MEarth data processing

For data taken 2008-2010 and 2010-2011 with the original D02 housed detectors, and 2011-2012 with the upgraded D09 housed detectors. Differences highlighted as appropriate.

Overview

Data reduction for MEarth follows the general philosophy described in Irwin & Lewis (2001) and Irwin² et al. (2007), and indeed shares most of the code too (see also Irwin 1996 for a more general overview of CCD reductions for wide-field imaging). However, there are a number of instrument-specific problems detailed below that require additional processing steps or modifications to the standard procedure. These are listed in roughly the order they are done by the software.

• Non-linearity correction

  Although this is not a modification to the normal procedure, it is worth stating that the MEarth detectors, like nearly all CCD systems, show some non-linearity. Non-linearity corrections were derived from sets of "dome" flats taken using daylight to illuminate the roof on a clear day, with careful attention given to monitoring the illumination level and correcting for variations.

• Dark correction

  Dark current is not negligible in MEarth data: although the overall level is quite low, there are quite a lot of hot pixels, particularly at the -15C operating temperatures used until the detector housings were upgraded during the 2011 summer monsoon. Experience shows the number of hot pixels is a function of device temperature. Moreover, the hot pixels do not seem to be completely stable from night to night.

  After the upgrade, the persistent image of the pre-flash lamp pattern is visible in the dark frames.

  Dark frames are taken every night (at the end of the night). This uses a fixed exposure time (it would be prohibitively expensive to do all exposure times used for targets). This means hot pixels are imperfectly corrected because they seem to be somewhat non-linear with exposure time.

  Dark frames before 2010-02-26 were affected by the persistent image of the dawn twilight flats (where taken) due to starting darks too soon after closedown. The persistent image would gradually decay away during the sequence of darks, but was still present in the average master dark used to correct the data. A delay to allow the persistent image to decay away before starting darks was added on 2010-02-26.

• Shutter shading correction

  The use of a leaf shutter necessitates a correction for the non-uniform exposure across the field-of-view due to shutter travel time. This is a ~5% effect on a 1 second exposure. The correction seems to be fairly stable over few month timescales, and was derived from sets of twilight flats. This gets close, but is not perfect due to the illumination change over twilight. Note that the number of shutter blades increased from 5 to 6 when the housings were upgraded at the 2011 summer monsoon, but the travel time is still similar.

• Flat fielding

  Experience shows that the standard (simple) method of taking twilight flats produces a result that is not uniform to better than the several percent level. Instead, as of 2008-10-29, twilight flat fields are taken on alternating sides of the meridian to mitigate most large-scale illumination effects (the detector rotates by 180 degrees relative to the sky upon crossing the meridian, so gradients tend to average out). Before the summer 2011 detector upgrade, the center was affected by a severe case of "sky concentration" / "scattered light" to the tune of about 10% (in 2008-2010) or 15% (in 2010-2011) of the sky level at peak. Some experimentation led to the current ad-hoc working solution, which is to divide out a Gaussian fit to just the center portions, to remove the "scattered light", followed by dividing out all the large-scale illumination components using a 2-D median filter (except these steps were reversed in 2008-2010 processing). This leaves essentially only (a) dust doughnuts, and (b) the pixel to pixel response variations. The "correct" illumination part was then derived photometrically and multiplied in to the final flat field used to process the images. This seems to be quite stable and was done by dithering a dense star-field around the detector, making an image of the photometric residuals putting down Gaussians of roughly the image FWHM in the corresponding places, and smoothing it (a lot) to produce the final map. The final flat-fielding appears to be good to about a percent, but there is clear evidence for residual errors in the form of "meridian offsets" in the light curves. The 2008-2010 illumination map was rather inferior quality compared to the 2010-2011 one since a lot of improvements were made to the methodology. The 2008-2010 map was derived from the same observations taken of target fields during loss of pointing refinement as the fringe maps (see below) rather than dedicated fields with a very high stellar density, so was lacking in terms of sampling.

