Chromatic exponential dip

[abhisrkckl]

Exponential dip model for things like profile change events.

This is different from the tempo2 implementation. In the tempo2 implementation, the exponential is multiplied by a Heaviside step function to construct the dip. This function is discontinuous at the event epoch and is hence not differentiable there. This will be a problem if the event epoch coincides with a TOA. Also, the design matrix becomes degenerate in that case.

Here, I have replaced the step function with a logistic function of a non-zero timescale. The explicit delay expression and its partial derivatives are given below. This function is normalized such that the amplitude A is the extremum value of the dip delay. Note that the extremum occurs at a time sliglthy after the event epoch.

(typos: tau -> tau_i, dt -> t-T_i)

The logistic function timescale is given by the epsilon parameter (EXPDIPEPS). It should be set to a value much less than the event decay timescale (e.g., fraction of a day), and is not a fittable parameter.

This will not work for the 2021 event of J1713+0747 since its chromatic behavior is not well-modeled by a powerlaw in observing frequency.

[abhisrkckl]

[abhisrkckl]

Since we need to fit for t0 also, we can't use the Heaviside theta function; it is not differentiable at 0. I am trying using the logistic function which tends to the Heaviside theta function in the appropriate limit.

[abhisrkckl]

Par file example:

RAJ            05:00:00                     1
DECJ           15:00:00                     1
POSEPOCH       55000
F0             100                          1
F1             -1e-15                       1
PEPOCH         55000
EXPDIPEP_1     54764.272428904194001        1      
EXPDIPAMP_1    1.6641670367524487e-06       1
EXPDIPTAU_1    112.00425959054773           1
EXPDIPIDX_1    -1.9148109887274356          1
EXPDIPEPS      0.01
TZRMJD         55000
TZRSITE        pks
TZRFRQ         1400
EPHEM          DE440
CLOCK          TT(BIPM2019)
UNITS          TDB

[abhisrkckl]

@jeremy-baier Please take a look.

[jeremy-baier]

this is awesome, @abhisrkckl .
to clarify, if the user sets epsilon to a very small value, this function converges to the heavy-side implementation in Tempo2 and enterprise extensions? i have been playing around with this and it seems like the parameters match up very well.
I typed up an achromatic very of this in desmos in case anyone else wants to play around with a comparison of @abhisrkckl ’s implementation vs the tempo2/enterprise extensions version. https://www.desmos.com/calculator/87yxg3muod

[jeremy-baier]

could epsilon be hardcoded to some small number to avoid user error? or perhaps a warning is appropriate if epislon is above some value.

[abhisrkckl]

to clarify, if the user sets epsilon to a very small value, this function converges to the heavy-side implementation in Tempo2 and enterprise extensions? i have been playing around with this and it seems like the parameters match up very well.

could epsilon be hardcoded to some small number to avoid user error? or perhaps a warning is appropriate if epislon is above some value.

As epsilon approaches 0, this approaches the tempo2 version. In practice, this means something much less than the decay timescale. I have checked that 0.001 day works well, and I have included this as the default value. If the user doesn't provide EXPDIPEPS in the par file, this default will be used.

[jeremy-baier]

yea, this looks great.
why specifically is it important to be differentiable in time rather than just differentiable w.r.t. the timing model parameters?

[abhisrkckl]

I meant differentiable in T. Since it appears as (t-T), differentiable in time == differentiable in T.

[dlakaplan]

What does the tempo2 par file look like? Are the shared parameters the same?

[dlakaplan]

I think the tempo2 model was implemented in mattpitkin/tempo2@2edd0bf

[abhisrkckl]

What does the tempo2 par file look like? Are the shared parameters the same?

EXPEP_1        54764.272428904194001       
EXPEP_2        57500.886666587524999       
EXPPH_1        1.6641670367524487e-06      
EXPPH_2        2.1964077367938582999e-06   
EXPTAU_1       112.00425959054773          
EXPTAU_2       26.57155879530331           
EXPINDEX_1     -1.9148109887274356         
EXPINDEX_2     -1.5060816448307765999

[abhisrkckl]

I have added the documentation and set up the parameter aliases such that tempo2 par files are now readable.

[dlakaplan]

This looks good to me. @scottransom : any comments or should we merge?

[scottransom]

Looks good to me, too! Nice job!