diff --git a/document/ms.tex b/document/ms.tex index 4f30d94..8ed2c38 100644 --- a/document/ms.tex +++ b/document/ms.tex @@ -87,9 +87,9 @@ \emph{Methods.} A generalization of line depth ratio analysis is developed and applied to a high-resolution near-IR IGRINS spectrum. The IGRINS spectrum is forward modeled with pre-computed \PHOENIX\ synthetic spectra using the open source spectral inference framework \texttt{Starfish} extended to include a two component mixture model comprised of ambient photosphere and starspot emission. -\emph{Results.} Spectral features attributable to both ambient photosphere $\teffa\sim4100$ K and cool starspots $\teffb\sim2750$ K are detected in the IGRINS spectrum. In some spectral regions, the starspot emission acts merely as a relatively featureless ``veiling'' continuum owing to the large rotational broadening ($\vsini\sim28$ km~s$^{-1}$) and heavy line-blanketing in cool star spectra. The filling factor of starspots is consistently large ($>50\%$) across all IGRINS spectral orders, with a best fit filling factor of $80\%$ starspots. The interpretation of the 20\% ambient photosphere as a ``hot spot'' is also possible, though there is no physical motivation for such a distinction. The spectral energy distribution, color variability, rotational modulation, and low-resolution optical/IR spectroscopy-- when registered to the filling factors at the epochs of their observations--strengthen the case for a large starspot filling factor and temperature contrast through time. The revised effective temperature and luminosity make LkCa 4 appear much lower mass and much younger compared to previous estimates based only on optical spectroscopy and not taking into account the large covering fraction of the cool spots. +\emph{Results.} Spectral features attributable to both ambient photosphere $\teffa\sim4100$ K and cool starspots $\teffb\sim2750$ K are detected in the IGRINS spectrum. In some spectral regions, the starspot emission acts merely as a relatively featureless ``veiling'' continuum owing to the large rotational broadening ($\vsini\sim28$ km~s$^{-1}$) and heavy line-blanketing in cool star spectra. The filling factor of of starspots at minimum light $\Delta V = 0.5$must be $>37\%$, and is likely much larger based on geometry and temperature considerations. The IGRINS spectra yield consistently large ($>50\%$) fill factors across all IGRINS spectral orders, with a best fit filling factor of $f=80\%$ starspots at the epoch of spectrum acquisition. The interpretation of the 20\% ambient photosphere as a ``hot spot'' is also possible, but then all previous studies have misclassified the K7 spectral type from an unseen persistent hot spot spectrum. The spectral energy distribution, color variability, rotational modulation, and low-resolution optical/IR spectroscopy-- when registered to the filling factors at the epochs of their observations--strengthen the case for a large starspot filling factor and temperature contrast through time. -\emph{Conclusions.} Emission from longitudinally symmetric starspots has likely caused the misplacement of spotted young stars on the observational HR diagram, leading to incorrect estimates for ages and masses and contributing, in part, to the apparent age spread observed towards young clusters. +\emph{Conclusions.} The revised effective temperature and luminosity make LkCa 4 appear much lower mass and much younger compared to previous estimates based only on optical spectroscopy and not taking into account the large covering fraction of the cool spots. Emission from longitudinally symmetric starspots has likely caused the misplacement of spotted young stars on the observational HR diagram, leading to incorrect estimates for ages and masses and contributing, in part, to the apparent age spread observed towards young clusters. \end{abstract} \keywords{stars: fundamental parameters --- stars: individual (LkCa 4) --- stars: low-mass -- stars: statistics} @@ -100,11 +100,11 @@ \section{Introduction}\label{sec:intro} %\subsection{Pre Main Sequence HR diagram spread} -Pre-main sequence stellar evolution is an unsolved problem \citep[see review by][]{soderblom14}. Systematic uncertainties in ages of pre-main sequence stars lead to uncertainties in dissipation timescales of envelopes and protoplanetary disks as evaluated in global studies. All clusters have large spreads in luminosity at any given temperature \citep[e.g.][]{reggiani11}, which frustrates the interpretation of individual star/disk properties as related to age. +Pre-main sequence stellar evolution is an unsolved problem \citep[see review by][]{soderblom14}. Systematic uncertainties in ages of pre-main sequence stars lead to uncertainties in dissipation timescales of envelopes and protoplanetary disks as evaluated in global studies. Young clusters also have large spreads in luminosity at any given