高精度位置天文解析手法の開発とシミュレーションによる検証

大澤亮

JASMINE project, National Astronomical Observatory of Japan河田大介 (UCL), 上塚貴史 (東京大学), 山田良透 (京都大学),Wolfgang Löﬄer, Michael Biermann (ARI/ZAH), JASMINE データ解析チーム

Overview
We developed the Plate Analysis algorithm for wide-area & relative astrometry.
The algorithm is built on the following assumptions.
▪ The apparent motions of sources are negligible within a dataset.
▪ The image distortion pattern is consistent within a dataset.

We developed software to obtain astrometric solutions for the Plate Analysis.
▪ The observation model is implemented with JAX and differentiable.
▪ Parameters are optimized with SVI, acceleratedwith JAX & numpyro.

The software successfuly provided a reasonable solution in a validation test.

Table of Contents
1. Introduction − Precise Astrometry
2. Plate Analysis
3. Observation model & implementation
4. Validation test − JASMINE mini-Mock survey
5. Results
6. Discussion

Power of Astrometry
Astrometry = Strong tool to reveal the history of the Milky Way (galactic archaeology, GA)
Gaia measures proper motions and parallaxes at ~10 μas/yr & 10 μas levels.
Several relics of satellite merges are discovered.

210VΦ [km/s]
220

230

240

0.5

1.060

The Gaia-Sausage-Enceladus in the inner halo
The remnant of a major merger that formed the inner halo
possibly occurred about 100 Gyr ago

40The phase spiral of the Galactic disk
The after effect of a satellite galaxy passage about 10 Gyr ago,
which disturbed the Galactic disk in a phase space

VZ*[km/s]

200-20-40-60-1.0-0.50.0
Z[kpc]

e.g., Belokurov et al. (2018), Antoja et al. (2023), Antoja et al. (2018)

Astrometric Frontier
Galactic center regions is not inspected by Gaia.
Sources at > 4 kpc are heavily obscured by interstellar dust.
Formation of the bar, bulge, and NSD can be inspectedby the Galactic center astrometry.

SIRIUS survey (Nishiyama+, 2006), Miras in NSD (Sanders+, 2024)

Precise Astrometry≫ size of FoV~ size of FoV

0.1′′

Global & AbsoluteWide-area & Relative

Local & Relative

Hipparcos & Gaia

UCAC4, VIRAC, gVIRAC, etc.

Keck, HST, Theia, TOLIMAN, etc.

SIRIUS (Nishiyama+, 2006); Keck/UCLA Galactic Center GroupPrecise Astrometry≫ size of FoVAdvanced data analysis~ size of FoV

New technology▪ Self-calibration▪ Global self-consistent solution▪ Focal plane monitoring▪ Phase masking

0.1′′

Global & AbsoluteWide-area & Relative

Local & Relative

Hipparcos & Gaia

UCAC4, VIRAC, gVIRAC, etc.

Keck, HST, Theia, TOLIMAN, etc.

~10 μas~0.5 mas/yr~1 μasSIRIUS (Nishiyama+, 2006); Keck/UCLA Galactic Center Group

Plate Analysis
A self-calibration method▪ Algorithm provides a self-consistent solution within a dataset.
▪ Observation model incorporates the telescope and instrument.
▪ Model parameters are tuned to reproduce the measurements.
▪ Image distortion is estimated using the measurements.

Reference sources

Anchoring with reference sources▪ Relative astrometry requires anchoring points to fix the fields.

Simplifications▪ Stellar motions are neglected within a sequence of exposures.
▪ Image distortion is regarded as consitant in a dataset.
▪ Scale (focal length) can be variable in this study.
Ohsawa+, 2024, SPIE Proc., 13101-1R

Plate Analysis

Pre-established calibration data may introduce systematic errors.
Self-calibration is required to achieve an ultimate precision.

A self-calibration method▪ Algorithm provides a self-consistent solution within a dataset.
▪ Observation model incorporates the telescope and instrument.
▪ Model parameters are tuned to reproduce the measurements.
▪ Image distortion is estimated using the measurements.

Reference sources

Anchoring with reference sources▪ Relative astrometry requires anchoring points to fix the fields.

Simplifications▪ Stellar motions are neglected within a sequence of exposures.
▪ Image distortion is regarded as consitant in a dataset.
▪ Scale (focal length) can be variable in this study.
Ohsawa+, 2024, SPIE Proc., 13101-1R

Astrometric Solution
The astrometric solution is derived in the Bayesian approach.
Construct a deterministic mapping function from celestial to detector coordinates.
celestial coordinates

detector coordinates
w/calibration parameters

The posteriors of (α, δ) are defined as follows.
Posteriors of α, δMeasurement errors

PriorsWe assume Gaussian distributions forthe likelihood function

Reference information is naturally
implemented as the priors

Differentiable Implementation
In general, we have to tackle with a huge model.

In this presentation~ 10,000 sources~300,000measurements

We take advantage of the differentiable programming techniques.
celestial coordinates

detector coordinates

Using the differentiable programming library, JAX,
the mapping is implemented as fully differentiable.

w/calibration parameters

The posteriors are approximated via (Stochastic) Variational Inference.
Posteriors of α, δpplication
aPD
gints
An intere
11b
found in P3

Measurement errors

PriorsMinimize the divergence
between these distributions.

