Hipparcos Modelling
This tutorial explains how to model Hipparcos IAD data. The first example reproduces the catalog values of position, parallax, and proper motion. The second uses Hipparcos to constrain the mass of a directly imaged planet.
Reproduce Catalog Values
This is the so-called "Nielsen" test from Nielsen et al (2020) and available in Orbitize!.
We start by using a system with a planet with zero mass to fit the straight line motion.
using Octofitter
using Distributions
using CairoMakie
hip_like = Octofitter.HipparcosIADLikelihood(;
hip_id=21547,
renormalize=true, # default: true
)
@planet b AbsoluteVisual{KepOrbit} begin
mass = 0.
e = 0.
ω = 0.
a = 1.
i = 0
Ω = 0.
tp = 0.
end
@system c_Eri_straight_line begin
M = 1.0 # Host mass not important for this example
rv = 0.0 # system RV not significant for this example
plx ~ Uniform(10,100)
pmra ~ Uniform(-100, 100)
pmdec ~ Uniform(-100, 100)
# It is convenient to put a prior of the catalog value +- 10,000 mas on position
ra_hip_offset_mas ~ Normal(0, 10000)
dec_hip_offset_mas ~ Normal(0, 10000)
dec = $hip_like.hip_sol.dedeg + system.ra_hip_offset_mas/60/60/1000
ra = $hip_like.hip_sol.radeg + system.dec_hip_offset_mas/60/60/1000/cosd(system.dec)
ref_epoch = Octofitter.hipparcos_catalog_epoch_mjd
end hip_like b
model = Octofitter.LogDensityModel(c_Eri_straight_line)LogDensityModel for System c_Eri_straight_line of dimension 5 and 104 epochs with fields .ℓπcallback and .∇ℓπcallback
Let's initialize the starting point for the chains to reasonable values
initialize!(model, (;
plx=34.,
pmra=44.25,
pmdec=-64.5,
ra_hip_offset_mas=0.,
dec_hip_offset_mas=0.,
))Chains MCMC chain (1000×18×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
Wall duration = 3.23 seconds
Compute duration = 3.23 seconds
parameters = plx, pmra, pmdec, ra_hip_offset_mas, dec_hip_offset_mas, M, rv, dec, ra, ref_epoch, b_mass, b_e, b_ω, b_a, b_i, b_Ω, b_tp
internals = logpost
Summary Statistics
parameters mean std mcse ess_bulk ess_tail ⋯
Symbol Float64 Float64 Float64 Float64 Float64 ⋯
plx 34.0000 0.0000 NaN NaN NaN ⋯
pmra 44.2500 0.0000 NaN NaN NaN ⋯
pmdec -64.5000 0.0000 NaN NaN NaN ⋯
ra_hip_offset_mas 0.0000 0.0000 NaN NaN NaN ⋯
dec_hip_offset_mas 0.0000 0.0000 NaN NaN NaN ⋯
M 1.0000 0.0000 NaN NaN NaN ⋯
rv 0.0000 0.0000 NaN NaN NaN ⋯
dec -2.4734 0.0000 0.0000 NaN NaN ⋯
ra 69.4004 0.0000 NaN NaN NaN ⋯
ref_epoch 48348.5625 0.0000 NaN NaN NaN ⋯
b_mass 0.0000 0.0000 NaN NaN NaN ⋯
b_e 0.0000 0.0000 NaN NaN NaN ⋯
b_ω 0.0000 0.0000 NaN NaN NaN ⋯
b_a 1.0000 0.0000 NaN NaN NaN ⋯
b_i 0.0000 0.0000 NaN NaN NaN ⋯
b_Ω 0.0000 0.0000 NaN NaN NaN ⋯
b_tp 0.0000 0.0000 NaN NaN NaN ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% ⋯
Symbol Float64 Float64 Float64 Float64 ⋯
plx 34.0000 34.0000 34.0000 34.0000 ⋯
pmra 44.2500 44.2500 44.2500 44.2500 ⋯
pmdec -64.5000 -64.5000 -64.5000 -64.5000 - ⋯
ra_hip_offset_mas 0.0000 0.0000 0.0000 0.0000 ⋯
dec_hip_offset_mas 0.0000 0.0000 0.0000 0.0000 ⋯
M 1.0000 1.0000 1.0000 1.0000 ⋯
rv 0.0000 0.0000 0.0000 0.0000 ⋯
dec -2.4734 -2.4734 -2.4734 -2.4734 ⋯
ra 69.4004 69.4004 69.4004 69.4004 ⋯
ref_epoch 48348.5625 48348.5625 48348.5625 48348.5625 483 ⋯
b_mass 0.0000 0.0000 0.0000 0.0000 ⋯
b_e 0.0000 0.0000 0.0000 0.0000 ⋯
b_ω 0.0000 0.0000 0.0000 0.0000 ⋯
b_a 1.0000 1.0000 1.0000 1.0000 ⋯
b_i 0.0000 0.0000 0.0000 0.0000 ⋯
b_Ω 0.0000 0.0000 0.0000 0.0000 ⋯
b_tp 0.0000 0.0000 0.0000 0.0000 ⋯
1 column omitted
We can now sample from the model using Hamiltonian Monte Carlo. This should only take about 15 seconds.
