# Physical observer-motion photon response — 2026-10-03

**Conclusion:** spacecraft motion predicts a real but small colour-dependent
photon-rate term for this field. Its nominal median/p95 science RMS is
3.54/7.47 ppm. Correcting science and references consistently changes the
existing relative light curves by median/p95 0.98/2.69 ppm RMS. This is too
small to explain the remaining hundred-ppm structure within the tested
spectral model. It is a prediction, not an independent detection or a
promoted calibration. The overall photometric goal remains unachieved.

Run: `/astro/armin/koji/syndiff/dev_runs/temporal_raw_epsf_20261001/`.
Worktree: `/home/kshukawa/syndiff_pipeline/dev/.worktrees/temporal-raw-epsf`,
local branch `codex/temporal-raw-epsf`. Nothing is pushed.

## Moving observer, not moving-source beaming

For the known observer velocity, define

$$D_{s,t}=\gamma_t\left(1+\frac{\mathbf v_t\cdot\mathbf n_s}{c}\right),
\qquad \gamma_t=\left(1-\frac{|\mathbf v_t|^2}{c^2}\right)^{-1/2}.$$

Here $D$ is the dimensionless received-to-barycentric photon frequency ratio,
$s$ labels a star and $t$ an exposure, $\mathbf v_t$ is the saved definitive
TESS barycentric velocity in km/s at the spacecraft-TDB midpoint, $\mathbf n_s$
is the unit direction to that star from its catalogue ICRS RA/Dec, and
$c=299792.458$ km/s. Motion towards a source increases $D$. These are the same
ephemerides used in `results/s24_wcs_physics/fields.npz`; no WCS coefficients
from another sector are transplanted.

Specific-intensity invariance and angular aberration give

$$I_{\nu,\mathrm{obs}}(\nu)=D^3 I_{\nu,0}(\nu/D),\qquad
d\Omega_{\mathrm{obs}}=D^{-2}d\Omega_0,\qquad
F_{\nu,\mathrm{obs}}(\nu)=D F_{\nu,0}(\nu/D).$$

Here $I_\nu$ is specific intensity per frequency and solid angle,
$d\Omega$ is a source solid-angle element in steradians, $F_\nu$ is flux
density integrated over an unresolved source in W m$^{-2}$ Hz$^{-1}$, and
$\nu$ is photon frequency in Hz. Subscript 0 denotes the fixed barycentric
radiation field and “obs” the observer frame at the same event. The area
transformation is why the familiar moving-source specific-frequency factor
must not be used unchanged. For a power law with spectral index
$\alpha=d\ln F_\nu/d\ln\nu$, the first-order observer coefficient is
$1-\alpha$, consistent with the observer-boost convention in the
[Ellis–Baldwin treatment](https://arxiv.org/abs/2404.07929), which uses the
opposite sign for its spectral index.

For a photon-counting detector, the corresponding rate is

$$C_s(D)=\frac{A_{\rm eff}}{hc}\int Q(\lambda)\lambda D^3
F_{\lambda,s}(D\lambda)\,d\lambda.$$

Here $C_s$ is detected electrons/second assuming the response includes
quantum efficiency, $A_{\rm eff}$ is the fixed collecting area in m$^2$,
$h$ is Planck's constant in J s, $c$ is now expressed in m/s, $\lambda$ is
wavelength in metres, $Q$ is dimensionless transmission times quantum
efficiency, and $F_\lambda$ is the fixed incident spectral flux density in
W m$^{-2}$ m$^{-1}$. The calculation uses only rate ratios, so area, $hc$ and
the fixed conversion from Gaia's per-nm units cancel. The factor of wavelength
weights photon counts. No light-curve amplitude is fitted.

## Spectra, support and numerical validation

The existing S24 C2K2 Gaia XP archive is from
`/astro/armin/koji/syndiff/dev_runs/xp_colour_20260925/`. A streaming extraction
retains the task's 431 source IDs and hashes the full 5.1-GB source file as
well as the selected CSV. GaiaXPy 2.1.4 in that archive's existing environment
calibrates 430 spectra at 2-nm sampling from 336 through 1020 nm. No spectra
are modified in the source archive. All are positive and finite over the
nonzero response support.

One reference, Gaia source 1629901678941155456, lacks XP and uses an explicitly
marked 6198.953-K GSP-Phot blackbody. All 165 eligible science sources have XP.
The copied TESS v2.0 response is the 2020-07-30 Vanderspek/POC curve; its
nonzero support extends to 1060 nm. Inputs, hashes and dependency information
are under `inputs/observer_photometry/`. The response definition follows
the [TESS detector-bandpass documentation](https://heasarc.gsfc.nasa.gov/docs/tess/docs/TESS_observatory_guide_v1.1.pdf).