  After the 2011 summer monsoon, the scattered light has been substantially reduced thanks to improvements made to the baffling close to Cassegrain focus, and upgraded anti-reflection coatings on the CCD window. The Gaussian fitting step is no longer needed. Otherwise, the flat fielding procedure is the same, since there is still some scattered light.

• Defringing

  The raw data show substantial fringing due to the combination of a very red bandpass and thinned, back-illuminated e2v detectors. However, before the detector upgrade the devices also showed a strong persistence effect (see below), which made it practically impossible to generate a usable dark sky flat ("fringe map") with which to correct this effect, since the stars always tend to stack up. I derived these maps from observations taken in 2008 December during a week when the pointing refinement code was not working, so our target fields drifted all over the detector, but they were getting long in the tooth by 2011 and the fringe correction was noticeably inferior during 2010-2011 after the filter had been changed (unsurprisingly!). The situation was substantially improved in 2011-2012 and later seasons, where fringe frames were much easier to derive and therefore could be updated more often.

  The defringing code works by first removing large-scale structure in the image (the smoothing box is 256 pixels in the code) to remove the "glow" (see below). The fringe map is then compared to this and an "optimal" scale factor determined. The scaling is first guessed using the ratio of sky levels and then refined adaptively, this is necessary to track variations in the (predominantly OH) emission that causes the fringes. Finally, the scaled fringe pattern is subtracted from the original image. The fringe pattern itself is found to be very stable over time, and appears to essentially be an intrinsic property of the device (from the thinning process).

• Astrometry / pointing

  The "blind" pointing of our commercial mounts (after application of a standard pointing model) is not sufficient to achieve the approx. few pixel rms accuracy required to mitigate the effect of flat-fielding errors in the photometry. The observing system uses short, binned exposures to refine the pointing (these are not saved) before taking the main science image. Sometimes, this can fail, and the science images are still taken regardless. A "blind" pointing solution based on the code from the astrometry.net project was added to allow even frames which are miles off where the FITS headers say they are to be solved properly. This is not very efficient, but it doesn't get called very often. The astrometric code was also modified to use the catalogue (i.e. desired) position reported in the CAT-RA and CAT-DEC FITS headers as the initial guess, since the RA and DEC keywords report where the telescope thinks it was pointed, as per their standard usage, whereas the pointing refinement already adjusted the pointing to be very close to the desired position. Final astrometric calibration is done from the 2MASS point source catalogue, and typically achieves RMS residuals of ~0.15 arcsec, depending on how windy it was (worse for long exposures in windy conditions due to wind shake). This is okay-but-not-great, normally better than 1/10 pixel should be possible, so we may be able to improve this (e.g. by switching to UCAC, which also has proper motions, if that is what the problem is).

  It should be noted that occasional losses of pointing refinement have been experienced due to failure of the Windows COM (common object model) communication between the telescope control software and the CCD control software after the system has been up for several weeks. It is not known why this happens, but it requires a reboot to fix. Procedures for ensuring this cannot affect science data by scheduling regular reboots and reporting errors to the operator in a more visible fashion have prevented the problem occurring in recent data, but occasional half-nights have been affected in earlier data, and there was one several night long stretch in 2008 December when the usual operator was on vacation (which has been used for calibration purposes as detailed above).

• Per-frame approximate photometric zero-points

  A useful trick for weeding out junk data, and for evaluating sky transparency in real-time, is to compute a magnitude zero-point for each individual observation, based on standard stars within the field. The only useful catalogue for performing this calibration when the system was originally implemented was 2MASS, since there are not always Tycho-2 stars within our field-of-view. The use of 2MASS is not perfect however; we must extrapolate from JHK to effectively I-band, and correct for the fact that most of the 2MASS stars within the plane are distant, and thus reddened. This is done using an empirical fit to the extinction values from Schlegel extinction maps. The results are decent at high Galactic latitude, but tend to become a bit scattery at low Galactic latitude. This should really be replaced with something based on APASS (to do).

