I investigate the errors one incurs when transferring the calibration of a bright star to fainter ones. This document concentrates on random errors, though it will mention a few sources of systematic error.
The goal is to go from very precise measurements of a star at V=5 to nearly-as-precise measurements of a star at V=25. I will use a target of 0.005 mag = 0.5 percent for the uncertainty in the relative measurements of the faintest star. In order to highlight the different sources of error for different methods of analysis, I will discuss three scenarios for the transfer process.
I do not yet consider the "ordinary" scenario, calibrating stars of a range of colors in multiple fields. However, one may extrapolate from the simpler situations to this most complicated one. It is certain that its errors will be larger than any in scenarios A, B, or C.
I perform calculations for a specific set of equipment:
The "narrow-band" filters are probably better for transferring the calibration, since they are less sensitive to the detailed shapes of the stellar spectra.
At the end of the document, I'll provide an example of an observing plan for summer 2002.
The goal is to move from V=5 to V=25. It would seem reasonable to do so in as few steps as possible, since both random and systematic errors may occur between steps. However, one is limited by the dynamic range of the detector; I consider CCDs in the optical.
Need to look up the appropriate numbers for near-IR devices.
If one wishes to compare a bright star with a faint star in the same image, one faces two competing effects:
With the MOSAIC and miniMOSAIC imaging cameras at WIYN, it turns out to be practically impossible to achieve a signal-to-noise (S/N) ratio of 100 for a star 5 magnitudes fainter than the brightest non-saturated star. There just isn't enough dynamic range on the chip. (Theoretically, defocusing to more than 3 arcsec FWHM and then using only the central few pixels portion of the PSF might give S/N = 100 for delta = 5 mag, but any spatial variations in the PSF would absolutely kill the measurement).
I therefore have chosen to restrict each transfer to a difference of 3 magnitudes between the "bright" and "faint" stars in question. Given this choice, we face the following steps:
from mag to mag name of step ------------------------------------------ 5.0 8.0 T1 8.0 11.0 T2 11.0 14.0 T3 14.0 17.0 T4 17.0 20.0 T5 20.0 23.0 T6 23.0 26.0 T7
There are seven steps. If we start with a perfect calibration of a V=5 star, demand a final goal of 0.005 magnitudes accuracy, and incur an equal (and independent!) error in each step, then each of the seven steps must yield a total error
error per step <= 0.0019 mag = 0.19 percent
If there are shared systematic errors between the steps, which seems very likely -- in fact, certain -- then the permitted error per step must be even smaller.
Suppose that we pick a field which has a large number of stars which are absolutely identical (color, temperature, spectral type), but span our desired range in magnitude from V=5 to V=25. We can use simple differential photometry between "bright" and "faint" stars in the same image to transfer the calibration. Thus, we avoid
We do not avoid having to apply extinction corrections. Suppose the field stretches 1 degree from top to bottom, and is observed at 30 degrees from the zenith. The difference in magnitude between two identical stars at the top and bottom of the field will be
m1 - m2 = k * (X1 - X2) = k * (0.012 airmass)
The first-order extinction coefficient k is largest in the blue, for optical measurements. In the B-band, it has a value approximately k=0.40 mag/airmass, and smaller values in the redder bands. We thus find
m1 - m2 = 0.005 mag in B-band = 0.0025 mag in V-band = 0.0015 mag in I-band
This is a systematic effect we must take into account.
I've worked through the numbers for some of the seven transfer steps, and come up with the following observational requirements. All exposure calculations made for the middle of the optical (V-band). I wrote my own code to perform the exposure-time calculations; you can use a fill-in-the-blank form or examine the Perl code directly.
