The LSST is a the nation's top priority next-gen ground-based astronomy project, with the objective ofconducting observations of the entire accessible sky, with about 800 visits per field. The scheduler for the projectwill determine the order in which these fields are observed, with the goal of maximizing some scientific merit function.A portion of that merit function has to do with Fourier coverage in the time domain.The variability in sky conditions (cloud cover, sky brightness and atmospheric seeing) makes the schedulingproblem non-trivial. It's a traveling salesman problem with a stochastic component, subject to certain constraints.Our goals are to1. devise a sensible quantitative framework that would accommodate various merit functions.2. assess whether an instantaneous (nightly) sequence of observations optimization will achieve a global optimum,3. build some numerical tools to make a toy model and try out some implementation schemes.
LSST synopsis
The telescope feeds a focal plane that spans a field of view of 9.6 square degrees, so it's 3.1 degrees on a side.
The default plan calls for a pair of 15 second exposures at each pointing of the telescope, and this pair is termed a "visit" to a field.
The LSST will be situated in Chile at a latitude of -30 degrees. From there it can usefully access the sky up to a declination of +30 degrees. The full sky is 41,253 square degrees, and LSST plans to conduct a survey over 20,000 (or 18,000?) square degrees in addition to a few "deeper" fields that will receive longer and more frequent observations.
This implies that we need to schedule observations in 6 bands, spread across 20,000/9.6~2100 fields over the course of 10 years.
Images are obtained in 6 different optical passbands, designated u,g,r,i,z and y, that span the atmospheric cutoff at 340 nm up to the longest wavelength that can be detected in silicon CCDs, about one micron.
The time it takes the telescope to slew from one place to another on the sky is determined by two upper limits, one is the maximum angular acceleration (about 3 deg/s^2) and the other is the maximum angular rate (3 deg/s).
Merit Function Ingredients.
Observational effectiveness is impacted by both deterministic and stochastic factors. A reasonable initial merit function is the signal to noise ratio for flux determination of an "unresolved" source (i.e. a star). The numerator contains the flux from the source. For the LSST case, the denominator (noise) is dominated by the Poisson noise from the sky background, within the footprint of the source. This footprint is turn dominated by time-variable turbulence in the atmosphere, which produces a Gaussian flux distribution with a FWHM that astronomers term "seeing". The surface brightness of the night sky depends on time of day, on the phase of the moon and distance to the moon, and on the optical filter.
The signal to noise ratio for measuring the flux phi (at the telescope input aperture) from an unresolved point source for an integration time t scales as
SNR~ (phi)(sqrt(t)) / (FWHM)(sqrt(sky)).
Both phi and FWHM depend on the "airmass" a of the observation. Looking straight up corresponds to a=1. Looking at an angle of 60 degrees from the zenith has a=2. The minimum airmass for any given field occurs on the meridian, which is the great circle on the sky that include the zenith and the celestial pole at declination = -90 degrees.
The atmosphere attenuates the light by an amount that depends on the airmass as well as the passband. So the SNR as a function of airmass and exposure time is a sensible merit function that we seek to optimize, summed over all fields and over the 10 year duration of the survey. This is subject to some science-driven constraints, such as the desire to achieve a uniform survey-integrated SNR across the survey area and the desire to achieve some temporal sampling cadence across the different portions of the sky.
The FWHM varies depending on atmospheric conditions, and is usually expressed in units of arcseconds. LSST expects to achieve a median seeing of 0.6 arcsec, with a long term distribution as shown in the Figure below.
The deterministic factors that influence the merit function are
- sky brightness, as a function of optical filter, phase of the moon and distance to the moon.