• Photometric zero-points

  The observing system tries to get one image of each of a set of 8 equatorial standard star fields containing multiple standard stars from Landolt (1992) every night. These are used to compute photometric zero-points in the usual way. Note that the code presently does not solve for the atmospheric extinction, so the accuracy of this procedure is limited by any variations. This needs to be fixed, and for any serious purposes it is currently necessary to manually go in and derive the extinction coefficient for the night in question. Note that the zero-points are computed regardless of weather, but the results will not be accurate in non-photometric conditions!

• Differential photometry and light curves

  This is essentially identical to the Monitor methodology (Irwin² et al. 2007), with a few minor exceptions. Because our stars are high proper motion, it is necessary to have the apertures follow them. This starts off with a guess based on the proper motion from LSPM, and then refined by allowing the centroids to move to mop up any errors before doing the final photometry.

  For the light curves, the most major change of relevance was the addition of a per-star "meridian flip offset", solved along with the per-star magnitudes so the weights (derived empirically from the RMS scatter in the light curves) used to compute frame zero-points are not artificially squashed at the bright end. Compared to the code described in the paper, there were numerous other small changes, but they are of little significance here. We do not use the polynomial fitting option (well, kind of, actually we do but with the degree set to zero), the frames are normalized using a simple per-frame constant (non spatially varying), fit in an iterative, clipped fashion. The per-star magnitudes' zero-point is tied down by forcing the median difference of these compared to the reference ("master") frame to be zero. If the master frame was taken in photometric conditions and was correctly photometrically calibrated, the magnitudes in the light curves should also be properly calibrated in an absolute sense.

  Aperture sizes: aperture photometry is computed in a series of concentric apertures, at factors of two in area, starting from a base aperture radius r of 5 pixels in the 2008-2010 data, and 4 pixels in the 2010-2011 data. The change was made because all telescopes were operated in focus during the entire 2010-2011 season, so there was less need for the extremely large apertures in use for the earlier season. The light curves use apertures of r, sqrt(2)*r, 2*r, and 2*sqrt(2)*r, and the one showing the lowest RMS scatter in the final light curve is adopted.

Other issues

• Focus, or lack thereof

  During the 2008-2009 season, the telescopes were operated intentionally out of focus, adjusted to achieve FWHM of approx. 5 pixels (where a normal in-focus FWHM is approx. 2.5 pixels) in order to be able to collect more photons per exposure before saturation. It was found to be very difficult to maintain a stable defocus, and the FWHM varied substantially with changes in ambient temperature. Stabilizing this was hindered by failure of the focus drive electronics on many of the telescopes. As a result, these data have rather unique systematics and many other related problems.

  For the 2009-2010 season, the decision was made to operate in focus, and to take multiple exposures per visit to a given target where required to achieve adequate signal to noise. Continued failures of the telescope focus drive electronics throughout this season severely hampered attempts to keep the telescopes in focus and stable, and these problems were not fully rectified until some replacement electronics were procured for telescopes 2 and 6 during the 2010-2011 season, although the situation was substantially improved already by 2010 startup.

• Stability of CCD operating temperature, or lack thereof

  During the 2008-2009 season, the temperature setpoint was -20C, which proved to be too optimistic, and became difficult to maintain during the summer. The setpoint was changed to -15C for the 2009-2010 season. During the 2010-2011 season, problems were experienced with the cooler on telescope 2, so the setpoint for this telescope was changed to -10C at the start of the season. The setpoint on telescope 5 was adjusted to -10C on 2010 December 10 due to concerns about condensation in the CCD chamber. The device operating temperature during these seasons was a particular concern due to persistence (see below). Since 2011 October, the operating temperature has been -30C, and has remained stable.

