Observation operators

The binary-aware likelihood (2) was derived assuming the mass-addition observation operator $m_{\mathrm{sys}} = m_1 + m_2$. This operator gives the largest possible binary distortion — it represents the case where the survey measures the true dynamical mass of every system. Real photometric and CMD-based surveys measure something different, with smaller distortions per unresolved binary, but the existence of bias remains: the naive single-star likelihood becomes confidently wrong at large $N$ for every operator, just at different sample sizes.

This chapter catalogues the three observation operators progenax supports, derives the photometric-mass formula that approximates flux-addition, and quantifies the bias at each level for the same underlying $\alpha = 2.35$ population.

Three operators¶

The default BinaryIMF uses mass addition because it is the simplest to expose to the marginalisation likelihood and provides the worst-case binary contamination. Photometric and CMD operators are provided for use cases where the survey forward model is needed — LSST-style cluster catalogues being the canonical target.

Mass addition: dynamical mass¶

For eclipsing binaries (which provide both component masses via light-curve fitting), SB2 spectroscopic orbits (which provide the mass ratio and mass sum), and astrometric binaries (which provide both component masses via orbital astrometry), the measured quantity is the true dynamical mass:

In this regime, the binary-aware likelihood (2) is exact. progenax’s default BinaryIMF uses it without modification.

Mass addition is the most consequential observation operator for benchmarking binary-aware inference because it produces the largest distortion. If you can demonstrate unbiased recovery under mass addition, the same machinery applied to a less-aggressive operator will perform at least as well.

Photometric: flux addition¶

Many published “mass functions” are actually photometric mass functions: stellar masses are inferred from observed flux through a mass-luminosity relation $L \propto m^s$. For an unresolved binary, the telescope measures the combined flux:

Inverting this through the single-star mass-luminosity relation yields the photometric mass:

For the rough main-sequence scaling $L \propto m^s$ with $s \approx 3$–4, this evaluates to

The key consequence: because $s$ is large, the secondary contributes appreciably to the flux only when $q$ is near 1.

Equal-mass binaries shift photometric mass by $\sim 22\%$ — still substantial. Unequal-mass binaries with $q \le 0.3$ are essentially invisible to photometry. This is why photometric IMF inferences are generally less contaminated than dynamical-mass inferences: the operator’s nonlinearity suppresses the contribution from low-$q$ companions, which are the most numerous in the Moe & Di Stefano (2017) distribution.

A future photometric likelihood layer can swap the mass-addition operator for (4) inside the marginalisation likelihood. The integral structure of (2) remains the same; only the relationship between $m_1$, $q$, and the observable changes. The current BinaryIMF does not export a with_photometric_operator() helper.

Multi-band CMD: full forward model¶

LSST and other modern surveys do not measure “mass” at all — they measure multi-band fluxes (or magnitudes) with uncertainties, plus incompleteness and crowding effects. For an unresolved binary at distance $d$:

Inferring “mass” from this requires:

1. Comparison to theoretical isochrones at assumed age $\tau$, metallicity $Z$, and distance $d$.

2. Extinction correction (dust column $A_V$ and reddening law $R_V$).

3. Color-magnitude diagram (CMD) fitting — turning per-band magnitudes into a position in CMD space and matching to a single-star isochrone or a binary-track curve.

Each step introduces additional model dependence. The binary distortion in CMD space is qualitatively different from mass-space: an unresolved equal-mass binary appears $\sim 0.75$ mag brighter at nearly the same colour, placing it above the main sequence in the CMD. This mimics a more massive or more evolved star — a degeneracy that observational binary-fraction studies have used to infer binary fractions from CMD residuals.

CMD-space observation operators are a planned layer, not a current public API in this checkout. The intended shape is:

from progenax.imf import BinaryIMF
from progenax.imf.observation_operators import CMDOperator  # planned

cmd_op = CMDOperator(
    isochrone_file="parsec_iso_logZ-0.5.dat",
    bands=("g", "r", "i"),
    distance_pc=10000.0,
    A_V=0.5,
)
binary_imf = BinaryIMF(primary_imf=primary_imf, observation_operator=cmd_op)
log_likelihood = binary_imf.logpdf(observed_magnitudes)  # planned

No CMDOperator implementation is currently exported from src/progenax/; this section is design guidance for a future survey-forward-model layer.

A bias hierarchy¶

Mass addition fails at $N \sim 10^4$. Photometric fails at $N \sim 10^5$. Multi-band CMD fails at $N \gtrsim 10^6$. Every operator eventually produces “confidently wrong” naive posteriors at large enough $N$; the binary-aware likelihood eliminates the bias at all $N$ regardless of operator.

Resolved/unresolved selection¶

The above analysis treats every binary as unresolved. In real catalogues, binaries with angular separations $\theta > \mathrm{PSF}$ are resolved into two detections rather than blended into one system. Whether a given binary resolves depends on:

• Orbital period (sets physical separation $a$).

• Cluster distance (sets angular separation $\theta = a/d$).

• PSF or seeing (sets the resolution threshold).

• Crowding (in dense fields, even resolvable binaries can be blended with neighbours).

The effective binary fraction relevant to the unresolved-photometry analysis is therefore

i.e. the integral of $f(m_1, P)$ over periods that produce unresolved configurations at the survey’s distance and resolution.

A future survey-selection layer should expose this kind of resolution cutoff and integrate over periods up to the resolution boundary using the Moe & Di Stefano (2017) period distribution. The current BinaryIMF does not provide a with_resolution_cutoff() convenience method. This refinement is most important for nearby clusters (where wide binaries are routinely resolved) and for adaptive-optics or space-based surveys (with resolution far below seeing-limited ground-based work).

Domain of validity¶

1. Mass-addition is exact for SB2 / eclipsing / astrometric binaries; an over-estimate of the bias for photometric surveys.

2. Photometric approximation $L \propto m^s$ is rough. Real isochrones depart from a single power law; for production work, use the multi-band CMD operator with a real isochrone.

3. Multi-band CMD requires isochrone selection. Mis-specifying age, metallicity, or extinction couples into the inferred $\alpha$ in non-trivial ways. The recommendation is to jointly infer $(\alpha, \tau, Z, d, A_V)$ rather than treat the latter as fixed.

4. All operators ignore stellar evolution. A binary with one evolved component (giant, white dwarf) has flux properties not captured by main-sequence isochrones. progenax flags this as a known limitation; full treatment requires coupling to a stellar-evolution code (gravax, startrax in the future).

Implementation, validation & references¶

• In code: the live mass-addition operator is built into src/progenax/imf/binary/imf.py (BinaryIMF, $m_{\mathrm{sys}} = m_1 + m_2$); the photometric and multi-band CMD operators on this page are planned survey-forward-model layers (no CMDOperator / with_photometric_operator() is exported yet). See the IMF API and the add-binary-population recipe.

• Validated in: binary-aware recovery exercises the mass-addition operator (the worst-case upper bound on the bias); the reduced-bias photometric / CMD operators have no dedicated validation page yet, since they are not implemented.

• Primary sources: the mass-addition operator follows from elementary dynamics; the flux-addition context is Maschberger (2013) §5, and the multi-band CMD analysis builds on standard isochrone fitting (PARSEC, MIST). Full notes in the bibliography.