temperature \citep[e.g.][]{reggiani11}, which further frustrate the interpretation of individual star/disk properties as related to age. The causes of global age uncertainties and of large luminosity spreads in individual clusters are controversial. Observationally, \citet{hartmann01} and \citet{slesnick08} argue that measurement uncertainties may mask any real differences in ages within a cluster. In models, \citet{hartmann97} and \citet{baraffe09} describe how variable accretion histories change the stellar contraction and hence luminosities at early times. Contraction rates also depend on the prescription for convection, which may vary with mass. -Magnetic activity is a likely source of significant uncertainty in both the models and observations of young low-mass stars. Convection at these young ages generates strong magnetic fields, as measured by Zeeman broadening and polarimetry \citep[e.g.][]{johnskrull07,donati09} and as seen in starspots \citep[e.g.][]{stauffer03,grankin08}. Evolutionary models are just now starting to implement new prescriptions for convection with magnetic fields (\citet{somers15}, \citet{feiden16}; see also \citet{baraffe15} for an updated treatment of convection without introducing magnetic fields). Stellar evolution models including the effect of starspots can make a coeval 10 Myr population exhibit apparent age spreads of 3$-$10 Myr, with standard mass estimates being biased towards lower masses \citep{somers15}. Starspots may also be responsible for biases in stellar effective temperatures derived by different methods. For example, the effective temperatures for 3493 young stars measured using the APOGEE spectrograph \citep[$1.5-1.70 \;\mu$m at $R=22,500$][]{wilson10} are offset by 200$-$500 K ($\sim$0.02$-$0.05 dex) and as high as 1000 K ($\sim$0.1 dex) relative to previous, usually optical, measurements \citep{cottaar14}. +Magnetic activity is a likely source of significant uncertainty in both the models and observations of young low-mass stars. Convection at these young ages generates strong magnetic fields, as measured by Zeeman broadening and polarimetry \citep[e.g.][]{johnskrull07,donati09} and as seen in starspots \citep[e.g.][]{stauffer03,grankin08}. Evolutionary models are just now starting to implement new prescriptions for convection with magnetic fields (\citet{somers15}, \citet{feiden16}; see also \citet{baraffe15} for an updated treatment of convection without introducing magnetic fields). Stellar evolution models including the effect of starspots can make a coeval 10 Myr population exhibit apparent age spreads of 3$-$10 Myr, with standard mass estimates being biased towards lower masses \citep{somers15}. Starspots may also be responsible for biases in stellar effective temperatures derived by different methods. For example, the effective temperatures for 3493 young stars measured using the APOGEE spectrograph \citep[$1.5-1.70 \;\mu$m at $R=22,500$][]{wilson10} exhibit systematic offsets from prior measurements predominantly that were predominantly made at optical wavelengths. Typical offsets range from $-400-400$ K, with a systematic dependence on $\teff$, with some offsets as large as 1000 K \citep{cottaar14}. % Theory of starspots % Jackson \& Jeffries 2014ab needs to go somewhere. @@ -116,9 +116,9 @@ \section{Introduction}\label{sec:intro} The WTTS LkCa 4 \citep{herbig86,strom89a,downes88,strom89b} is an ideal exemplar for a spotted pre-MS star because it does not have any veiling \citep[\emph{e.g.}][]{hartigan95} or mid-IR or mm excess \citep[\emph{e.g.}][]{andrews05,furlan06,buckle15}, and is not actively accreting \citep[\emph{e.g.}][]{edwards06,cauley12}. LkCa 4 has no evidence for a close companion from both direct imaging searches\citep{karr10,kraus11,daemgen15}\footnote{The status for wide companions is less clear; see \citet{stauffer91,itoh08,kraus09,kraus11,herczeg14}} and RV searches \citep{nguyen12,donati14}. This single star demonstrates a large amplitude of sinusoidal photometric variability indicative of large spots \citep{grankin08,xiao12}. The variability amplitude cannot arise from eclipsing binarity, since the large RV variations would have been seen in spectroscopic monitoring. All of these observations indicate that the spectrum of \name should be devoid of complicating factors like near-IR excess veiling, accretion excess, or close binaries. -LkCa 4 has recently been examined with ZDI \citep{donati14}, revealing a complex distribution of cool and warm spots in brightness map reconstructions. Dark polar spots extend to about $30^\circ$ from the pole, with about 5 appendages reaching down to about $60^\circ$ from the pole. There is also evidence for a hot spot in the reconstructed ZDI map. Still, unanswered questions about the large disagreement ($\Delta \teff \sim$ 500 K) in LkCa 4's effective temperature \citep{herczeg14, donati14} remain. +LkCa 