using Pigeons
chain,pt = octofit_pigeons(model, n_rounds=6)(chain = MCMC chain (64×21×1 Array{Float64, 3}), pt = PT(checkpoint = false, ...))Plot the posterior values:
octoplot(model,chain,show_astrom=false,show_astrom_time=false)
We now visualize the model fit compared to the Hipparcos catalog values:
using LinearAlgebra, StatsBase
fig = Figure(size=(1080,720))
j = i = 1
for prop in (
(;chain=:ra, hip=:radeg, hip_err=:e_ra),
(;chain=:dec, hip=:dedeg, hip_err=:e_de),
(;chain=:plx, hip=:plx, hip_err=:e_plx),
(;chain=:pmra, hip=:pm_ra, hip_err=:e_pmra),
(;chain=:pmdec, hip=:pm_de, hip_err=:e_pmde)
)
global i, j, ax
ax = Axis(
fig[j,i],
xlabel=string(prop.chain),
)
i+=1
if i > 3
j+=1
i = 1
end
unc = hip_like.hip_sol[prop.hip_err]
if prop.chain == :ra
unc /= 60*60*1000 * cosd(hip_like.hip_sol.dedeg)
end
if prop.chain == :dec
unc /= 60*60*1000
end
if prop.hip == :zero
n = Normal(0, unc)
else
mu = hip_like.hip_sol[prop.hip]
n = Normal(mu, unc)
end
n0,n1=quantile.(n,(1e-4, 1-1e-4))
nxs = range(n0,n1,length=200)
h = fit(Histogram, chain[prop.chain][:], nbins=55)
h = normalize(h, mode=:pdf)
barplot!(ax, (h.edges[1][1:end-1] .+ h.edges[1][2:end])./2, h.weights, gap=0, color=:red, label="posterior")
lines!(ax, nxs, pdf.(n,nxs), label="Hipparcos Catalog", color=:black, linewidth=2)
end
Legend(fig[i-1,j+1],ax,tellwidth=false)
fig
Constrain Planet Mass
We now allow the planet to have a non zero mass and have free orbit. We start by specifying relative astrometry data on the planet, collated by Jason Wang and co. on whereistheplanet.com.