To transfer calibration in a single passband transfer indiv. N total time step mags telescope filter exp length exp exp + readout ------------------------------------------------------------------------ T1 5 - 8 0.9m w/mask narrow 10 sec 20 1.2 hours T2 8 - 11 0.9m narrow 10 sec 12 0.7 hours T3 11 - 14 0.9m narrow 100 sec 13 1.0 hours T4 14 - 17 3.5m narrow 300 sec 12 1.5 hours T5 17 - 20 3.5m narrow 1000 sec 173 57 hours broad 150 64 6.4 T6 20 - 23 ? T7 23 - 26 ? ------------------------------------------------------------------------
The very first transfer, T1, requires a telescope smaller than the WIYN 0.9m -- the table above postulates an effective diameter of 0.15 meters. We can easily place a temporary mask of this size on the WIYN 0.9m telescope. Scintillation noise in a 10-second exposure with a 0.15-m telescope is only 0.005 mag, less than the 0.008 mag due to photon noise.
The transfers T6 and T7, to the faintest stars, must be done from space. The background sky signal becomes the limiting factor, and even a giant telescope would be unable to reach the required S/N ratio for V=26 stars in any reasonable time. A ground-based 8-m telescope, for example, would require 4500 hours of exposure through a broad-band filter to yield a magnitude measurement accurate to 0.0019 mag.
Remember that the table above provides time estimates for calibration in a single passband. If one wished to transfer at, say, 10 different wavelengths across the optical, one would multiply the times above by a factor of 10.
In this scenario, I consider the systematic errors one makes by comparing stars of slightly different colors, without any correction. I will also consider the errors after having made a correction based upon theoretical models of atmospheric extinction.
When target stars are not identical, one must take into account the differential extinction between them, due to the different shapes of their continuum flux. Blue stars will suffer more extinction at the same airmass than red ones. The effect is usually described as
observed mag m = m0 + k'*(B-V)*Xwhere X is the airmass and k' is the second-order, color-dependent extinction term, and (B-V) is a representative color. Is it possible for this effect to produce significant errors in our transfer from one star to another?
The extinction coefficient k' is roughly 0.03 in the B-band. At longer wavelengths, it is small enough that few have attempted to measure it. Consider the effect in the B-band as a worst-case scenario.
I calculate the DIFFERENCE between the color-dependent extinction for the faint and bright stars to be 0.0007 mag (if bright G8), or 0.020 mag (if bright K4). Making no correction incurs no significant error if the two stars are as close in color as G8 and K0, but a signficant error if the stars are as far apart in color (about 0.50 mag in B-V) as K4 and K0. Comparing a blue star (say, an A0) with a red one (say, a K0) will require a much larger correction.
Making the correction requires that we know the value of the extinction coefficient k'. One can calculate it based on the spectrum of extinction by the Earth's atmosphere and the continuum shape of a star, but variations in air properties from night to night will occur. Note that if one assumes a value for k' which is in error by only 10%, one will cause a systematic error in relative magnitudes for stars as similar as K0 and K4 which is as large as the entire error budget (0.0019 mag) for a single transfer step.
This color-dependent extinction depends on the width of the passband. As indicated by the calculations above, we must be careful to correct for it when reducing broad-band measurements of stars in the B and probably V bands. The effect will be much smaller, probably negligible, if one observes through narrow-band filters with widths of order 50 Angstroms.
It seems clear to me that one must rely on stars of very nearly the same color to carry magnitudes from bright stars to faint ones, even in the same field.
In this scenario, I consider the systematic errors one makes by comparing identical stars in different fields. One must make large corrections for first-order extinction, which depend sensitively on the accuracy with which extinction can be determined.
Recall that we must apply the extinction corrections even for stars within a single wide-angle field, as shown above.
First-order extinction is usually described as follows:
observed mag m = m0 + k*Xwhere X is the airmass and k is the first-order extinction coefficient. In this model, stars of all colors are affected equally, and the value of k is the same for broad-band and narrow-band filters with the same central wavelength. Typical values for k are
passband k ----------------------------------- U 0.5 mag/airmass B 0.4 V 0.2 R 0.1 I 0.08
Because the atmosphere changes from night to night, so does the amount of extinction. A good source for empirical measurements of the variations both night-to-night, and within a night, is Rufener, A&A, 165, 275 (1986). Rufener found that the extinction at La Silla showed correlations over periods of several nights and had clear seasonal trends. He indicates that the determination of k on any single night has an uncertainty of between 0.005 and 0.008 mag/airmass.
