Fit Radial Velocity
You can use Octofitter to fit radial velocity data, either alone or in combination with other kinds of data. Multiple instruments (up to 10) are supported, as are gaussian processes to model stellar activity..
Radial velocity modelling is supported in Octofitter via the extension package OctofitterRadialVelocity. To install it, run pkg> add OctofitterRadialVelocity
For this example, we will fit the orbit of the planet K2-131 to perform the same fit as in the RadVel Gaussian Process Fitting tutorial.
We will use the following packages:
using Octofitter
using OctofitterRadialVelocity
using PlanetOrbits
using CairoMakie
using PairPlots
using CSV
using DataFrames
using DistributionsWe will start by downloading and preparing a table of radial velocity measurements, and create a StarAbsoluteRVLikelihood object to hold them.
The following functions allow you to directly load data from various public RV databases:
HARPS_DR1_rvs("star-name")HARPS_RVBank_observations("star-name")Lick_rvs("star-name")HIRES_rvs("star-name")
Basic Fit
Start by fitting a 1-planet Keplerian model.
Calling those functions with the name of a star will return a StarAbsoluteRVLikelihood table.
If you would like to manually specify RV data, use the following format:
rv_like = StarAbsoluteRVLikelihood(
# epoch is in units of MJD. `jd` is a helper function to convert.
# you can also put `years2mjd(2016.1231)`.
# rv and σ_rv are in units of meters/second
(;inst_idx=1, epoch=jd(2455110.97985), rv=−6.54, σ_rv=1.30),
(;inst_idx=1, epoch=jd(2455171.90825), rv=−3.33, σ_rv=1.09),
# ... etc.
)For this example, to replicate the results of RadVel, we will download their example data for K2-131. and format it for Octofitter:
rv_file = download("https://raw.githubusercontent.com/California-Planet-Search/radvel/master/example_data/k2-131.txt")
rv_dat_raw = CSV.read(rv_file, DataFrame, delim=' ')
rv_dat = DataFrame();
rv_dat.epoch = jd2mjd.(rv_dat_raw.time)
rv_dat.rv = rv_dat_raw.mnvel
rv_dat.σ_rv = rv_dat_raw.errvel
tels = sort(unique(rv_dat_raw.tel))
rv_dat.inst_idx = map(rv_dat_raw.tel) do tel
findfirst(==(tel), tels)
end
rvlike = StarAbsoluteRVLikelihood(
rv_dat,
instrument_names=["harps-n", "psf"],
)StarAbsoluteRVLikelihood Table with 4 columns and 70×1 rows:
epoch rv σ_rv inst_idx
┌───────────────────────────────────
1,1 │ 57782.2 -6682.78 3.71 1
2,1 │ 57783.1 -6710.55 5.95 1
3,1 │ 57783.2 -6698.96 8.76 1
4,1 │ 57812.1 -6672.32 4.0 1
5,1 │ 57812.2 -6672.0 4.22 1
6,1 │ 57812.2 -6690.31 4.63 1
7,1 │ 57813.0 -6701.27 4.12 1
8,1 │ 57813.1 -6700.71 5.34 1
9,1 │ 57813.1 -6695.97 3.67 1
10,1 │ 57813.1 -6705.87 4.23 1
11,1 │ 57813.1 -6694.27 5.12 1
12,1 │ 57813.2 -6699.99 5.0 1
13,1 │ 57813.2 -6696.17 10.43 1
14,1 │ 57836.0 -6675.4 7.03 1
15,1 │ 57836.0 -6665.92 7.48 1
16,1 │ 57836.0 -6661.9 6.03 1
17,1 │ 57836.1 -6657.92 5.08 1
⋮ │ ⋮ ⋮ ⋮ ⋮Now, create a planet. We can use the RadialVelocityOrbit type from PlanetOrbits.jl that requires fewer parameters (eg no inclination or longitude of ascending node). We could instead use a Visual{KepOrbit} or similar if we wanted to include these parameters and visualize the orbit in the plane of the sky.