astrom_like1 = PlanetRelAstromLikelihood(
(;epoch=57009.1, sep=454.24, σ_sep=1.88, pa=2.98835, σ_pa=0.00401426),
(;epoch=57052.1, sep=451.81, σ_sep=2.06, pa=2.96723, σ_pa=0.00453786),
(;epoch=57053.1, sep=456.8 , σ_sep=2.57, pa=2.97038, σ_pa=0.00523599),
(;epoch=57054.3, sep=461.5 , σ_sep=23.9 ,pa=2.97404, σ_pa=0.0523599 ,),
(;epoch=57266.4, sep=455.1 , σ_sep=2.23, pa=2.91994, σ_pa=0.00453786),
(;epoch=57332.2, sep=452.88, σ_sep=5.41, pa=2.89934, σ_pa=0.00994838),
(;epoch=57374.2, sep=455.91, σ_sep=6.23, pa=2.89131, σ_pa=0.00994838),
(;epoch=57376.2, sep=455.01, σ_sep=3.03, pa=2.89184, σ_pa=0.00750492),
(;epoch=57415.0, sep=454.46, σ_sep=6.03, pa=2.8962 , σ_pa=0.00890118),
(;epoch=57649.4, sep=454.81, σ_sep=2.02, pa=2.82394, σ_pa=0.00453786),
(;epoch=57652.4, sep=451.43, σ_sep=2.67, pa=2.82272, σ_pa=0.00541052),
(;epoch=57739.1, sep=449.39, σ_sep=2.15, pa=2.79357, σ_pa=0.00471239),
(;epoch=58068.3, sep=447.54, σ_sep=3.02, pa=2.70927, σ_pa=0.00680678),
(;epoch=58442.2, sep=434.22, σ_sep=2.01, pa=2.61171, σ_pa=0.00401426),
)PlanetRelAstromLikelihood Table with 5 columns and 14 rows:
epoch sep σ_sep pa σ_pa
┌────────────────────────────────────────────
1 │ 57009.1 454.24 1.88 2.98835 0.00401426
2 │ 57052.1 451.81 2.06 2.96723 0.00453786
3 │ 57053.1 456.8 2.57 2.97038 0.00523599
4 │ 57054.3 461.5 23.9 2.97404 0.0523599
5 │ 57266.4 455.1 2.23 2.91994 0.00453786
6 │ 57332.2 452.88 5.41 2.89934 0.00994838
7 │ 57374.2 455.91 6.23 2.89131 0.00994838
8 │ 57376.2 455.01 3.03 2.89184 0.00750492
9 │ 57415.0 454.46 6.03 2.8962 0.00890118
10 │ 57649.4 454.81 2.02 2.82394 0.00453786
11 │ 57652.4 451.43 2.67 2.82272 0.00541052
12 │ 57739.1 449.39 2.15 2.79357 0.00471239
13 │ 58068.3 447.54 3.02 2.70927 0.00680678
14 │ 58442.2 434.22 2.01 2.61171 0.00401426We specify our full model:
@planet b AbsoluteVisual{KepOrbit} begin
a ~ truncated(Normal(10,1),lower=0.1)
e ~ Uniform(0,0.99)
ω ~ Uniform(0, 2pi)
i ~ Sine()
Ω ~ Uniform(0, 2pi)
θ ~ Uniform(0, 2pi)
tp = θ_at_epoch_to_tperi(system,b,58442.2)
mass = system.M_sec
end astrom_like1
@system cEri begin
M_pri ~ truncated(Normal(1.75,0.05), lower=0.03) # Msol
M_sec ~ LogUniform(0.1, 100) # MJup
M = system.M_pri + system.M_sec*Octofitter.mjup2msol # Msol
rv = 12.60e3 # m/s
plx ~ Uniform(20,40)
pmra ~ Uniform(-100, 100)
pmdec ~ Uniform(-100, 100)
# It is convenient to put a prior of the catalog value +- 1000 mas on position
ra_hip_offset_mas ~ Normal(0, 1000)
dec_hip_offset_mas ~ Normal(0, 1000)
dec = $hip_like.hip_sol.dedeg + system.ra_hip_offset_mas/60/60/1000
ra = $hip_like.hip_sol.radeg + system.dec_hip_offset_mas/60/60/1000/cos(system.dec)
ref_epoch = Octofitter.hipparcos_catalog_epoch_mjd
end hip_like b
model = Octofitter.LogDensityModel(cEri)LogDensityModel for System cEri of dimension 13 and 118 epochs with fields .ℓπcallback and .∇ℓπcallback