• Condensation / ice crystals and CCD repairs

  Numerous problems were experienced with condensation in the CCD chambers leading to formation of ice crystals on the CCD at the start of the 2009-2010 and 2010-2011 seasons. After the detector housing upgrade in 2011-2012, all of the shutter driver transistors eventually failed, and were dealt with as needed during the season. There have also been occasional shutter failures and readout electronics failures. All of these incidents required removing the detector from the telescope and shipping it for repair, and these were occasionally returned with the electronics adjusted, meaning the gain was altered and it was necessary to re-do all of the calibrations. Additionally, the gain was intentionally altered to make better use of the full-well capacity of the CCDs during the 2011 Summer detector housing upgrade.

  A separate log has been maintained showing the dates of all changes for each telescope, and the data files contain an "instrument version number" that increments by one every time a detector was taken off and put back on. The light curves are split every time there was such a change, and the software solves for separate per-star magnitudes and meridian offsets on either side of the change due to the expected discontinuity in flat-fielding error. The influence of these changes on the photometry is unfortunately exacerbated by a lack of precision and repeatability in the adapter used to align the detector axes with the cardinal axes (and telescope optics), which is approx. 0.5 degree at best.

• Wind shake

  The telescopes are extremely prone to wind shake as used in the present enclosure (which has minimal protection from wind). This results in distorted PSFs in moderate wind, becoming severely distorted in high winds approaching the close-down limit for the enclosure. Due to the design of the building, this affects each telescope differently, and also depends on wind direction, with east generally being more problematic than west due to the lack of shelter. Winds from the northeast and occasionally northwest can also result in extremely poor seeing. This is noticeable in MEarth images even though the intrinsic PSF FWHM is approx. 2 arcsec, which normally renders them relatively insensitive to seeing variations, and can often be distinguished from wind shake by lower winds and a lack of high image ellipticities, as well as the detailed PSF shape.

  Images tend to show visible PSF distortions in winds exceeding 25 km/h, and it is likely there are astrometric distortions at even lower wind speeds. The effect on image quality also depends on the nature of the wind with gusty conditions tending to produce a mixture of undistorted and highly distorted frames. Because aperture photometry is used with large apertures, the light curves are remarkably insensitive to these problems, but there is a noticeable trend of large scatter (often caused by a poor zero-point solution) in high wind. These images can usually be detected by examining the FWHM and ellipticity parameters, although the most distorted frames are known to fool the source classification procedure, which can result in no estimates of FWHM or ellipticity being made because it didn't think there were any stellar sources on the frame, which appears in the output as FWHM = -1.0. It should also be noted that the achieved pointing error is significantly degraded in high wind.

• Periodic error

  The mounts do natively (before correction) show a small amount of periodic error, which can cause image elongation in the y (RA) direction for long exposures. The worm period is 2.5 minutes, so these effects are most pronounced for exposures longer than 30 seconds or so. However, the effects can also show up in high-cadence continuous sequences of exposures taken for transit followup or similar, where it leads to excess scatter in the y coordinate time-series. It is possible for the periodic error to interact badly with the pointing stabilization software loop used during these observations, and this can increase the size of the error since the feedback attempting to fix the worm error occurs out of phase with the error itself.

  Periodic error correction has been used since 2010 January to address this problem. The periodic error curves are based on measurements of the error averaged over approx. 20-25 worm cycles, using a dedicated set of very high-cadence short exposures on a bright, equatorial star taken during low wind. However, it is found that the solutions need to be updated quite frequently (every few months) as there is some drift in the worm error over time. This has occasionally lagged due to the special conditions (extremely low wind) needed to obtain a good calibration or because it was not noticed quickly. The periodic error can change dramatically when the mounts are re-lubricated, so the calibration is always re-run at these times (annual, at startup after the monsoon).

• Meridian flips during exposure sequences or acquisition not handled very well

  The difficulty here is really that it is not possible to tell the mount decided to flip until it is too late, because the only way to know is by solving astrometry on an image. The acquisition code attempts to detect flips between the first and second acquisition exposures and starts over if one is detected. But flips after the second acquisition exposure show up in the science image, because they evade detection until it is too late (the telescope has already slewed to the next target by the time the science image can be analysed). Worse still, the pointing error vector is typically reversed on flip, so the resulting error is roughly double the native mount pointing error because the correction ended up applied backwards. Thankfully, this doesn't happen very often.