Four red-tail continuations are tested beyond 1020 nm: power-law slopes fitted
over the final 40 and 80 nm, and 2500/15000-K blackbody tails. Each joins the
last measured spectral value continuously. These are sensitivity scenarios,
not posterior uncertainties. The nominal extrapolated photon fraction has
5th/median/95th percentiles 0.94/1.37/1.74% among eligible science sources.
Mean Gaia spectra, finite resolution, mean-frame conventions, temporal SED
changes and spectral calibration uncertainty remain limitations.

The first physical run failed its declared accuracy guard: uniform 0.5/0.25-nm
integration differed by 0.0243 ppm for source zero. No source was removed and
no result accepted. Integration now splits at all response knots and all
Doppler-shifted spectral knots, including the red-tail join, then uses Gaussian
quadrature inside each smooth interval. Nominal-model four/eight-point results agree to
floating-point precision at every source's minimum, median and maximum actual
Doppler factor. This is numerical agreement, not detector-truth accuracy.

Tests independently verify power-law spectra; an explicit moving spectral
derivative break against adaptive integration; zero velocity and sign; the
angular-area Jacobian; and the all-wavelength photon and energy scaling
limits. The first-order log-rate approximation differs from the exact
integral by at most 0.00084 ppm here. The reported result uses the integral.
Midpoint velocities and fixed mean spectra are retained approximations.

![Measured Gaia spectra and predicted spectral changes](/astro/armin/koji/syndiff/dev_runs/temporal_raw_epsf_20261001/results/observer_photometry/input_spectra.png)

The left panels show the actual Gaia-derived individual photon spectra and
their medians. The vertical line marks the XP support limit. The right panels
are calculated last-minus-first spectral changes, not measured time-resolved
spectra. Narrow wiggles in that derivative inherit the finite-resolution XP
representation; they are not independently detected variable spectral lines.

## Photon-rate and relative-photometry effect

The same 431 sources and all 1,224 exposures are retained. The observer
coefficient has science 5th/median/95th percentiles 0.85/1.88/2.60. For the
nominal 40-nm power-law tail, the maximum science peak-to-peak photon-rate
change is 23.79 ppm.

The conditional photometric control divides both science and reference
rates by their respective predicted observer responses and recomputes the
reference precision weights. The aperture-ratio model, pixel WCS/shape and
physical count-loss predictions remain fixed. The original reference method
is first reproduced exactly. Applying the correction only to science targets
would double-count much of a signal already tracked by references.

| Red-tail model | Science rate RMS median/p95 [ppm] | Post-reference curve change RMS median/p95 [ppm] |
|---|---:|---:|
| Power law, last 40 nm | 3.54 / 7.47 | 0.98 / 2.69 |
| Power law, last 80 nm | 3.51 / 7.42 | 0.97 / 2.66 |
| Blackbody tail, 2500 K | 3.55 / 7.49 | 0.94 / 2.64 |
| Blackbody tail, 15000 K | 3.45 / 7.29 | 0.93 / 2.59 |

The largest nominal post-reference RMS change is 4.29 ppm. The maximum
temporal RMS sensitivity to these tail choices is 0.285 ppm in the direct
photon-rate prediction. That does not include a full Gaia calibration-error
posterior or certify every possible spectral model.

![Colour, time and detector-position predictions](/astro/armin/koji/syndiff/dev_runs/temporal_raw_epsf_20261001/results/observer_photometry/predicted_photometry.png)

![Actual unresolved histories before and after the conditional control](/astro/armin/koji/syndiff/dev_runs/temporal_raw_epsf_20261001/results/observer_photometry/actual_curve_comparison.png)

All three figures were inspected. The actual large residual histories barely
change, while the expanded lower panel resolves the small colour-dependent
change. The calculation bounds this mechanism within its stated assumptions;
it is not an observational measurement of the effect and does not establish
photon-noise dominance. No actual instrument model was refitted to flatten
the histories, and no replacement science product is promoted.

## Provenance and next action

Frozen jobs on plscience3: preparation 1166 (two CPUs/4,000 MB, exit 0, 56 s),
failed uniform-grid calculation 1186 (four CPUs/4,000 MB, exit 1, 16 s), and
accepted piecewise calculation 1194 (four CPUs/4,000 MB, exit 0, 124 s).
Code snapshots and exact commands are in the corresponding `condor/` paths.
Outputs: `results/observer_photometry/`; input spectra remain task-local under
`inputs/observer_photometry/` on `/astro`.

Next return to independent detector-calibration evidence. In particular,
test time-dependent row bias against held-out overclock pixels before
propagating a correction to science apertures. Archive reconstruction alone
does not establish true bias accuracy. Existing row/smear ambiguities,
PSF/scene errors, hp_d missing-light repair, broader populations and the full
noise budget remain unresolved requirements.