Erratum: Exploring the M-dwarf Luminosity–Temperature–Radius relationships using Gaia DR2

errata, addenda, techniques: photometric, stars: fundamental parameters, stars: late-type, stars: low-mass

1 INTRODUCTION

This is an erratum for the paper ‘Exploring the M-dwarf Luminosity–Temperature–Radius relationships using Gaia DR2’, published in MNRAS 489, 2615–2633 (2019). We have discovered two important issues compromising the synthetic photometry underlying the fitting performed in this work. However, these issues act in the opposite sense, and effectively work to counteract each other. As a result, after being corrected, the radii of 99 per cent of the targets in the revised catalogue presented here differ by less than 1 per cent from the values we originally presented. However given the precision to which we measured stellar radii in the original work we thought it prudent to publish revisions to the derived catalogues and relations. The full revised catalogue is available at CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?VI/156, the University of Exeter ORE repository (https://doi.org/10.24378/exe.1683) and the GitHub repository of supplementary material for Morrell & Naylor (2019).¹ We also provide a revised version of the supplementary material, which are the tabulated temperature-radius-luminosity relationships. In light of these revisions, we thought it prudent to also include, and describe, improvements we have made to the process through which the uncertainties of our stellar parameters are calculated. We will now detail the issues, and the remediations which we have applied.

As presented in Evans et al. (2018), and originally described in Jordi et al. (2010), the zero-point flux |$f^\circ _\lambda$| for the Gaia Data Release 2 (DR2) photometric system is set using Vega (see equation 2 of Evans et al. 2018). The reference SED for this zero-point is that of an A0V star from the Kurucz/ATLAS9 Vega spectrum with T_eff = 9550 K, log (g) = 3.95 dex, [Fe/H]  = −0.5 and ν_micro = 2 km s⁻¹. This SED is normalized such that the flux density at λ = 550 nm is λ₅₅₀ = 3.66 × 10⁻¹¹ W m⁻² nm⁻¹. Due to a misconfiguration in the code that produced the grids used for the fitting, the zero-point flux for the synthetic photometry in our original work was instead provided by the Vega spectrum of Bohlin & Gilliland (2004). This introduced a systematic zero-point error into the Gaia DR2 synthetic photometry of ΔZP_BP = −0.030 and ΔZP_RP = −0.028. In the revised catalogue, the fitting is performed on grids with the zero-point corrected to be consistent with that prescribed in Evans et al. (2018).

The synthetic photometry were generated from synthetic stellar spectra from the BTSettl model atmosphere grid. The CIFIST grid was used for the primary grid, and was supplemented by the AGSS2009 grids for super-solar and sub-solar metallicities. From the definition of effective temperature, the total radiant spectral intensity of these stellar atmospheres should satisfy

$$\begin{eqnarray*} \frac{\sigma T_{\rm eff}^4}{ \int _\lambda I_\lambda d\lambda } = C, \end{eqnarray*}$$

(1)

where I_λ is the mean disc intensity at the stellar surface of the synthetic atmosphere, provided in units of energy per unit area per unit wavelength, and the correction factor C ≃ 1. We found C to deviate from unity by 1 per cent, increasing to around 5 per cent for very low-mass stars, across the M-dwarf regime for the CIFIST model grid, regardless of log (g). The issues are more evident in the AGSS2009 grid, which commonly has deviations on the order of 2–4 per cent across the low-mass regime. Hence, we amended our methodology for generating grids, presented in section 2.2 of Morrell & Naylor (2019), to multiply each intensity bin in the synthetic spectra by the correction factor C prior to folding through the system responses. This approach ensures that the overall colour and spectral features of the synthetic atmosphere remain unaffected, while ensuring that equation (1) is satisfied.

The correction of both the zero point and normalization errors results in a revisions in T_SED of between −40 and 80 K (±2 per cent) and R of between −0.027 and 0.019 |$R_\odot \left(^{+3}_{-5}\ \rm {per\ cent}\right)$|⁠. However, these are the extreme cases, with 99 per cent and 96 per cent remaining within 1 per cent of their original T_SED and R respectively, with 84 per cent remaining in the same or adjacent 10 K temperature bins.

