TTVFaster

First order eccentricity transit timing variations (TTVs) computed in Agol & Deck (2015)

This implements equation (33) from that paper by computing the Laplace coefficients using a series solution due to Jack Wisdom, computing the f_{1,j}^{(+-k)} coefficients given in equation (34) using the functions u and v_+- with coefficients given in Table 1.

Here is an example of using the code in IDL:

first_order$ idl
IDL Version 8.4, Mac OS X (darwin x86_64 m64). (c) 2014, Exelis Visual Information Solutions, Inc.
Installation number: 97443-1.
Licensed for use by: University of Washington

IDL> call_ttv,10
% Compiled module: CALL_TTV.  
% Compiled module: COMPUTE_TTV.  
% Compiled module: TTV_SUCCINCT.  
% Compiled module: LAPLACE_COEFFICIENTS3.  
% Compiled module: LAPLACE_WISDOM.  
% Program caused arithmetic error: Floating illegal operand  
IDL>

This computes the TTVs for a system similar to Kepler-62e/f stored in the file kepler62ef_planet.txt. The TTVs will be plotted to the screen.

Here is an example of using the code in Julia:

Julia$ julia  
               _
   _       _ _(_)_     |  A fresh approach to technical computing  
  (_)     | (_) (_)    |  Documentation: http://docs.julialang.org  
   _ _   _| |_  __ _   |  Type "help()" for help.  
  | | | | | | |/ _` |  |
  | | |_| | | | (_| |  |  Version 0.3.0 (2014-08-20 20:43 UTC)  
 _/ |\__'_|_|_|\__'_|  |  Official http://julialang.org/ release  
|__/                   |  x86_64-apple-darwin13.3.0  

julia> data=readdlm("kepler62ef_planets.txt",',',Float64)  
1x10 Array{Float64,2}:
 3.02306e-5  122.386  -16.5926  -0.00127324  0.0026446  1.67874e-5  267.307  155.466  -0.0025544  0.00117917

julia> include("test_ttv.jl")  
test_ttv (generic function with 1 method)

julia> @time ttv1,ttv2=test_ttv(5,40,20,data);  
elapsed time: 0.345652398 seconds (13941760 bytes allocated)  
julia> @time ttv1,ttv2=test_ttv(5,40,20,data);  
elapsed time: 0.000526126 seconds (20404 bytes allocated)

This computes the TTVs for a system similar to Kepler-62e/f stored in the file kepler62ef_planet.txt. The TTVs will be written to the files inner_ttv.txt and outer_ttv.txt, as well as stored in the variables ttv1 and ttv2. The test_ttv.jl routine accepts jmax (the maximum j to sum to, in this example 5), ntime1 (number of transits of the inner planet), ntime2 (the number of transits of the outer planet), and data which contains the parameters of the planet.

The file kepler62ef_planets.txt contains a comma-separated set of 10 parameters that describe the system: \mu,t0,Period, e cos(omega), e sin(omega) for each planet, where \mu is the mass ratio of the planet to the star, t0 is the initial transit time (of the averaged orbit), Period is the mean orbital period, e is the eccentricity, and omega is the longitude of periastron.

An example of using the C version of this code:

to compile:

gcc -o predict_formula predict_formula.c -lm -O3 -Wall

to run:

./predict_formula 2 test_ic2 0 1600 6 output_test_ic2
./executable    n_planets   param_file t0 tfinal jmax output_file

param_file format:

mstar m1 p1 e1*cos(arg peri1) i1 Omega1 e1*sin(arg peri1) TT1
+ repeat for remaining planets

Units are hardwired into predict_formula.c in the definition of G (global variable, at top) to be masses/solar masses distance/AU and time/day.

all angles are in radians

TT1 = the initial time of transit (according to the mean ephemeris). Make sure that your t0 in the input line matches the t0 your initial transits are refered to in the parameter file!

orbital elements are averaged orbital elements

inclinations do not enter the formula at all; they are in fact never used by the code. Omega is used only to determine the true anomaly of the planet at transit, according to the (hardwired) reference direction that at transit the true longitude is 0.0.

Hence Omega = -PI/2 and i=PI/2 implies an edge on orbit with the ascending node in the plane of the sky.

the output file is: Planet # (zero based) Transit # (zero based) Transit time