision on the target alone in ADU units is represented by the following equation from Southworth et al. (2009):

$$\begin{eqnarray}&&\begin{array}{l}{({{\boldsymbol{\sigma }}}_{{\bf{total}}})}_{{\bf{target}}}\,=\\ \sqrt{{{\boldsymbol{\sigma }}}_{{\bf{bias}}}^{{\bf{2}}}+{{\boldsymbol{\sigma }}}_{{\bf{dark}}}^{{\bf{2}}}+({{\boldsymbol{\sigma }}}_{{\bf{flat}}}{)}_{{\bf{target}}}^{{\bf{2}}}+{\left({{\bf{I}}}_{{\bf{rms}}}\right)}_{{\bf{target}}}^{{\bf{2}}}+{{\boldsymbol{\sigma }}}_{{\bf{sky}}}^{{\bf{2}}}+{{\boldsymbol{\sigma }}}_{{\bf{target}}}^{{\bf{2}}}},\end{array}\end{eqnarray} \tag{ 1 }$$

where σ_bias is the bias noise and is a constant for a given camera; its value is established from calibration measurements and is defined by Equation (2). σ_dark is the dark current noise for the CCD camera and increases linearly with the user defined exposure time (${{\boldsymbol{t}}}_{\exp }$) as defined by Equation (3); σ_flat is the flat-field noise for the CCD camera obtained from dome calibrations and is defined by Equations (4) to (5); I_rms is scintillation noise from air turbulence as defined by Equations (6) to (7); σ_sky is sky noise at a specific observation location is defined by Equation (8); and σ_target is the target noise defined by Equation (9).

The bias and dark biases for a particular camera are established from calibrations, and represent the variation in each pixel in units of ADU as outlined in Beck (2018), where t_exp is the user-selected exposure time,

$$\begin{eqnarray}&&{{\boldsymbol{\sigma }}}_{{\bf{bias}}}={\bf{constant}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{\boldsymbol{\sigma }}}_{{\bf{dark}}}={\dot{{\boldsymbol{\sigma }}}}_{{\bf{dark}}}\ast {{\bf{t}}}_{{\bf{\exp }}}\end{eqnarray} \tag{ 3 }$$

The flat field for the particular CCD camera, the scintillation noise, and the target error depend on the target count (${{\boldsymbol{N}}}_{{\bf{target}}}$). For the flat field, a count of 1 gives the flat-field noise for ${{\boldsymbol{N}}}_{{\bf{target}}}={\bf{1}}$, where σ_flat is the standard deviation per pixel averaged over the flat images, ${({{\boldsymbol{N}}}_{{\bf{target}}})}_{\mathrm{flat}}$ is the total ADU count per pixel averaged over the flat images,

$$\begin{eqnarray}&&{({{\boldsymbol{\sigma }}}_{{\bf{flat}}})}_{{{\boldsymbol{N}}}_{{\bf{target}}}={\bf{1}}}={{\boldsymbol{\sigma }}}_{{\bf{flat}}}/{({{\boldsymbol{N}}}_{{\bf{target}}})}_{{\bf{flat}}}\end{eqnarray} \tag{ 4 }$$

Thus, the general case for the flat-field noise with any count (${{\boldsymbol{N}}}_{\mathrm{target}}$) is

$$\begin{eqnarray}&&{({{\boldsymbol{\sigma }}}_{{\bf{flat}}})}_{{\bf{target}}}={({{\boldsymbol{\sigma }}}_{{\bf{flat}}})}_{{{\boldsymbol{N}}}_{{\bf{target}}={\bf{1}}}}{{\bf{N}}}_{{\bf{target}}}.\end{eqnarray} \tag{ 5 }$$