Let us try a sample calculation to see the magnitude of the effect when comparing two stars in different fields at different airmasses. Suppose we observe in B-band stars A and B, at airmass 1.2 and 1.5, respectively. Suppose further that the true extinction coefficient k is 0.40, but we mistakenly determine it to be 0.39 for the current night. Then
true extinction correction 0.40 * (1.5 - 1.2) = 0.120 mag applied " " 0.39 * (1.5 - 1.2) = 0.117 mag ------------------------------------------------------------------- error in mag diff (A-B) 0.003 mag
This error is larger than the budget of 0.0019 mag per transfer step. It does not help to measure many stars in each field instead of a single star: the same error will appear between any two stars in the two different fields.
How can we reduce this source of error?
If one is setting up a net of standard stars in a new passband, then one MUST observe over at least one full year anyway, in order to "close the gap" around the sky.
The errors we make in correcting for first-order extinction (like those for second-order, color-dependent extinction) will be largest at the blue end of the visible range: atmospheric extinction is largest in the blue.
The main goals of an observing run in the near future should be
It is quite possible that a successful run would yield no measurements which would actually be used for the transfer process.
The documentation for the WIYN miniMOSAIC camera indicates that someone has tested the accuracy of its shutter. I could find no similar indication for the MOSAIC camera. My first priority at WIYN would be to test, test and test again the repeatability and relative accuracy of the shutters on the two cameras. One can run tests during the day: take dome flats with a range of exposure times (being careful to take sequences of mixed times, like this: "10 sec, 20, 10, 30, 10, 40, 20, 30, 20, 40, etc."), take dome flats with the 0.9-m telescope with and without an aperture mask. Scattered light will plague these tests, but one will still learn the gross characteristics of the instrument. One can take dome flats at night, especially when the weather is poor, and get more accurate results. These tests would also allow one to check the linearity of the CCDs.
One can make a second set of shutter tests on good nights by taking similar sequences of exposures of a star field with a large number of stars over a wide range of magnitudes. The outer regions of M13 or M3, or open clusters NGC 6633 and M39, would be good choices. By measuring differential magnitudes for stars over a large range of brightness in a single field, and comparing the results for a range of exposure times, one can test for shutter effects and non-linearity ... and get a good estimate for the size of random errors.
For example, comparing the relative magnitudes of two bright stars of very similar brightness in a large number of (say) 10-second exposures might reveal that a scatter of 0.03 mag. That's too large to be photon noise, so one might conclude that the shutter's motion varies by +/- 3 percent, from 9.97 to 10.03 seconds. Making a set of long exposures (say, 300 seconds), one might find a pair of similar bright stars to have a scatter of 0.005 mag. Again, one can use this result to place constraints on the shutter's repeatability; but one might also start to place constraints on the throughput of the telescope, since photon noise might start to become a significant source of error.
I have calculated above the exposure times which ought to be appropriate for each step in the transfer process. One of the tasks for the observing run would be to observe stars of known magnitudes with each telescope, through both narrowband and broadband filters, to make sure that the calculations accurately predict the actual measurements. That is, verify that a 10-second exposure through a narrowband filter on the WIYN 0.9m telescope doesn't saturate a star of V=5 and provides the expected signal-to-noise ratio for a star of V=8, and so on for the other transfer steps.
Finally, I suggest that one use the WIYN telescopes with both broadband and narrowband filters to take images of the northern SNAP field: the North Ecliptic Pole. The idea is to make a first census of the stellar population in the area. How many red and blue stars are there per square degree at tenth magnitude? At fifteenth magnitude? At what point -- if ever -- does crowding become an issue for ground-based measurements? There are a number of fifth magnitude stars near the NEP; where should one pick the SNAP field in order to stay as far as possible from all of them?
If one picks the right combination of narrowband filters, one can distinguish giants from dwarf based on photometry alone. See Geisler, PASP, 96, 723 (1984) for an explanation of the technique, and Morrison et al., AJ, 121, 283 (2001) for one application. The WIYN filter set includes one filter which is similar to the "DDO 51" filter used in the Washington system: