Computing the Intensity Upper Limit and Upper Value for Flux Uncertainty for an Unresolved Source

Overview

Synopsis:

The tool aprates, computes values and bounds for source intensity quantities (net counts, source rate, photon flux, energy flux) using counts and exposure data obtained in source and background apertures, as shown in the Compute Net Counts, Rate, or Flux for Point Sources thread. aprates uses Bayesian statistics to compute the background-marginalized, posterior probability distribution for source intensity, assuming non-informative prior distributions for background and source intensity (for details on the algorithm, see the Background Marginalized X-ray Source Intensity memo). The posterior distribution can be used to determine intensity value and confidence bounds or intensity upper limit.

Upper limits vs. upper value for flux uncertainty

If the lower bound for the uncertainty of the flux reaches 0, only the upper bound is reported. This number is sometimes erroneously used as the upper flux limit for an undetected source, but strictly speaking the uncertainty on a flux measurement and the upper limit of an undetected source are two different questions, even though the numbers often come out to be in a similar range. When talking about the upper limit for an undetected sources (the maximal flux that a source could have without being detected), a single number for conf is not sufficient. Instead, two numbers are needed: The probability of detecting a source in background fluctuations (false positive) and the probability of missing a real source because it is compatible with background (false negative). The aplimits tool performs this calculation.

The aplimits tool estimates the upper limit for a detection. This is a property of the detection process and depends on the values chosen for two error conditions: First, the maximum probability of a false detection (false positive), i.e. that a background fluctuation alone exceeds the calculated upper limit, and second the probability that a source with a flux of exactly the upper limit will be missed (false negative), because Poisson fluctuation of background and source counts lead to an observed count rate that is below the detection limit. The value of the upper limit depends only on the background rate and the probabilities chosen for the two errors, not on the observed number of counts in the source region. This tool is based on the following article: On computing upper limits to source intensities (Kashyap et al. 2010, ApJ, 719, 900).

The aprates steps in this thread have been automated in the srcflux script. This thread shows how to get the upper value on the flux uncertainty this way.

The thread then shows how to run aprates and aplimits step-by-step to determine how x-ray bright a source could be at the given location in the observation. The intensity upper limit is calculated in photon units in this thread.

Purpose:

To calculate the intensity upper limit for a source within a specified aperture.

Last Update: 10 May 2023 - Updated to make the distinction between true upper limits, now available using the new aplimits tool, and the upper value on the flux uncertainty computed by srcflux and aprates.

Parameters for /home/username/cxcds_param/ulimit.par


src_cnts,r,h,0.0,,,
src_cnts_err_lo,r,h,INDEF,,,
src_cnts_err_up,r,h,4.17198,,,
src_cnts_conf,r,h,0.684242,,,
src_cnts_status,i,h,0,,,
src_cnts_signif,r,h,0,,,
src_cnts_mode,r,h,0,,,
src_rate,r,h,0.0,,,
src_rate_err_lo,r,h,INDEF,,,
src_rate_err_up,r,h,5.775e-05,,,
src_rate_conf,r,h,0.684242,,,
src_rate_status,i,h,0,,,
src_rate_signif,r,h,0,,,
src_rate_mode,r,h,0,,,
photflux_aper,r,h,0.0,,,
photflux_aper_err_lo,r,h,INDEF,,,
photflux_aper_err_up,r,h,1.58745e-07,,,
photflux_aper_conf,r,h,0.680054,,,
photflux_aper_status,i,h,0,,,
photflux_aper_signif,r,h,0,,,
photflux_aper_mode,r,h,0,,,
flux_aper,r,h,0.0,,,
flux_aper_err_lo,r,h,0,,,
flux_aper_err_up,r,h,0,,,
flux_aper_conf,r,h,0,,,
flux_aper_status,i,h,4,,,
flux_aper_signif,r,h,0,,,
flux_aper_mode,r,h,0,,,
eflux_aper,r,h,0.0,,,
eflux_aper_err_lo,r,h,0,,,
eflux_aper_err_up,r,h,0,,,
eflux_aper_conf,r,h,0,,,
eflux_aper_status,i,h,4,,,
eflux_aper_signif,r,h,0,,,
eflux_aper_mode,r,h,0,,,
# ---- Inputs below ----
_n,i,h,15,,,
_A_s,r,h,78.5398,,,
_alpha,r,h,1,,,
_T_s,r,h,72242.2,,,
_E_s,r,h,2.60217e+07,,,
_eng_s,r,h,0,,,
_flux_s,r,h,0,,,
_m,i,h,132,,,
_A_b,r,h,628.318,,,
_beta,r,h,0,,,
_T_b,r,h,72242.2,,,
_E_b,r,h,2.60147e+07,,,
_eng_b,r,h,0,,,
_flux_b,r,h,0,,,

Parameters for /home/username/cxcds_param/ulimit_2.par


min_counts_detect,i,h,22,,,"Detection threshold"
upper_limit,r,h,8.537294343113899e-05,,,"Upper limit"