Atmospheric extinction and scintillation can be one of the largest sources of error in photometric observations (Balona et al. 1995; Gilliland & Brown 1992;Hartman et al. 2005; Ryan & Sandler 1998). The scintillation error with a small telescope and a bright target (Castellano et al. 2004) is shown to be the dominant source of error. It is important to maximize the exposure time, because the longer exposure time, the lower noise due to scintillation (Dravins et al. 1998; Young 1992) as the normalized scintillation noise is given by

$$\begin{eqnarray}&&{{\boldsymbol{\sigma }}}_{{\bf{scint}}}={\bf{0.004}}{{\boldsymbol{D}}}^{-\tfrac{2}{3}}{{\boldsymbol{X}}}^{\tfrac{7}{4}}{{\boldsymbol{e}}}^{-\tfrac{h}{H}}{({\bf{2}}{{\boldsymbol{t}}}_{\exp })}^{-0.5},\end{eqnarray} \tag{ 6 }$$

where D is the telescope aperture (m), X is the air mass, h is the altitude of the telescope (m), H  = 8000 (m) is the scale height of the atmosphere.

(I_rms)_target is the scintillation noise from air turbulence for the target obtained by scaling σ_scint for the total ADU count for the target

$$\begin{eqnarray}&&{({I}_{\mathrm{rms}})}_{\mathrm{target}}={N}_{\mathrm{target}}{\sigma }_{\mathrm{scint}}\end{eqnarray} \tag{ 7 }$$

The sky SD (σ_sky) is established from values obtained from an analysis of areas of sky in several images in the absence of a target. It is recognized that from this equation it will give a "typical" sky condition of the images that are analyzed to give

$$\begin{eqnarray}&&{{\boldsymbol{\sigma }}}_{{\bf{sky}}}=\sqrt{{{\boldsymbol{\sigma }}}_{{\bf{sky}}}^{{\prime} {\bf{2}}}-{{\boldsymbol{\sigma }}}_{{\bf{cal}}}^{{\bf{2}}}-({{\boldsymbol{I}}}_{{\bf{rms}}}{)}_{{\bf{sky}}}^{{\bf{2}}}},\end{eqnarray} \tag{ 8 }$$

where ${\sigma }_{{cal}}^{{\bf{2}}}={\sigma }_{\mathrm{bias}}^{2}+{\sigma }_{\mathrm{dark}}^{2}+{\sigma }_{\mathrm{flat}(\mathrm{sky})}^{2}$, and (${{\boldsymbol{I}}}_{\mathrm{rms}}{)}_{\mathrm{sky}}^{2}={({{\boldsymbol{N}}}_{\mathrm{sky}}{\sigma }_{\mathrm{scint}})}^{2}$, where ${{\boldsymbol{N}}}_{\mathrm{sky}}={constant}* {t}_{\exp }$ and where the constant is the ADU count of the sky background for 1 second, ${\sigma }_{\mathrm{sky}}^{{\prime} }$ is the background noise taken from a region of sky essentially devoid of stars.

The standard assumption is made that the target signal follows Poisson noise characteristics and consequently the target noise is given by

$$\begin{eqnarray}&&{{\boldsymbol{\sigma }}}_{{\bf{target}}}=\sqrt{{{\boldsymbol{N}}}_{{\bf{target}}}}\end{eqnarray} \tag{ 9 }$$

The derivation of a predicted value for ${{\boldsymbol{N}}}_{\mathrm{target}}$ has been condensed down to a series of equations that after working out the constants for the equipment just need the catalog magnitude of the target star, exposure time, and the air mass. The key relationship between catalog magnitude and instrument magnitude for an air mass of 1.0 and an exposure time of 1.0 s for a particular telescope/camera combination is obtained from a series of observations.