@planet b RadialVelocityOrbit begin
e = 0
ω = 0.0
# To match RadVel, we set a prior on Period and calculate semi-major axis from it
P ~ truncated(Normal(0.3693038/365.256360417, 0.0000091/365.256360417),lower=0.00001)
a = cbrt(system.M * b.P^2) # note the equals sign.
τ ~ UniformCircular(1.0)
tp = b.τ*b.P*365.256360417 + 57782 # reference epoch for τ. Choose an MJD date near your data.
# minimum planet mass [jupiter masses]. really m*sin(i)
mass ~ LogUniform(0.001, 10)
end
@system k2_132 begin
# total mass [solar masses]
M ~ truncated(Normal(0.82, 0.02),lower=0) # (Baines & Armstrong 2011).
# HARPS-N
rv0_1 ~ Normal(0,10000) # m/s
jitter_1 ~ LogUniform(0.01,100) # m/s
# FPS
rv0_2 ~ Normal(0,10000) # m/s
jitter_2 ~ LogUniform(0.01,100) # m/s
end rvlike bSystem model k2_132
Derived:
Priors:
M Truncated(Distributions.Normal{Float64}(μ=0.82, σ=0.02); lower=0.0)
rv0_1 Distributions.Normal{Float64}(μ=0.0, σ=10000.0)
jitter_1 Distributions.LogUniform{Float64}(a=0.01, b=100.0)
rv0_2 Distributions.Normal{Float64}(μ=0.0, σ=10000.0)
jitter_2 Distributions.LogUniform{Float64}(a=0.01, b=100.0)
Planet b
Derived:
e, ω, a, τ, tp,
Priors:
P Truncated(Distributions.Normal{Float64}(μ=0.001011081092683449, σ=2.4914008313533153e-8); lower=1.0e-5)
τy Distributions.Normal{Float64}(μ=0.0, σ=1.0)
τx Distributions.Normal{Float64}(μ=0.0, σ=1.0)
mass Distributions.LogUniform{Float64}(a=0.001, b=10.0)
Octofitter.UnitLengthPrior{:τy, :τx}: √(τy^2+τx^2) ~ LogNormal(log(1), 0.02)
StarAbsoluteRVLikelihood Table with 4 columns and 70×1 rows:
epoch rv σ_rv inst_idx
┌───────────────────────────────────
1,1 │ 57782.2 -6682.78 3.71 1
2,1 │ 57783.1 -6710.55 5.95 1
3,1 │ 57783.2 -6698.96 8.76 1
4,1 │ 57812.1 -6672.32 4.0 1
5,1 │ 57812.2 -6672.0 4.22 1
6,1 │ 57812.2 -6690.31 4.63 1
7,1 │ 57813.0 -6701.27 4.12 1
8,1 │ 57813.1 -6700.71 5.34 1
9,1 │ 57813.1 -6695.97 3.67 1
10,1 │ 57813.1 -6705.87 4.23 1
11,1 │ 57813.1 -6694.27 5.12 1
12,1 │ 57813.2 -6699.99 5.0 1
13,1 │ 57813.2 -6696.17 10.43 1
14,1 │ 57836.0 -6675.4 7.03 1
15,1 │ 57836.0 -6665.92 7.48 1
16,1 │ 57836.0 -6661.9 6.03 1
17,1 │ 57836.1 -6657.92 5.08 1
⋮ │ ⋮ ⋮ ⋮ ⋮
Note how the rvlike object was attached to the k2_132 system instead of the planet. This is because the observed radial velocity is of the star, and is caused by any/all orbiting planets.
The rv0 and jitter parameters specify priors for the instrument-specific offset and white noise jitter standard deviation. The _i index matches the inst_idx used to create the observation table.
Note also here that the mass variable is really msini, or the minimum mass of the planet.