Initialize the starting points, and confirm the data are entered correcly:
init_chain = initialize!(model, (;
plx=34.,
pmra=44.25,
pmdec=-64.5,
ra_hip_offset_mas=0.,
dec_hip_offset_mas=0.,
))
octoplot(model, init_chain)
Now we sample:
using Pigeons
chain,pt = octofit_pigeons(model, n_rounds=8, explorer=SliceSampler())
chainChains MCMC chain (256×24×1 Array{Float64, 3}):
Iterations = 1:1:256
Number of chains = 1
Samples per chain = 256
Wall duration = 472.02 seconds
Compute duration = 472.02 seconds
parameters = M_pri, M_sec, plx, pmra, pmdec, ra_hip_offset_mas, dec_hip_offset_mas, M, rv, dec, ra, ref_epoch, b_a, b_e, b_ω, b_i, b_Ω, b_θ, b_tp, b_mass
internals = loglike, logpost, logprior, pigeons_logpotential
Summary Statistics
parameters mean std mcse ess_bulk ess_tail ⋯
Symbol Float64 Float64 Float64 Float64 Float64 ⋯
M_pri 1.7390 0.0473 0.0039 148.7037 137.3135 ⋯
M_sec 9.8555 17.7061 1.3836 193.9058 108.8616 ⋯
plx 33.9609 0.3415 0.0265 166.7147 165.4535 ⋯
pmra 44.4331 0.5267 0.0406 186.8091 167.0159 ⋯
pmdec -64.4064 0.3122 0.0225 210.1332 184.4831 ⋯
ra_hip_offset_mas -2.2972 4.1344 0.3209 171.2353 108.8616 ⋯
dec_hip_offset_mas -0.2278 0.5002 0.0979 78.7606 89.7199 ⋯
M 1.7484 0.0511 0.0041 160.1364 113.6174 ⋯
rv 12600.0000 0.0000 NaN NaN NaN ⋯
dec -2.4734 0.0000 0.0000 171.2353 108.8616 ⋯
ra 69.4004 0.0000 0.0000 78.7606 89.7199 ⋯
ref_epoch 48348.5625 0.0000 NaN NaN NaN ⋯
b_a 10.3245 0.5027 0.0658 55.8552 62.2894 ⋯
b_e 0.5764 0.0551 0.0052 160.6194 86.2594 ⋯
b_ω 1.8610 1.1776 0.1432 169.1137 76.4464 ⋯
b_i 2.4499 0.0695 0.0082 62.8152 67.3279 ⋯
b_Ω 1.5590 1.3227 0.1418 166.6806 98.9104 ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
2 columns and 3 rows omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% ⋯
Symbol Float64 Float64 Float64 Float64 ⋯
M_pri 1.6551 1.7026 1.7413 1.7704 ⋯
M_sec 0.1186 0.5503 1.7445 7.9406 ⋯
plx 33.3021 33.7332 33.9505 34.1827 ⋯
pmra 43.7083 44.0545 44.3348 44.6960 ⋯
pmdec -65.1464 -64.6099 -64.3859 -64.1823 - ⋯
ra_hip_offset_mas -15.3784 -1.8820 -0.4571 -0.1770 ⋯
dec_hip_offset_mas -1.5949 -0.3369 -0.1007 0.0641 ⋯
M 1.6629 1.7113 1.7468 1.7813 ⋯
rv 12600.0000 12600.0000 12600.0000 12600.0000 126 ⋯
dec -2.4734 -2.4734 -2.4734 -2.4734 ⋯
ra 69.4004 69.4004 69.4004 69.4004 ⋯
ref_epoch 48348.5625 48348.5625 48348.5625 48348.5625 483 ⋯
b_a 9.3849 9.9990 10.3327 10.6806 ⋯
b_e 0.4518 0.5564 0.5873 0.6073 ⋯
b_ω 0.9434 1.0863 1.5106 1.8898 ⋯
b_i 2.3117 2.4119 2.4477 2.4859 ⋯
b_Ω 0.1785 0.5082 1.2699 2.0023 ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋱
1 column and 3 rows omitted
octoplot(model, chain, show_mass=true)
We see that we constrained both the orbit and the parallax. The mass is not strongly constrained by Hipparcos.