How many temperatures do three infrared colours reveal?

A catalog point already contains a spectral convention

A SPIRE detector receives light over a band, not at one exact wavelength. Its catalog value quotes the monochromatic flux density that a reference spectrum $S_\nu\propto\nu^{-1}$ would need to produce that band signal. A different source spectrum must therefore be integrated through the band before comparison.

Call the resulting response operator $\mathcal B_b$ for band $b$. It includes the relative spectral response, aperture efficiency and declared beam treatment. This is an energy-response integral in frequency, not a photon counter with an extra $1/(h\nu)$. The tutorial keeps a point-response comparison and two extended-map beam approximations separate.

NGC4254 is our fixed example. Its three SPIRE entries have no source-quality flag. We compare them to the same saved row in every scenario. The two PACS entries have minor coverage flags and lack matching response curves in this exercise, so they enter only a separately labeled monochromatic sensitivity case. Their conditional use is carried through into the comparison.

Reading: Clark et al., DustPedia photometry, arXiv:1708.05335v1 · Griffin et al., SPIRE photometer calibration, arXiv:1306.1778v1 · ESA SPIRE context 14.3 RSRF and aperture-efficiency products

What one temperature actually assumes

Choose an optically thin modified blackbody with fixed opacity slope $\beta$. Define its shape $f_\nu(T)$ to equal one at the reference frequency $\nu_r$ corresponding to $250\,\mu\mathrm m$. Its amplitude $A$ is then the monochromatic model normalization in janskys, before the band operator.

A single component predicts $A\mathcal B_b[f_\nu(T)]$. For each candidate temperature the best nonnegative amplitude follows from a one-parameter weighted linear fit. Searching temperature completes a two-parameter conditional fit. It does not prove that every emitting grain has this temperature.

In the main extended-map approximation with $\beta=2$, the three clean bands give an observer-frame colour parameter near $23.54\,\mathrm K$. With $\beta=1.5$, the same saved data give about $37.12\,\mathrm K$. The material slope and the location of the thermal spectrum can compensate one another. Neither fit licenses us to hide the choice of slope inside an apparently measured temperature.

Reading: Dust absorption, heat and polarization: standalone science · Clark et al., DustPedia photometry, arXiv:1708.05335v1

Add a second component without changing the data

Now predict $S_\nu=A_1f_\nu(T_1)+A_2f_\nu(T_2)$ with nonnegative amplitudes. Even with $\beta$ fixed, there are four parameters: two amplitudes and two temperatures. Three measured bands supply only three numbers. The local three-by-four Jacobian cannot have rank four. Ordering $T_1<T_2$ removes the label swap, not the physical degeneracy.

The real-data program searches a declared $8$ to $80\,\mathrm K$ temperature grid with $0.25\,\mathrm K$ spacing. At each pair it solves the two linear amplitudes and retains physically allowed candidates. Mixtures near $11.75+33\,\mathrm K$ and $15.5+67.5\,\mathrm K$ both closely match the same three fluxes under the main response scenario. They are examples of different solutions, not two alternative discoveries of an actual cold and hot phase.

There are $13\,607$ sampled temperature pairs whose squared whitened distance lies within one of the grid minimum in that scenario. This count depends on the grid, support, response and weights. It is not a confidence region or a posterior probability: no prior measure or validated real-data likelihood has been supplied.

The practical lesson is stronger than “more parameters fit better.” The measurements leave directions in model space weakly or entirely undetermined. A very narrow numerical optimizer output can hide that fact if it reports only one convenient point.

Reading: Dust absorption, heat and polarization: standalone science

Worked boundary: a second component with no light

A two-component model includes the one-component limit $A_2=0$. At that boundary, changing $T_2$ does nothing to any predicted band. The second temperature is unidentified, not zero kelvins and not the midpoint of its allowed interval.

Our initial strictly positive discrete grid sometimes produced a slightly worse residual than the continuous one-component optimum. That was a search limitation, not evidence that adding a component worsens the nested model. The retained grid result is now compared with the explicit zero-amplitude boundary. The best reported candidate cannot be worse than that one-component solution.

For the five-band minor-PACS sensitivity cases, this boundary often wins the candidate comparison. The output then records a zero amplitude and no identified second temperature. It does not invent a cold or hot component simply because the code has a slot for one. A winning grid point remains a candidate; finding the exact continuous global optimum would require a further search.

Reading: Dust absorption, heat and polarization: standalone science

Shared calibration is not extra independent noise

The published errors already combine aperture noise and adopted calibration in quadrature. Suppose we explore a common calibration component $c_b$ across SPIRE bands. To preserve each quoted marginal variance $\sigma_b^2$, remove $c_b^2$ from its independent diagonal before adding the common outer product. The remaining diagonal must be positive.

The conditional comparison uses squared whitened distance $D^2=(\boldsymbol y-\boldsymbol m)^{\mathsf T}C^{-1}(\boldsymbol y-\boldsymbol m)$. It is a useful scale for comparing declared candidates. The saved row supplies no full covariance, however, so calling this a calibrated chi-square significance would add information we do not possess.

A common calibration shift tends to move several bands together. A change in spectral shape moves their ratios. These affect normalization and colour differently, which is why a correlated scenario cannot be replaced by simply inflating every independent error. The program retains both scenarios without pretending to measure the true covariance.

Reading: Clark et al., DustPedia photometry, arXiv:1708.05335v1 · Griffin et al., SPIRE photometer calibration, arXiv:1306.1778v1 · Dust absorption, heat and polarization: standalone science

What temperature, and whose mass?

No redshift or distance is fixed for these galaxies in this exercise. The spectral parameter is an observer-frame colour temperature. For the same redshifted modified-blackbody shape, $T_{\mathrm{rest}}=(1+z)T_{\mathrm{colour}}$. Without fixing $z$, we do not relabel the fitted value as a physical rest-frame temperature.

Similarly, converting a normalization to dust mass requires a distance, an opacity per stated mass and a physical geometry/emission model. The fitted Jy amplitude is not that mass. A distribution of equilibrium cell temperatures in the Lab and a two-component fit to unresolved galaxy fluxes answer different questions.

The next article shows how to acquire the fixed source rows yourself, inspect flags and run these same-record comparisons. Its most useful output is not one best temperature. It is a visible record of which changes the same observations still allow.

Reading: Clark et al., DustPedia photometry, arXiv:1708.05335v1 · Dust absorption, heat and polarization: standalone science