We can now prepare our model for sampling.
model = Octofitter.LogDensityModel(k2_132)LogDensityModel for System k2_132 of dimension 9 with fields .ℓπcallback and .∇ℓπcallback
Sample:
using Random
rng = Random.Xoshiro(0)
chain = octofit(rng, model)Chains MCMC chain (1000×27×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
Wall duration = 28.07 seconds
Compute duration = 28.07 seconds
parameters = M, rv0_1, jitter_1, rv0_2, jitter_2, b_P, b_τy, b_τx, b_mass, b_e, b_ω, b_a, b_τ, b_tp
internals = n_steps, is_accept, acceptance_rate, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt, loglike, logpost, tree_depth, numerical_error
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
M 0.8192 0.0202 0.0006 1125.1426 489.3575 1.0021 ⋯
rv0_1 -6693.4505 2.2889 0.0812 805.2573 651.2369 0.9995 ⋯
jitter_1 13.7535 1.9467 0.0683 881.4580 567.3051 0.9992 ⋯
rv0_2 1.4364 3.4984 0.1319 705.3459 472.7397 1.0139 ⋯
jitter_2 17.8337 2.9363 0.1122 759.9206 520.5404 1.0019 ⋯
b_P 0.0010 0.0000 0.0000 1084.4514 691.7446 1.0012 ⋯
b_τy 0.7520 0.1615 0.0066 732.9331 376.7255 1.0001 ⋯
b_τx -0.6232 0.1906 0.0066 866.5706 623.8066 0.9996 ⋯
b_mass 0.0419 0.0087 0.0003 683.6193 362.7727 0.9991 ⋯
b_e 0.0000 0.0000 NaN NaN NaN NaN ⋯
b_ω 0.0000 0.0000 NaN NaN NaN NaN ⋯
b_a 0.0094 0.0001 0.0000 1126.2965 489.3575 1.0022 ⋯
b_τ -0.1099 0.0373 0.0015 632.9820 423.8635 0.9992 ⋯
b_tp 57781.9594 0.0138 0.0006 633.0129 423.8635 0.9992 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 0.7812 0.8047 0.8190 0.8334 0.8560
rv0_1 -6698.1343 -6694.9449 -6693.3785 -6691.9918 -6689.1428
jitter_1 10.6321 12.4279 13.5143 14.9226 18.2011
rv0_2 -5.7378 -0.8855 1.3597 3.7745 8.1148
jitter_2 13.0378 15.7607 17.5097 19.6121 24.0961
b_P 0.0010 0.0010 0.0010 0.0010 0.0010
b_τy 0.4018 0.6615 0.7614 0.8646 1.0262
b_τx -0.9751 -0.7507 -0.6261 -0.5025 -0.2279
b_mass 0.0236 0.0363 0.0423 0.0478 0.0580
b_e 0.0000 0.0000 0.0000 0.0000 0.0000
b_ω 0.0000 0.0000 0.0000 0.0000 0.0000
b_a 0.0093 0.0094 0.0094 0.0095 0.0096
b_τ -0.1838 -0.1338 -0.1101 -0.0860 -0.0365
b_tp 57781.9321 57781.9506 57781.9594 57781.9683 57781.9865
Excellent! Let's plot the maximum likelihood orbit:
using CairoMakie: Makie
fig = OctofitterRadialVelocity.rvpostplot(model, chain) # saved to "k2_132-rvpostplot.png"
Create a corner plot:
using PairPlots
using CairoMakie: Makie
octocorner(model, chain, small=true)
Gaussian Process Fit
Let us now add a Gaussian process to model stellar activity. This should improve the fit.
We start by writing a functino that creates a Gaussian process kernel from a set of system parameters. We will create a quasi-periodic kernel.
using AbstractGPs
function gauss_proc_quasi_periodic(θ_system)