  Flips during continuous observations (high cadence followup) are more of a problem. Once a flip is detected, the code tries to treat the situation as an acquisition, but this was buggy for many years due to lack of use and consequently, limited testing. The pointing stabilization loop uses a proportional-integral-derivative filter to prevent oscillation (the additional complexity over an auto-guider, which usually only does the proportional part, is needed because the images being used are the science images, so the feedback is very slow, and the gradual "drift in" that results from a pure proportional feedback turned down low enough to stop oscillating was not adequate when tested). The impulse response of such a loop is ideally similar to that of a critically damped harmonic oscillator, but tuning errors are inevitably present in a real implementation, particularly as the loop has not been individually adjusted for each mount. This is clearly seen in the pixel coordinate time-series after a meridian flip during a continuous observation and can temporarily disrupt the photometry until the loop settles to its new equilibrium.

  The author also notes he was somewhat amused to find that some continuous observations have flipped three times when they arrived at the meridian, because the requested pointing after the first flip and acquisition exposure with the new pointing error correction vector applied happened to be on the other side of the meridian again (temporarily!). Murphy (and his law) are alive and well.

• Time stamps

  The use of commercial data acquisition software and the Windows operating system causes difficulty with achieving accurate time stamps. Consequently, they are only good to a few seconds at best, and it should also be noted that the acquisition software writes them out to the FITS headers only to one second precision (it does not seem to be possible to change this).

  The first data release presented the light curve time stamps in Heliocentric Julian Date (HJD) in the UTC time-system, correcting only for the motion of the Geocenter relative to the Sun using the Stumpff (1980) routine to compute the appropriate state vector, and did not allow for source proper motion between the master frame and the frame being reduced. This has now been changed (for the present release, and all subsequent releases until further notice) to use Barycentric Julian Date (BJD) in the TDB time-system. The treatment used to compute this quantity includes many more corrections, and uses the JPL DE405 solar system ephemeris. It should be at least three orders of magnitude more accurate than the input UTC time stamps from the data acquisition system, given the astrometric quantities stated in the file headers.

Example calibration curves/frames

All taken from tel05 on 2011-02-21 and 2011-10-22 (nights of), where the former date was before the housing change, and the latter after the housing change. In cases where the calibration has changed significantly after the housing upgrade, both are shown (before the change, then after the change; most browsers should display them side-by-side with before on the left, if the window is wide enough), otherwise only the one from before the housing change. Only the bias, dark, and high frequency component of flat are derived per night, all other calibrations were those considered current when that night was reduced.

Non-linearity

The quantity on the vertical axis in this plot is the ratio of measured sky level to the reference sky level, scaled by the ratio of exposure times. The reference frames were exposed around 13,000 ADU (as you can see from the plot) and interleaved between the target frames to track variations in the illumination of the roof. Counts in this diagram are raw, not de-biased, but were corrected for the shutter shading (this is important!). The curve is a simple polynomial, and in the code is parameterized as:

actual = raw (1 + c_1 * raw + c_2 * raw² + c_3 * raw³)

This assumes the two (should) agree at 0. It doesn't matter exactly where things are normalized, as this is removed by the photometric calibration, and is factored into the gain measurement by computing the gain from dome flats that have already had the non-linearity correction applied.

Please note that saturation on the CCD (the full-well capacity) was below 65535 counts on many of the detectors before the housing upgrade, and all of them afterwards.

Bias

Dark

Shutter shading

(derived from twilight flat sequences; low-pass filtered to remove noise; note change in number of shutter blades post-upgrade)

Flat

Raw

De-glow'd

(Elliptical Gaussian fit and subtracted, 2008-2010 and 2010-2011 seasons only)

Low frequency structure removed

(2-D median filter with median box=171, linear box=131)

The first of these is a bad (or good, depending on your perspective) example due to the unfortunately placed doughnut that messes up the glow removal. This is one of the major problems with having to manipulate the flats so much before using them.

