Exposure Time Calculator

1. The core relationship

Peak pixel count scales with flux (which falls off exponentially with magnitude) and linearly with exposure time. Empirically, for MuSCAT3:

where coef is a per-band, per-focus calibration constant (the peak ADU a mag=0 star would produce in a 60s exposure at airmass 1.1) and mag_eff is the target's magnitude corrected for airmass (step 3). Solving this the other way gives the exposure needed to reach a target peak:

"Calculate exposure times" uses the second form (solves for exptime given a target ADU); "Calculate peak counts" uses the first form (solves for peak ADU given a fixed exptime).

2. Where coef comes from

For MuSCAT3, coef is calibrated directly from real science frames already in the database: for each usable frame, the peak pixel value is measured (99.9th percentile above the background median — robust against cosmic rays and hot pixels), the target's magnitude in that band is looked up from Pan-STARRS DR1 or SkyMapper DR2, and the formula above is solved for coef. Results are averaged per (band, focus) bin and stored in the database, then interpolated between the calibrated focus points for any focus you select. Instruments without their own frame-based calibration (shown as empirical in the status badges above) fall back to the MuSCAT3 coefficients, scaled by relative telescope aperture, pixel scale, and gain — a rough estimate until enough frames accumulate for a direct calibration.

3. Airmass and focus corrections

Atmospheric extinction dims a target as airmass increases, at a band-dependent rate k (mag per airmass; e.g. 0.15 for g, 0.09 for r, tabulated per band). This is folded into an effective magnitude:

Defocusing the telescope spreads a star's light over more pixels, lowering its peak count for the same total flux — and also grows its FWHM. Both effects are captured by calibrating coef and FWHM at seven focus offsets (0–6mm) and interpolating between them for whatever focus you set.

4. Narrowband filters

Narrowband filters (g_narrow, r_narrow, i_narrow, z_narrow, Na_D) reuse their broadband parent's magnitude and coefficient, offset by log10(narrow bandwidth / broadband bandwidth) — a ~10–100× narrower bandpass lets through proportionally fewer photons, so the same star produces a much lower peak count.

5. Saturation and the recommended exposure

Peak ADU is converted to electrons via the instrument's gain and compared against its full-well capacity to get the % of full well shown per band (green/orange/red below 70%/100%). In "Calculate exposure times" mode, each band's target ADU is set from the chosen saturation limit (50/80/100% of full well, or a custom ADU), and the recommended exposure is the minimum exposure time across all requested bands — whichever band saturates first sets the ceiling for every other band, since MuSCAT instruments expose all bands simultaneously.