# θ_system is a named tuple of variable values for this posterior draw.
η₁ = θ_system.gp_η₁ # h
η₂ = θ_system.gp_η₂ # λ
η₃ = θ_system.gp_η₃ # θ
η₄ = θ_system.gp_η₄ # ω
kernel = η₁^2 *
(SqExponentialKernel() ∘ ScaleTransform(1/(η₂))) *
(PeriodicKernel(r=[η₄]) ∘ ScaleTransform(1/(η₃)))
gp = GP(kernel)
return gp
endgauss_proc_quasi_periodic (generic function with 1 method)Now provide this function when we construct our likelihood object:
rvlike = StarAbsoluteRVLikelihood(rv_dat,
instrument_names=["harps-n", "psf"],
gaussian_process = gauss_proc_quasi_periodic
)StarAbsoluteRVLikelihood Table with 4 columns and 70×1 rows:
epoch rv σ_rv inst_idx
┌───────────────────────────────────
1,1 │ 57782.2 -6682.78 3.71 1
2,1 │ 57783.1 -6710.55 5.95 1
3,1 │ 57783.2 -6698.96 8.76 1
4,1 │ 57812.1 -6672.32 4.0 1
5,1 │ 57812.2 -6672.0 4.22 1
6,1 │ 57812.2 -6690.31 4.63 1
7,1 │ 57813.0 -6701.27 4.12 1
8,1 │ 57813.1 -6700.71 5.34 1
9,1 │ 57813.1 -6695.97 3.67 1
10,1 │ 57813.1 -6705.87 4.23 1
11,1 │ 57813.1 -6694.27 5.12 1
12,1 │ 57813.2 -6699.99 5.0 1
13,1 │ 57813.2 -6696.17 10.43 1
14,1 │ 57836.0 -6675.4 7.03 1
15,1 │ 57836.0 -6665.92 7.48 1
16,1 │ 57836.0 -6661.9 6.03 1
17,1 │ 57836.1 -6657.92 5.08 1
⋮ │ ⋮ ⋮ ⋮ ⋮We create a new system model with added parameters gp_η₁ etc for the Gaussian process fit.
gp_explength_mean = 9.5*sqrt(2.) # sqrt(2)*tau in Dai+ 2017 [days]
gp_explength_unc = 1.0*sqrt(2.)
gp_perlength_mean = sqrt(1. /(2. *3.32)) # sqrt(1/(2*gamma)) in Dai+ 2017
gp_perlength_unc = 0.019
gp_per_mean = 9.64 # T_bar in Dai+ 2017 [days]
gp_per_unc = 0.12
# No change to planet model
@planet b RadialVelocityOrbit begin
e = 0
ω = 0.0
P ~ truncated(Normal(0.3693038/365.25, 0.0000091/365.25),lower=0.00001)
a = cbrt(system.M * b.P^2)
mass ~ LogUniform(0.001, 10)
τ ~ UniformCircular(1.0)
tp = b.τ*b.P*365.25 + 57782 # reference epoch for τ. Choose an MJD date near your data.
end
@system k2_132 begin
M ~ truncated(Normal(0.82, 0.02),lower=0) # (Baines & Armstrong 2011).
# HARPS-N
rv0_1 ~ Normal(0,10000)
jitter_1 ~ LogUniform(0.01,10)
# FPS
rv0_2 ~ Normal(0,10000)
jitter_2 ~ LogUniform(0.01,10)