Illumination map (raw)

(or, "photometric flat"; determined by dithering a star-field around the detector, by random offsets, many times, and doing photometry of all the stars. this image contains a Gaussian scaled proportional to the measured flux ratio wherever there was a measurement)

Illumination map (smoothed)

(2-D median filter with median box=301, lin box=211)

Final

Fringe

(also filtered, low-pass to remove noise, and high-pass to remove scattered light; most of the change below results from adjustments to the filtering parameters, and not in the device itself)

Bad pixel masks

It does also use these, they were derived mostly by hand. There is no point in me showing a binned one, you wouldn't see much. They are used by the source detection and all later stages in the processing.

Image example

(t05.obj.20110221.00022.fit, before on the left, after on the right)

Notice particularly two clearly visible defects that are not fixed by the standard processing. (1) the "glow" in the centre of the field, caused by scattered light. (2) the "pulldown" in rows containing bright stars.

If we had not removed the "glow" from the flat, we would have divided most of it out of the image, so it would look flat, but would not actually be flat (in a photometric sense). The "glow" (scattered light) is thought to be an additive effect, not a multiplicative one (essentially, a fraction of sky is scattered into a Gaussian-like image on-axis).

This is what the de-skied image it does source detection and photometry on looks like (and plotted with detected sources overlaid on the right):

(see Irwin 1985 for details of how this works; sky background following box size was 64 pix)

The source classification has been used to colour the symbols overlaid on the image, and is used to select potential comparison stars. This essentially works by determining the locus of stellar sources in flux vs flux ratios between different sized apertures, and then folds in ellipticities. In this image, red = stellar (PSF-like), blue = non-stellar (i.e. galaxies), yellow = blended (or more accurately, sources with overlapping isophotes), and green = junk-like (usually "cosmics" or hot pixels, essentially sources that are too sharp to have been through convolution with the PSF as determined from the stellar images).

I have not tried to fix the "pulldown", because of concerns about making the correction robust against real sources in the same lines, given that there is no overscan region to use for a line-by-line overscan subtraction (or similar). But it seems to rarely be a problem for photometry of targets.

Persistence

While no corrections are attempted for this effect (it depends on the full illumination history of the pixel, and for one thing we do not save our field acquisition exposures), I thought it might be useful to show it anyway.

The quantity on the vertical axis of this plot is the fraction of the initial counts (shown at the top) that accumulate in the dark current per second of integration time. This is the integral over a photometric aperture (it was done on stars) so in practice includes a "flux-weighted" range of illumination levels. The model fit assumes the flux decays by a simple exponential law exp(-t/τ). This τ value (20 minutes) is fairly typical of our detectors. The level might seem small, but 43 ADU x 60 seconds (typical exposure) is a quite detectable source above typical sky.

Note that both the normalization, and timescale, in this plot are thought to be temperature dependent. The sense is apparently that with decreasing device temperature, the normalization decreases, and the timescale increases. The effect becomes negligible at LN₂ temperatures for the same device operated in non inverted mode, according to some sources at major observatories.

During the 2011 summer monsoon, the detector housings were upgraded to Apogee's "high cooling" D09 housing, which allowed a lower operating temperature (-30C) and also added a preflash feature using IR LEDs, which is used on all data taken since the upgrade. The combination of these has rendered persistence to be practically no longer a concern for all data taken 2011 October onwards (it would not be true to say it removed persistence, of course, because we have really done the opposite - the preflash floods the detector with a big persistent image, but in doing so makes it stable; the persistent image then behaves simply as if it was an elevated dark current and is removed by dark frame subtraction).