Many historical observations are needed that cover a wide range of target magnitudes, exposure times, values of air mass and observing conditions. The only prerequisite in the choice is that there are no clear problems with the image and that there were no obvious filling of a pixel well to beyond its linearity limit:

$$\begin{eqnarray}&&{[{{\bf{m}}}_{{\boldsymbol{t}}={\bf{1}}{\bf{s}},{\boldsymbol{X}}={\bf{1.0}}}^{{\prime} }]}_{{\bf{predicted}}}={\boldsymbol{gradient}}\ast {{\boldsymbol{m}}}_{{\bf{target}}}+{\boldsymbol{Bias}},\end{eqnarray} \tag{ 10 }$$

where gradient and Bias come from the trendline of the Catalog to Instrument Magnitude plot, ${{\boldsymbol{m}}}_{\mathrm{target}}$ is the catalog magnitude of the target,

[${{\rm{m}}}_{{\rm{t}}=1\,{\rm{s}},{\rm{X}}=1.0,\mathrm{target}}^{{\prime} }{]}_{\mathrm{predicted}}$ is the predicted instrument magnitude for the target with an exposure time of 1.0 s and air mass of 1.0.

$$\begin{eqnarray}&&{\left[{{\boldsymbol{m}}}_{{\boldsymbol{t}}={\bf{1}}{\bf{s}},{\boldsymbol{X}}={\boldsymbol{X}}}^{{\prime} }\right]}_{{\bf{predicted}}}={\left[{{\boldsymbol{m}}}_{{\boldsymbol{t}}={\bf{1}}{\bf{s}},{\boldsymbol{X}}={\bf{1.0}}}^{{\prime} }\right]}_{{\bf{predicted}}}-\in ({\bf{1.0}}-{\boldsymbol{X}}),\end{eqnarray} \tag{ 11 }$$

where ∈ is the airmass extinction coefficient. ${[{{\boldsymbol{m}}}_{{\boldsymbol{t}}={\bf{1}}{\boldsymbol{s}},{\boldsymbol{X}}={\boldsymbol{X}}}^{{\prime} }]}_{\mathrm{predicted}}$ is the predicted instrument magnitude for the target with an exposure time of 1.0 s, and the airmass is the observed airmass:

$$\begin{eqnarray}&&{\left[{({{\boldsymbol{N}}}_{{\bf{target}}})}_{{\boldsymbol{t}}={{\boldsymbol{t}}}_{\exp },{\boldsymbol{X}}={\boldsymbol{X}}}\right]}_{{\bf{predicted}}}={{\boldsymbol{t}}}_{\exp }{{\bf{10}}}^{-{\left[{{\boldsymbol{m}}}_{{\boldsymbol{t}}={\bf{1}}{\boldsymbol{s}},{\boldsymbol{X}}={\boldsymbol{X}}}^{{\prime} }\right]}_{{\bf{predicted}}/2.5}},\end{eqnarray} \tag{ 12 }$$

where ${[{({{\boldsymbol{N}}}_{\mathrm{target}})}_{t={t}_{\exp },{\boldsymbol{X}}=X}]}_{\mathrm{predicted}}$ is the predicted ADU total count for the target with an exposure time of t_exp and airmass of X. The best-fit line gives the equation that can then be used to predict the value of instrument magnitude for the target for t = 1.0 s and X = 1.0 based solely on the target's catalog magnitude.

Having derived the best-fit line (Equation (10)), the formulation was then verified by using the catalog magnitude to back calculate a predicted value for instrument magnitude for each image and then subtract the predicted instrument magnitude from the measured instrument magnitude to give (Δm'). The average and standard deviation of the values of Δm' gave an estimate of the intrinsic error in the estimate of 0.00045 mag and 0.195 mag, respectively.

It was considered essential to establish how robust Equation (10) was, as the derivation of ${[{{\bf{m}}}_{{\rm{t}}=1{\rm{s}},{\rm{X}}=1.0,\mathrm{target}}^{{\prime} }]}_{\mathrm{predicted}}$ was considered to be fundamental. Consequently, a significant exercise was undertaken to take an even larger sample of 211 completely independent observations from the Bayfordbury Observatory archive of images with different targets covering a wide range of catalog magnitude, exposure time, and implicitly air mass and observing conditions to validate the range of conditions that the formulation would hold. Again the catalog magnitude was used to back calculate a predicted value for instrument magnitude for each image and then this value was compared with the measured instrument magnitude to give (Δm').