2 REVISED TEMPERATURE–LUMINOSITY–RADIUS RELATIONS

In line with the revised catalogue, we have revised the T_SED − R and L_SED − R relations. The relations were derived using methodology identical to the original paper, using the parameters in the revised catalogue. The revised temperature–radius relation is given by

$$\begin{eqnarray*} R_{\rm fit}(T_{\rm SED}) &=& -3.842 + 2.046\times 10^{-3}\ T_{\rm SED} -\ 2.328\times 10^{-7}\ T_{\rm SED}^2, \\ 3000\ {\rm K} &\le& T_{\rm SED} \le 4400\ {\rm K}, \end{eqnarray*}$$

(2)

with the upper and lower radius bounds being given by

$$\begin{eqnarray*} R_\mathrm{high}\ (T_{\rm SED}) \!&=&\! -3.336 \!+\! 1.835\!\times \! 10^{-3}\ T_{\rm SED} \!-\! 2.090 \!\times \! 10^{-7}\ T_{\rm SED}^2, \\ 3000\ {\rm K} &\le& T_{\rm SED} \le 4400\ {\rm K}, \end{eqnarray*}$$

(3)

$$\begin{eqnarray*} R_\mathrm{low}\ (T_{\rm SED}) \!&=& \!-3.258 \!+\! 1.674\!\times \! 10^{-3}\ T_{\rm SED} \!-\! 1.792 \!\times \! 10^{-7}\ T_{\rm SED}^2, \\ 3000\ {\rm K} &\le& T_{\rm SED} \le 4400\ {\rm K}. \end{eqnarray*}$$

(4)

The revised T_SED − R relationship is tabulated in Table 1.

Table 1.

The revised version of table 3 from Morrell & Naylor (2019), which provides a tabulated form of the T_SED − R relation presented in equation (2) with bounds from equations (4) and (3). The full table is available in the online electronic supplementary material.

TSED (K)	Rfit(TSED) (R⊙)	Rlow(TSED)	Rhigh(TSED)
3000	0.199	0.151	0.286
3001	0.200	0.152	0.286
3002	0.201	0.152	0.287
3003	0.201	0.153	0.287
3004	0.202	0.153	0.288

The revised correction to the Dotter et al. (2008) 4 Gyr isochrone is given by

$$\begin{eqnarray*} R_{\rm fit}(L_{\rm SED}) &=& R_{\textrm {D}08}(L_{\rm SED}) + 0.0136 + 0.7087 L_{\rm SED} - 7.6924 L_{\rm SED}^2, \\ 0.003\ \mathrm{L}_{\odot } &\le& L_{\rm SED} \le 0.1\ \mathrm{L}_{\odot }, \end{eqnarray*}$$

(5)

with the upper and lower bounds in radius being given by

$$\begin{eqnarray*} R_\mathrm{high}\ (L_{\rm SED})\! &=& \!R_\textrm {D08}(L_{\rm SED}) \!+\! 0.0288 \!+\! 0.7662 L_{\rm SED} \!-\! 5.4204 L_{\rm SED}^2, \\ 0.003\ \mathrm{L}_{\odot } &\le& L_{\rm SED} \le 0.1\ \mathrm{L}_{\odot } \end{eqnarray*}$$

(6)

$$\begin{eqnarray*} R_\mathrm{low}\ (L_{\rm SED}) \!&=&\! R_\textrm {D08}(L_{\rm SED}) \!+\! 0.0026 \!+\! 0.2696 L_{\rm SED} \!-\! 4.2515 L_{\rm SED}^2, \\ 0.003\ \mathrm{L}_{\odot } &\le& L_{\rm SED} \le 0.1\ \mathrm{L}_{\odot }. \end{eqnarray*}$$

(7)

The revised L_SED − R relationship is tabulated in Table 2.

Table 2.

The revised version of table 4 from Morrell & Naylor (2019), which provides a tabulated form of the L_SED − R relationship presented in equation (5) as the sum of the Dotter et al. (2008) isochronal radius with a correction ΔR. We also include the bounds from equations (7) and (6), and the applied correction. The full table is available in the online electronic supplementary material.

LSED (L⊙)	ΔR	ΔRlow	ΔRhigh	Rfit (R⊙)	Rlow	Rhigh
0.003	0.016	0.003	0.031	0.193	0.181	0.209
0.004	0.016	0.004	0.032	0.218	0.205	0.233
0.005	0.017	0.004	0.033	0.239	0.226	0.254
0.006	0.018	0.004	0.033	0.257	0.243	0.273
0.007	0.018	0.004	0.034	0.273	0.260	0.289

The interpolation points for F(L_SED), presented in table 5 of Morrell & Naylor (2019), also see revision. The revised F(L_SED) values are tabulated in Table 3. Throughout their domain of validity, the radius that is prescribed by these relations remains within 1 per cent of those that were originally published.

Table 3.

The revised version of table 5 from Morrell & Naylor (2019), which presents the interpolation points for the values of F(L_SED). This table is available in the online electronic supplementary material.