Astrometric calibration

The standard astrometric analysis uses the 2MASS all sky survey as an astrometric reference, and thus is tied (indirectly) to ICRS. This is not the ideal astrometric catalogue, because it does not have proper motions; thus, the solutions will degrade gradually as the observing epoch departs from when the 2MASS was done. I intend to update this to UCAC once I have ironed out the kinks (it currently seems to give solutions with larger scatter).

There is relatively little of note for the calibration itself.

I have found essentially zero radial distortion in the data, as expected given the optical system. The derived values vary from telescope to telescope, but they are all at negligible levels and do not need to be corrected for most purposes.

There does seem to be a significant "magnitude equation" effect (the measured positions depend on the brightness of the star) in the "default" astrometric measurements in the catalogues. These are derived using standard intensity weighted moments (this is often called a "centroid").

The usual solution to this is PSF-fitting. There are good reasons why this is difficult to make work on MEarth data, namely we often have very few usable reference stars for constructing the empirical PSF (especially as it likely needs to be spatially varying due to the wide-ish field of view), and our PSF is also very poorly behaved, so would not be well-fit by analytic models, and is time-variable (e.g. wind shake). While I have played with PSF fitting astrometry on the data I eventually abandoned the attempt as it didn't seem to be getting very far.

Photometric calibration (2010-2011 only)

These steps are performed for the other data, too, but the coefficients differ, and the solutions against Landolt have a larger scatter. Due to the non-standard bandpass produced by the RG715 filter + CCD used in 2008-2010 and from 2011-2012 onwards, the photometry likely cannot be standardized. I have not included equations for it below in order to avoid giving a misleading impression in this regard.

2MASS

There are two interacting components here. The first is the colour equation relating instrumental I_715-895 mags to J-H:

And the second is the extinction correction, which is derived by fitting the residual error (after applying the colour equation) as a function of E(B-V) from Schlegel et al. (1995)'s maps.

The result is:

I_715-895 = J + 1.288 (J - H) + 0.083 + 0.665 E(B-V), scatter ~0.06 mag

Note particularly above that there is a degeneracy in I_715-895-J for J-H colours around 0.6. This is of course caused by the M-dwarfs (this diagram is essentially an I-J vs J-H colour-colour plot; see Leggett 1992 or one of the other tables of empirical M-dwarf colours). The code clips them out of the fit when deriving solutions for individual frames. Most of the giants should have already been eliminated by a J-K cut (these are often heavily extinguished and are not useful in practice for solving frames).

Landolt

This plot shows the difference between our magnitudes and Landolt's I-mags as a function of Landolt's V-I colours. This is how the colour equation used for the absolute calibrations was derived. The result was:

I_715-895 = I + 0.036 (V - I), valid approx. 0 < V-I < 2, maybe 2.5; NOT VALID FOR REDDER STARS OR FOR THE RG715 FILTER.

The scatter about this fit estimated using a robust 1.48*MAD estimator was approx. 0.012 mag over the range of validity, although it should be noted that this may underestimate the true uncertainty if some of the outliers that were rejected are real.

One of the problems with the equatorial fields is that there are not very many red stars. It is not uncommon to find the transformations become significantly different or non-linear for very red stars, which is precisely the regime we wish to operate in for MEarth. This simple linear relationship is thus only valid over the data range shown in the plot. This suffices for calibration of the zeropoints of the instrumental magnitude system (by construction - we are using the same stars as were fit), but for the purposes of deriving standard I-mags rather than I_715-895 mags, we need to supplement it at the red end by fitting empirical measurements of M-dwarfs. One such equation (using 2MASS K to provide the other magnitude needed to make an observed colour) appears in Irwin et al. (2011), although it is only valid for red stars, and used a limited quantity of the available data. I intend do re-do this eventually, time permitting, but it has not been a high priority given that we are no longer using this filter.

---

Document prepared by Jonathan Irwin (jirwin at cfa.harvard.edu).

Substantial contributions and assistance from Mike Irwin, Christopher J. Burke, Philip Nutzman, and the entire MEarth team are gratefully acknowledged.