# Add priors on GP kernel hyper-parameters.
gp_η₁ ~ truncated(Normal(25,10),lower=0)
gp_η₂ ~ truncated(Normal(gp_explength_mean,gp_explength_unc),lower=0)
gp_η₃ ~ truncated(Normal(gp_per_mean,1),lower=0)
gp_η₄ ~ truncated(Normal(gp_perlength_mean,gp_perlength_unc),lower=0)
end rvlike b
model = Octofitter.LogDensityModel(k2_132; autodiff=:ForwardDiff)LogDensityModel for System k2_132 of dimension 13 with fields .ℓπcallback and .∇ℓπcallback
Sample from the model as before:
using Random
rng = Random.Xoshiro(0)
chain = octofit(
rng, model,
adaptation = 100,
iterations = 100,
)Chains MCMC chain (100×31×1 Array{Float64, 3}):
Iterations = 1:1:100
Number of chains = 1
Samples per chain = 100
Wall duration = 1907.53 seconds
Compute duration = 1907.53 seconds
parameters = M, rv0_1, jitter_1, rv0_2, jitter_2, gp_η₁, gp_η₂, gp_η₃, gp_η₄, b_P, b_mass, b_τy, b_τx, b_e, b_ω, b_a, b_τ, b_tp
internals = n_steps, is_accept, acceptance_rate, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt, loglike, logpost, tree_depth, numerical_error
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
M 0.8208 0.0194 0.0022 74.9873 34.3628 1.0306 ⋯
rv0_1 -6563.4520 0.4181 0.3630 1.4059 18.3939 2.1313 ⋯
jitter_1 0.0687 0.0136 0.0091 2.2833 17.2397 1.4087 ⋯
rv0_2 -116.9768 0.2423 0.1871 1.5713 12.8043 1.8356 ⋯
jitter_2 1.7523 0.9953 0.9110 1.4188 21.5943 2.1313 ⋯
gp_η₁ 65.6337 5.5222 1.0836 28.4422 66.9638 1.0285 ⋯
gp_η₂ 14.6994 1.1476 0.1764 43.6140 51.5563 0.9985 ⋯
gp_η₃ 9.5627 1.0615 0.4806 5.5638 22.4187 1.1405 ⋯
gp_η₄ 0.4083 0.0200 0.0019 123.2051 78.3393 0.9968 ⋯
b_P 0.0010 0.0000 0.0000 107.2863 51.5338 1.0019 ⋯
b_mass 0.0143 0.0044 0.0040 1.4444 40.5549 2.0821 ⋯
b_τy 0.6837 0.1415 0.0387 14.2879 21.0246 1.1262 ⋯
b_τx -0.6515 0.1372 0.0340 17.4281 20.6246 1.0849 ⋯
b_e 0.0000 0.0000 NaN NaN NaN NaN ⋯
b_ω 0.0000 0.0000 NaN NaN NaN NaN ⋯
b_a 0.0094 0.0001 0.0000 74.2250 34.3628 1.0308 ⋯
b_τ -0.1214 0.0297 0.0075 16.7500 14.7365 1.2178 ⋯
b_tp 57781.9552 0.0110 0.0028 16.7500 14.7365 1.2262 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 0.7897 0.8065 0.8215 0.8354 0.8584
rv0_1 -6563.9381 -6563.8013 -6563.6235 -6563.1571 -6562.6608
jitter_1 0.0511 0.0578 0.0667 0.0769 0.1009
rv0_2 -117.4854 -117.1010 -116.9954 -116.8309 -116.4809
jitter_2 0.5133 0.7950 1.6374 2.6769 3.4244
gp_η₁ 57.3208 61.4333 65.6576 69.0235 76.9867
gp_η₂ 12.4763 13.9981 14.7889 15.3610 16.7696
gp_η₃ 7.5877 8.6108 9.8695 10.1704 11.2740
gp_η₄ 0.3722 0.3934 0.4080 0.4232 0.4466
b_P 0.0010 0.0010 0.0010 0.0010 0.0010
b_mass 0.0085 0.0100 0.0135 0.0188 0.0215
b_τy 0.3666 0.6108 0.6922 0.7811 0.9181
b_τx -0.8858 -0.7315 -0.6618 -0.5714 -0.3875
b_e 0.0000 0.0000 0.0000 0.0000 0.0000
b_ω 0.0000 0.0000 0.0000 0.0000 0.0000
b_a 0.0093 0.0094 0.0094 0.0095 0.0096
b_τ -0.1878 -0.1378 -0.1226 -0.1062 -0.0640
b_tp 57781.9307 57781.9491 57781.9547 57781.9608 57781.9764
For real data, we would want to increase the adaptation and iterations to about 1000 each.
Plot the results:
fig = OctofitterRadialVelocity.rvpostplot(model, chain) # saved to "k2_132-rvpostplot.png"
Corner plot:
octocorner(model, chain, small=true) # saved to "k2_132-pairplot-small.png"
It is expensive (per iteration) to fit Guassian processes. If you have many cores (or a cluster) available you might also try starting julia with multiple threads (julia --threads=auto) and using octofit_pigeons. Pigeons is less efficient with single-core, but has better parallelization scaling:
using Pigeons
chain, pt = octofit_pigeons(
model,
n_rounds=9
)
display(chain)