This validation process produced Figure 1, where a matching set of predicted instrument magnitudes (in green) and measured instrument magnitudes (in red) were produced from a large number of archived images. Inspection of Figure 1 shows that a good match was achieved over a range of catalog magnitudes, exposure times, and air masses.

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Section 2.1 identified a series of equations that have been formulated to predict the precision for each measurement of instrument magnitude from a telescope whose performance has been characterized. Light curves have been produced from the Chris Kitchen Telescope (CKT), James Hough Telescope (JHT), and Robert Priddey Telescope (RPT), which are robotically controlled telescopes at the Bayfordbury Observatory. The CKT configuration had been used to quantify the parameters in the SD equations, and is equipped with an SBIG STL-6303E CCD camera and a Meade LX200 series telescope. The JHT also has a Meade LX200 series telescope and has the same camera designation as the CKT, consequently the SD parameters should be close to those for the CKT. However, while the RPT telescope also has a Meade LX200 series telescope, it has a significantly different scientific camera (Moravian Instruments G4-9000). Consequently, any tolerances derived for the RPT using the CKT equations will not be fully representative, as several of the telescope specific parameters have not been derived for the RPT. The field of view of the CKT and JHT telescopes being 23.4' × 15.6', whereas that for the RPT is noticeably wider at 31.2' × 31.2'. This implies that the FWHM for a star with the RPT occupies a relatively smaller area on the chip compared with that with same field of view as the CKT/JHT telescopes, and consequently would be expected to reach a linearity limit sooner.

The CKT, JHT, and RPT robotically controlled telescopes were used to independently simultaneously capture the light curve of a transit of the deeply eclipsing cataclysmic variable UX Uma on the 2017 June 1 to demonstrate consistency with measurements taken by different telescopes.

Nominally, the light curves from the three telescopes should be identical, as the equipment is similar and the observations have been conducted at the same time. However, tolerances will mean that individual ADU values will be different. The objective of this exercise was to overlay the three light curves and verify that their tolerance values typically encompass the other light curves. Achieving such a match verifies that the calculated tolerances are consistent with the observations. By using the results from different telescopes, and knowing the tolerances on individual values, offers the prospect of weighting each data point according to its SD value and in turn generate a composite plot. Consequently the eclipse of UX UMa could have a light curve with significantly more data points than from a single telescope, plus the tolerances will enable the user to determine if a variation in the light curve is a tolerance effect or indicating a genuine physical effect. Figure 2 presents the overplotted light curves from the different telescopes.

Despite the significant variation in magnitude during the eclipse, Figure 2 shows that the CKT and JHT light curves and associated tolerances display a good degree of correlation. In the case of the JHT telescope, three anomalous data points have been removed that occurred shortly after the telescope conducted a re-focus and consequently the JHT light curve is drawn with two separate lines. Consequently, it is concluded that the predicted precision is consistent with the observed precision. Further, simultaneous observations of the same star would warrant plotting the interpolated differences between the observations, with the width of the distribution compared to the predicted uncertainty for several examples though the opportunities to have multiple telescopes on one target are limited.

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The transit follow-up tool includes the equations defined by the system model (Section 2.1) and provides the recommended observing times for known and yet to be confirmed exoplanet transits like targets of interest from TESS. Although predicted transits are available from websites such as the Exoplanet Transit Database website of the Czech Astronomical Society (Poddany et al. 2010; 2013), the recommended transits from the Transit Follow-up Tool exclude any that cannot be satisfactorily observed in their entirety and in particular do not meet one or more of the following constraints:

• (a)  
  Observing must be in the period between the scheduled start and end of robotic observing (the Sun must be below the horizon by at least 12° of the horizon).
• (b)  
  Observing must include the user-specified duration to capture the images both before and after the end of a transit.
• (c)  
  Must not exceed the allowed telescope elevation angle.
• (d)  
  Must still have sufficient predicted precision after allowing for variation in air mass during the observing period.
• (e)  
  The predicted precision must be sufficiently small for a specified transit depth.

