Introduction

In section 3.2 of Dunham et. al. (2008), we obtained a crucial wavelength for measuring internal luminosity, $L_\text{int}$, of embedded, low-luminosity protostars, allowing astronomers to circumvent the traditional multi-wavelength observations typically required for studies of these objects. Specifically, Dunham et. al. discovered that the flux values at 70 microns, $F_{70}$, were tightly correlated with $L_\text{int}$ when simulating embedded, low-luminosity systems using radiative transfer models. A linear fit at that wavelength subsequently yielded coefficients of $m=1.06$ and $b=-9.02$.

However, the motivation behind this project stems from the observation that the correlation was only derived for protostars whose envelopes' variable masses were between 1-10 solar masses, implying that the study failed to consider protostars whose cores contained less than 1 solar mass. In other words, it was only tested for younger protostars whose mass distributions were still heavily centered around their surrounding envelopes (so-called class 0 objects). To add insult to injury, astronomers have been widely misusing the original 70 micron correlation by frequently applying it to non-class 0 protostars, and in doing so, implicitly introducing an error factor into their calculations that has gone largely unnoticed.

With that in mind, the goals of this project are threefold; by repeating the derivations performed by Dunham et. al., this time considerably extending our considerations, we aim to:

1. Determine if the original 70 micron correlation still holds for older protostellar systems (so-called class 1a and 1b objects).
2. Compute a correction factor for studies that misused the Dunham et. al. (2008) result.
3. Provide a suite of observing windows (in wavelength space) where $F$ and $L_\text{int}$ remain sufficiently correlated, rather than a singular entry point.

*for clarity, we'll refer to the original fit as dfit (a.k.a. Dunham's fit), and the updated fit as yfit (a.k.a. Yek's fit).

Results

Upon inspection of the available data closest to 70 microns, we notice a strong positive correlation between $F$ and $L_\text{int}$ for class 0 targets. This is supported by a correlation coefficient of 0.98, and is consistent with the findings of Dunham et. al. (2008).

Intriguingly, while slightly weaker for class 1a and 1b objects, this correlation appears to hold firm, yielding near identical correlation coefficients of 0.95. A combined analysis of the data at all 3 classes also yields a correlation coefficient of 0.95.

Next, we compute the percent error introduced by using dfit as opposed to yfit as

$$ \text{Percent Error}=\frac{\left|L_\text{yfit}-L_\text{dfit}\right|}{L_\text{yfit}}\times100\% $$

We find that for class 0 objects, dfit underpredicts by roughly 2% - 50%; roughly 60% - 65% for class 1a objects; roughly 68% - 78% for class 1b objects. In other words, the higher the expected flux, and the older the protostellar object, the larger the percent error in predicted luminosity introduced from dfit.

Finally, we expand our considerations to all available wavelengths by setting an upper-threshold value for the 1 $\sigma$ standard deviation of model variances $V$, which is defined as

$$V=\log_{10}\left(L_\text{pred}\right)-\log_{10}(L)=\log_{10}\left(\frac{L_\text{pred}}{L}\right)$$

where $L_\text{pred}$ denotes the predicted luminosity, and $L$ the actual luminosity.

We find that for a somewhat arbitrary threshold of 0.333, we obtain an optimal wavelength range between $10^1$ and $10^2$ microns. In other words, any measurement of internal luminosity for embedded, low-luminosity targets can be performed within this range with a high-degree of confidence (and correspondingly low-degree of uncertainty), using the appropriate linear fit provided by yfit. Naturally, more generous thresholds yield larger wavelength ranges, and vice versa.

Conclusion

1. Determine if the original 70 micron correlation still holds for older protostellar systems (so-called class 1a and 1b objects).
   • Yes, though the correlation is slightly weaker w.r.t. class 0
2. Compute a correction factor for studies that misused the Dunham et. al. (2008) result.
   • 2% - 78% percent error, depending on the input flux and protostellar age
3. Provide a suite of observing windows (in wavelength space) where $F$ and $L_\text{int}$ remain sufficiently correlated, rather than a singular entry point.
   • Between $10^1$ and $10^2$ microns

*relevant analyses alongside detailed documentation can be found in src/; higher resolution images can be found in fig/; raw data available upon request.