Poisteriorsare approximated by a combination of Normal distributions.
The Bayesian inference problem is reduced to an optimazation problem.

Overview of the validation process
Ground Truth coordinates

Resample

Ground Truth catalog

PropagateCompare

Generate

Mock measurements
Astrometric Analysis

Reference catalog

Estimated coordinates
Ohsawa+, 2024, SPIE Proc., 13101-R1

JASMINE
JASMINE: Japan Astrometry Satellite Mission INfrared Exploration
An satellite mission for precise photometry and astrometry in 1.0-1.6 μm
Measureing the proper motions and parallaxes of stars around the Galactic center.
Image-based and small-field space astrometry mission.

~110,000 target stars~40 μas for parallax
125 μas/yr for proper motons

1 degreeA preliminary design of the JASMINE satellite

See the SPIE proc. (Kataza+, 2024, 13092-9; Isobe+, 2024, 13092-186; Suematsu+, 2024, 13092-185)

JASMINE mini survey
Actual mission
Survey field
Epoch# of orbits# of exposurestarget stars
referencemeasurement error
relativistic aberration
focal length
image distortion
stellar color

2.1° × 1.2° region>3 years> 6,000~ 50 per fieldmany bulge stars
foreground Gaia starsmagnitude dependent
Earth + satellite
can vary every exposure
can be changed
widely distributed

Mini survey
around (l, b) = (-0.3°, 0.1°)
3 years100
μ=0&π=024 exp × 4 fields
Gaia DR3 + artificial sources~ 10,000 Gaia stars~ 0.01 pixel (4 mas)
Earth only
can very every exposure
fixed (polynomial)
not included

JASMINE mini-survey dataset is available in Zenodo (zenodo.org/records/10403895)

Results − Datasets
Footprints of the first orbit

Single field of view
24 exposures~8,600 unique sources~263,000 measurements

Results − Plate Analysis
The mock measurements were reproduced with an accuracy of 0.01 pixels.
The estimated coordinates wer consistent with the ground truth at a 1-mas level.
Residuals on the sky (RA)±1mas

Residuals on the sky (Dec)

O-C scatter scatter plots on the detector

1σ (0.01pix)
2σ (0.02pix)
3σ (0.03pix)±1mas

Results − Stellar Motion
Stellar motions via the 100-orbit analysis
The estimated coordinates well reproduced the proper motion + parallax motion.
Artificial sources with μ=0 & π=0 did not stay within uncertainties asexpected.

○ Ground Truth○ Estimation

Artificial source with μ = 0 and π = 0

Ohsawa et al. (2024), SPIE Proc., 13101-64

Discussion
The celestial coordinates of the artificial sources were estimated by the weighted means.
The deviations from the ground truth are consistent with the measurement errors.

~70 μas at 100 orbits

Discussion
Some reference sources were highly affected by wrong priors.
The prior errors are mainly attributable to the propagation of the proper motion errors.
The proper motions and parallaxes are less affected.
The sources around such errorneous reference sources should be carefully treated.
○ Ground Truth○ Reference (Priors)○ Estimationreference source 1

reference source 2

reference source 3

Ohsawa et al. (2024), SPIE Proc., 13101-1R

Conclusion
We developed the Plate Analysis algorithm for wide-area & relative astrometry.
Plate Analysis is based on the following assumptions.
▪ The apparent motions of sources are negligible within a dataset.
▪ The image distortion pattern is consistent within a dataset.

We developed software to obtain astrometric solutions for the Plate Analysis.
▪ The observation model is implemented with JAX and differentiable.
▪ Parameters are optimized with SVI, acceleratedwith JAX & numpyro.

The software successfuly provided a reasonable solution in a validation test.
Possible future update▪ Loosen the assumption of the Plate Analysis to the global solution.
▪ Adopt more flexible distributions in the Stochastic VariationalInference.

Basic observation sequence
The satellite is in the sun-synchronous orbit on the day-night boundary (P ~ 100 min).
The telescope is pointed toward the target for the half of the satellite orbit.
The satellite attitude is controlled to stabilize the temperature for the other half.

AutumntowardGalactic Center

SpringStabilization Phase

Observation Phase

Kataza et al. (2024) SPIE Proc.,13092-9

Implementation
SourceParameters

TelescopeParameters

DistortionParameters

DetectorParameters

Priors
SourceParameters

TelescopeParameters

DistortionParameters

DetectorParameters

Optimization
The models are optimized using the ADAM optimizer.

α = 10⁻³, 10⁻⁴, ..., 10⁻¹⁰

The parameters are iteratively updated with decreasing learning rates.

We used the GPU cluster at CfCA/NAOJ (dgx-full). Each job was parallelly executed and accelerated by a single GPU (A100, 40 GB).

JAX (Bradbury+, 2018); numpyro (Phan+, 2019)

Mission Concepts of JASMINE
The key features for the mission success:Optics made of extremely low-thermal-expansion materials (CTE < 10-8/K).
Sophisticated thermal design and control to achieve an effectively large thermal inertia.
Sharp PSFs (SR > 0.9) over the entire field of view.
Foreground sources as achors to fix and align the coordinate frame to the ICRS.
Estimating the image distortion patterns using partially overlapped images.

+See Kataza+ (13092-9), Isobe+ (13092-186), and Suematsu+ (13092-185) in SPIE 2024