LSED (L⊙)	F(LSED)	LSED (L⊙)	F(LSED)	LSED (L⊙)	F(LSED)
0.0035	−0.0151	0.0365	−0.0668	0.0695	−0.0758
0.0045	−0.0188	0.0375	−0.0874	0.0705	−0.0436
0.0055	−0.0280	0.0385	−0.0857	0.0715	−0.0573
0.0065	−0.0342	0.0395	−0.0539	0.0725	−0.0480
0.0075	−0.0339	0.0405	−0.0811	0.0735	−0.0652
0.0085	−0.0411	0.0415	−0.0810	0.0745	−0.0584
0.0095	−0.0434	0.0425	−0.0641	0.0755	−0.0388
0.0105	−0.0436	0.0435	−0.0843	0.0765	−0.0432
0.0115	−0.0508	0.0445	−0.0675	0.0775	−0.0589
0.0125	−0.0553	0.0455	−0.0805	0.0785	−0.0278
0.0135	−0.0527	0.0465	−0.0602	0.0795	−0.0639
0.0145	−0.0563	0.0475	−0.0763	0.0805	−0.0604
0.0155	−0.0584	0.0485	−0.0742	0.0815	−0.0287
0.0165	−0.0630	0.0495	−0.0740	0.0825	−0.0409
0.0175	−0.0622	0.0505	−0.0643	0.0835	−0.0397
0.0185	−0.0676	0.0515	−0.0682	0.0845	−0.0606
0.0195	−0.0653	0.0525	−0.0601	0.0855	−0.0440
0.0205	−0.0695	0.0535	−0.0881	0.0865	−0.0219
0.0215	−0.0728	0.0545	−0.0593	0.0875	−0.0525
0.0225	−0.0751	0.0555	−0.0549	0.0885	−0.0344
0.0235	−0.0696	0.0565	−0.0607	0.0895	−0.0328
0.0245	−0.0756	0.0575	−0.0656	0.0905	−0.0501
0.0255	−0.0791	0.0585	−0.0598	0.0915	−0.0416
0.0265	−0.0769	0.0595	−0.0645	0.0925	−0.0185
0.0275	−0.0691	0.0605	−0.0702	0.0935	−0.0584
0.0285	−0.0767	0.0615	−0.0600	0.0945	−0.0371
0.0295	−0.0812	0.0625	−0.0543	0.0955	−0.0448
0.0305	−0.0665	0.0635	−0.0485	0.0965	−0.0211
0.0315	−0.0838	0.0645	−0.0396	0.0975	−0.0375
0.0325	−0.0881	0.0655	−0.0468	0.0985	−0.0257
0.0335	−0.0736	0.0665	−0.0671	0.0995	−0.0238
0.0345	−0.0812	0.0675	−0.0702
0.0355	−0.0697	0.0685	−0.0175

3 REVISED UNCERTAINTY BOUNDS

We have also made some small revisions to our uncertainty estimates. To account for distance uncertainty in our radius bounds, we convolved each row along the radius axis in the 2D T_SED − R plane of |$\chi^2$| with a Gaussian, whose |$\sigma$| was the symmetrized distance uncertainty from Bailer-Jones et al. (2018). As this Gaussian represents a posterior, we should have performed this convolution in probability space; as opposed to |$\chi^2$| space. While implementing this change, we also discovered a minor issue in the marginalization routines for the 2D R − T_eff PDF. Both of these issues have been resolved in our revised catalogue.

Whilst remedying these issues we improved the process by which uncertainty bounds are found. Our bounds, in both T_SED and R were drawn from the pixel nearest the |$1\sigma$| confidence level in the marginalized 1D PDF for each parameter. We now perform an additional linear interpolation between the nearest pixel, and that which bounds the |$1\sigma$| level, to find the crossing point – meaning that we are now able to estimate the bounds to within about 1 per cent of their true value, regardless of relative pixel size. The final result of these modifications is that the median uncertainty of our sample for which uncertainties were drawn has increased from 1.64 per cent to 1.88 per cent; remaining well within the observed spread of 3–7 per cent.

4 DISCUSSION AND CONCLUSION

We have revised our catalogue of stellar parameters and their uncertainties, and corrected our T_SED − R and L_SED − R relations. The resulting changes are sufficiently small that the conclusions drawn in Morrell & Naylor (2019) remain unaffected.

SUPPORTING INFORMATION

Table 1. The revised version of table 3 from Morrell & Naylor (2019), which provides a tabulated form of the T_SED − R relation presented in equation (2) with bounds from equations (4) and (3).

Table 2. The revised version of table 4 from Morrell & Naylor (2019), which provides a tabulated form of the L_SED − R relationship presented in equation (5) as the sum of the Dotter et al. (2008) isochronal radius with a correction ΔR.

Table 3. The revised version of table 5 from Morrell & Naylor (2019), which presents the interpolation points for the values of F(L_SED).

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

ACKNOWLEDGEMENTS

The authors would like to thank Yang Chen, of Padova University, for bringing our attention to the need to re-normalize the BTSettl atmospheres.

Footnotes

1

https://github.com/sammorrell/mn19-supplementary-material