Analytical vs Physical Scaling

When ForMoSA evaluates the likelihood, it compares a transformed model spectrum to the observed flux. Part of that transformation is a scaling step that brings the model to the same flux level as the data. ForMoSA offers two approaches.

Physical scaling: r + d

Physical scaling applies the inverse-square law:

\[F_\text{obs}(\lambda) = F_\text{model}(\lambda) \times \left(\frac{r}{d}\right)^2\]

where r is the companion radius in Jupiter radii and d is the distance in parsecs. This scaling is physically motivated and lets you retrieve the radius as a free parameter.

from ForMoSA.config.global_config import ConfigParameters

config_parameters = ConfigParameters(
    par1 = ["uniform", "800",  "2000"],   # Teff
    par2 = ["uniform", "3.0",  "5.5"],    # log g
    r    = ["uniform", "0.5",  "3.0"],    # radius (R_Jup) — free
    d    = ["constant", "27.7"],          # distance fixed to Gaia value (pc)
)

Use physical scaling when:

• Your flux calibration is reliable (flux-calibrated spectrum or photometry).

• You want to constrain the companion’s physical radius.

• You can fix the distance (e.g. from Gaia parallax).

Analytical scaling: alpha

Analytical scaling multiplies the model by a constant factor:

\[F_\text{obs}(\lambda) = F_\text{model}(\lambda) \times \alpha\]

This is a pure nuisance parameter: it absorbs any flux-level offset without making any physical claim about the radius or distance.

from ForMoSA.config.global_config import ConfigParameters

config_parameters = ConfigParameters(
    par1  = ["uniform", "800",  "2000"],
    par2  = ["uniform", "3.0",  "5.5"],
    alpha = ["uniform", "0.0",  "10.0"],  # free scaling factor
    # r and d are NOT set
)

Use analytical scaling when:

• The absolute flux calibration of your data is uncertain.

• You are fitting contrast spectra (e.g. from integral-field unit observations) where the flux level is not physically meaningful.

• You want a quick exploratory fit without committing to a radius prior.

Side-by-side comparison

	Physical (r + d)	Analytical (alpha)
Physical meaning	Yes — retrieves radius	No — nuisance parameter
Requires distance	Yes (can be fixed)	No
Requires flux calibration	Yes	No
Adds free parameter?	Yes (r), or fix d	Yes (alpha)
Recommended for	Photometry, flux-calibrated spectra	Contrast spectra, exploratory fits

Combining both

You can set both r+d and alpha simultaneously — in that case ForMoSA applies the physical scaling first and then multiplies by alpha. This is occasionally useful when testing for residual systematic offsets on top of a physical model, but in most cases you should pick one or the other.