FRED Light Curve

Definition

The Fast Rise Exponential Decay (FRED) light curve is defined mathematically as follows:

import numpy as np 
F(t) = F0*np.exp(-t/tau) # if t > t_start 

F(t) = 0 # otherwise

Integrated Flux

import numpy as np
F_int = F0*tau*(np.exp(-max(T_start,t_start)/tau) - np.exp(-T_end/tau))/(T_end - T_start)

Probability Contours

Sample probability contours for the FRED light curve

Lines

Each probability contour plot has three vertical lines or curves:

Shortest observation

The leftmost vertical line in the above image represents the shortest observation input

Longest Gap

The second line is from the left is a curve representing the shortest possible duration for a transient that will always be detected. Mathematically:

F_int = S_obs 
# But we can substitute for F_int
F0*tau*(np.exp(-max(T_start,t_start)/tau) - np.exp(-T_end/tau))/(T_end - T_start) = S_obs
F0 = S_obs*(T_end - T_start)/tau/(np.exp(-max(T_start,t_start)/tau) - np.exp(-T_end/tau))/tau)))

We now have F0 as a function of tau. We simply need to figure out what T_end, T_start, t_start, and S_obs are.

S_obs is the sensitivity of the observation that would detect the transient that would have fallen in the longest gap:

sens_maxgap = obs['sens'][np.where((gaps[:] == max(gaps)))[0]+1][0]

We can set t_start=0 and set this as the beginning of our "epoch."

By definition of the problem, the transient starts at the beginning of a gap and gets detected at the end of the gap. Therefore T_start is the duration of the longest gap. T_end is T_start plus the duration of the observation.

Longest Timescale

The third line from the left is a curve representing the longest duration for a transient before it is detected as constant.

We follow the same equation as before:

F0 = S_obs*(T_end - T_start)/tau/(np.exp(-max(T_start,t_start)/tau) - np.exp(-T_end/tau))/tau)))

But the variables change: S_obs is the sensitivity of the last observation.

We can set t_start=0 and set this as the beginning of our "epoch."

By definition of the problem, the transient starts at the beginning of the first observation and gets non-detected at the end of the observations. Therefore T_start is equal to the duration of the observations minus the duration of the last observation. T_end is the duration of the survey.