- time lost during telescope slews and focal plane readout (~3 seconds per image)
- zenith angle dependence of the signal to noise ratio
- exposure time spent on each field
Stochastic factors are
- seeing
- cloud cover
Oct 18 2013, C. Stubbs
Observations obtained at angles z from zenith suffer from two effects:
1) additional optical attenuation due to increased atmospheric path length
2) degraded "seeing".
| band | central wavelength | extinction (mag per airmass) | seeing degradation compared to r band at zenith |
|---|---|---|---|
| u | 350 nm | 0.40*a | (1.1)*a0.6 |
| g | 450 nm | 0.18*a | (1.07)*a0.6 |
| r | 650 nm | 0.10*a | (1.0)*a0.6 |
| i | 750 nm | 0.08*a | (0.97)*a0.6 |
| z | 850 nm | 0.05*a | (0.94)*a0.6 |
| y | 1000 nm | 0.04*a | (0.91)*a0.6 |
plot of seeing degradation vs. airmass, and polynomial fit:
Decent approximation to seeing degradation vs. airmass a is
S=0.35+0.72a^2-0.07a
Signal to Noise degradation vs. airmass:
SNR scales as source flux in the numerator and (for unresolved objects) as seeing in the denominator. Flux at an airmass "a" is reduced by a factor f(a)=10^(x*(a-1)/2.5) where x is the extinction coefficient listed in the table above. So SNR vs. airmass at fixed exposure time for unresolved point sources scales as SNR(a)~10^(x*(a-1)/2.5)/(0.35+0.72a^2-0.07a).
| airmass | seeing degradation | SNR_u | SNR_g | SNR_r | SNR_i | SNR_z | SNR_y |
|---|---|---|---|---|---|---|---|
1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |
| 1.1 | 1.06 | 0.91 | 0.93 | 0.94 | 0.94 | 0.94 | 0.94 |
| 1.2 | 1.11 | 0.83 | 0.87 | 0.88 | 0.88 | 0.89 | 0.89 |
| 1.3 | 1.17 | 0.77 | 0.81 | 0.83 | 0.84 | 0.84 | 0.85 |
| 1.4 | 1.22 | 0.71 | 0.77 | 0.79 | 0.79 | 0.80 | 0.81 |
| 1.5 | 1.27 | 0.65 | 0.72 | 0.75 | 0.76 | 0.77 | 0.77 |
| 1.6 | 1.32 | 0.61 | 0.68 | 0.71 | 0.72 | 0.73 | 0.74 |
| 1.7 | 1.37 | 0.56 | 0.65 | 0.68 | 0.69 | 0.71 | 0.71 |
| 1.8 | 1.42 | 0.53 | 0.62 | 0.65 | 0.66 | 0.68 | 0.68 |
| 1.9 | 1.46 | 0.49 | 0.59 | 0.62 | 0.64 | 0.65 | 0.66 |
| 2.0 | 1.52 | 0.46 | 0.56 | 0.60 | 0.61 | 0.63 | 0.64 |
| fits | Seeing=0.35+0.72a^2-0.07a | SNR_u=2.1-1.4*a+0.30*a^2 | SNR_g=1.9-1.1*a+0.23*a^2 | SNR_r=1.8-a+0.21*a^2 | SNR_i=1.8-0.98*a+0.20*a^2 | SNR_z=1.7-0.94*a+0.19*a^2 | SNR_y=1.7-0.93*a+0.19*a^2 |
Slew times.
Oct 19 2013, CWS.
angular distance moved is limited by both the maximum angular rate and the maximum angular acceleration. If we imagine the max angular rate is 3 deg/s and max angular acceleration is 3 deg/s/s, then to move one field width requires a slew of 3 degrees in angle (gives small overlap). For no coast phase this takes a time given by t=2*sqrt(2*1.5deg/alpha)=2 seconds. The maximum angular rate achieved is omega=alpha(1)=3 deg/sec. So for these parameters for any slew larger than a field width, we are angular-rate-limited, and the slew requires a time t_slew~2+dtheta/3 seconds. IF we can slew during readout, the overhead between images separated by an angle theta then is approximately (2+theta/3) seconds.
Arguably the system operation is optimized if the slew time is exactly equal to the (unavoidable) 2 second readout time. This suggests that we should tile the sky with overlapping observations that slew by half a field width between successive exposures.
Operating on the meridian in this mode, with 15 second exposures we'll assume it would take 20 seconds total on average, per exposure. It would take 24*(3.1/360)~12 minutes for the sky to move by one field width, on the equator. In the course of 12 minutes we can acquire 12minutes*3images/min=36 images on the meridian. At this half-overlap rate we could cover 36*3.1/2~56 degrees.
Sky rotation.
The position of objects on the sky changes in right ascension direction at an angular rate of 15 degrees per hour times cos(declination).