Consequently, this subset of predicted transits identifies a limited number of key occasions when the transits are particularly suitable for observing.

The main programming for the online transit follow-up tool is done in PHP and JS and as a website also HTML. The source code and documentation is included at the software repository GitHub (https://github.com/lukerobson/followup). Beyond providing a variety of tools to plan future exoplanet transit observations, the software also produces RTML code that telescope software can use to operate a chosen telescope automatically. Below, we illustrate some of the prediction and plotting tools the software provides. The data for potential exoplanet transit events for the online tool comes from the NASA Exoplanet Archive (2019) as an aggregator of exoplanets both confirmed and tentative, while data for TESS targets of interest comes separately from ExoFOP (2019).

In Figure 3, we show two examples of the use of the tool for observed and simulated observations for the transits of WASP-10b and WASP-52b, with the online tool providing a number of other examples. The predicted observation is based on the known transit depth, airmass, length of transit, and exposure time, including the determination of the standard deviation for each simulated point in the light curve using the air mass during the observations. The observed error bars are derived following the data reduction of the recorded images using AstroimageJ (Collins et al. 2017) as described in the online tool.

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The observed light curves appear to be reasonably matched by the simulated data and are consistent at approximately 2σ as measured by the online tool provided by (Poddany et al. 2010). The predicted plot is based on the reference values as imported from the NASA exoplanet archive. It should be noted that the predicted values for depth and duration are sensitive to the precise details of the chosen sampling. The simulated data assume a constant detector readout time, but for a variety of reasons the observed data is rarely so uniform. It is also likely that some of the difference between the observed and predicted curves are intrinsic to the stars. For example, various studies have shown evidence for stellar activity modifying observed transits in terms of both the depth and duration of the transit. The scenario is that the transiting planet can be occulted by starspots, leading to a modulation of transit depth with stellar rotation phase, e.g., Croll et al. (2015). For example, Maciejewski et al. (2011) identify a worst case of a 3 mmag variation in WASP-10 in a 4 hour period, due to spots compared to the transit depth of 39.4 mmag. Oshagh et al. (2013) find that transit-timing variations of up to 200 s can be produced when a spot is completely dark and has the size of the largest spot on the Sun. With multiple observation of the same transit, we would be sensitive to such effects. However, our rather sparse data set, in particular lacking multiple transits of the same star, does not warrant such investigations. Where the information is available, the online tool modifies the predicted errors account for the activity of target star or gives a warning. Oshagh et al. (2013) assert that even a sophisticated fitting procedure for the depth and time of each individual transit will not produce accurate results for active stars. In practice the prediction will always be limited by the quality of the input assumptions.

The key feature of the online tool is not the analysis of the transit event per se, but its ability to offer a useful prediction of what the transit might look like before attempting to capture the transit. This helps the observer to properly plan their observations and for example to adjust the exposure time to see if a relatively short or long exposure time is better when considering transit duration, image cadence, and other photometric uncertainties. This functionality of the tool is illustrated in Figure 4 that provides a breakdown of the error with the CKT telescope over a range of catalog magnitudes, using appropriate exposure times for the magnitudes, and where more than one exposure time is available taking the average value. The change in appearance of the percentage error after 12th magnitude is caused by fixing the exposure time at 300 s corresponding to the approximate length of time that a small telescope might guide for without intervention.

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In Figure 5, we show how the online tool may be used to predict system performance in mmags as a route to adjusting observations in order that a particular precision is obtained. Figure 5(a) illustrates the predicted performance, without a contribution to the error from the reference star, over a range of catalog magnitudes and exposure times. For clarity, the performance has been set to zero where there is no significant predicted performance. Figure 5(b) illustrates the corresponding predicted performance where the precision is better than 3 mmag to illustrate the operating region where 3 mmag precision is required to adequately capture an exoplanet transit. Alternative plots can be generated with different precision requirements.

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