4 has recently been examined with ZDI \citep{donati14}, revealing a complex distribution of cool and warm spots in brightness map reconstructions. Dark polar spots extend to about $30^\circ$ from the pole, with about 5 appendages reaching down to about $60^\circ$ from the pole. There is also evidence for a hot spot in the reconstructed ZDI map. Still, unanswered questions about the large disagreement in LkCa 4's effective temperature remain, with 4100 K measured from holistic analysis of ESPaDOnS spectra yielding $\teff=4100$K \citep{donati14}, and analysis of the TiO bands of the same spectra yielding $\teff=3600$ K \citep{herczeg14}. -In this work, we constrain the starspot properties of \name with 4 complementary techniques. Section \ref{sec:obs} quantifies the optical variability of \name over the last 31 years based on all available photometric monitoring; a collection of spectral observations at various phases of variability is introduced. Section \ref{sec:methods} and Appendix \ref{methods-details} describe a spectral inference technique aimed at generalizing line depth ratio analysis, which is applied to a high resolution panchromatic near-IR spectrum of \name in Section \ref{sec:two_tempIGRINS} . Section \ref{sec:GJHsection4} shows results from SED fitting, polychromatic photometric monitoring, and optical TiO line-depth ratio fitting. All lines of evidence are ultimately combined to build a consistent picture of the spectral and temporal evolution of \name, and what can be understood about pre-MS stellar evolution from this exemplar. +In this work, we constrain the starspot properties of LkCa 4 with four complementary techniques. Section \ref{sec:obs} quantifies the optical variability of LkCa 4 over the last 31 years based on all available photometric monitoring; a collection of spectral observations at various phases of variability is introduced. Section \ref{sec:methods} and Appendix \ref{methods-details} describe a spectral inference technique aimed at generalizing line depth ratio analysis, which is applied to a high resolution $H$ and $K$ near-IR spectrum of LkCa 4 in Section \ref{sec:two_tempIGRINS}. Section \ref{sec:GJHsection4} shows results from SED fitting, polychromatic photometric monitoring, and optical TiO line-depth ratio fitting. All lines of evidence are ultimately combined to build a consistent picture of the spectral and temporal evolution of LkCa 4, and what can be understood about pre-MS stellar evolution from this exemplar. \section{Observations}\label{sec:obs} @@ -155,7 +155,7 @@ \subsection{Photometric monitoring} \begin{figure*} \centering \includegraphics[width=0.95\textwidth]{figures/Vband_22s.pdf} - \caption{Phase-folded lightcurves constructed assuming $P=3.375$ days for all 22 observing seasons. The blue solid lines show a ``multiterm'' regularized periodic fit, that is, keeping the first $M_{\rm max}=4$ Fourier components \citep{vanderplas15a}. The vertical lines show the epochs at which spectra or ancillary photometry were obtained, with the same line styles and colors as Figure \ref{fig:PhotTime}. LkCa 4 shows secular changess in its light curve morphology.} + \caption{Phase-folded lightcurves constructed assuming $P=3.375$ days for all 22 observing seasons. The blue solid lines show a ``multiterm'' regularized periodic fit, that is, keeping the first $M_{\rm max}=4$ Fourier components \citep{vanderplas15a}. The vertical lines show the epochs at which spectra or ancillary photometry were obtained, with the same line styles and colors as Figure \ref{fig:PhotTime}. LkCa 4 shows secular changess in its light curve morphology. The IGRINS spectrum was acquired near the median flux level, not the extrema.} \label{fig:PhotPhase} \end{figure*} @@ -190,13 +190,18 @@ \section{FITS TO HIGH RESOLUTION SPECTRA}\label{sec:Starfish} Starspot line depth ratio analysis has traditionally been limited to isolated portions of spectrum that possess spectral lines attributable only to starspots and spectral lines attributable only to ambient photosphere \citep[\emph{e.g.}][]{neff95, oneal01}. The apparent veiling of the lines and their respective temperature dependences can be combined to solve for the starspot and ambient photosphere temperatures and relative areal coverages. This line depth ratio method suffers from the need to identify portions of the spectrum that possess such strong lines, and from the need to assemble large atlases of observed spectral templates to which the spotted star spectrum can be compared. In this section, we introduce a generalization of the line depth ratio analysis which employs pixel-by-pixel modeling of all spectral orders. By using all the spectral data, this strategy has the power to constrain starspot properties at relatively low filling factors and to identify weak lines that originate from the starspots. -We took two approaches to characterizing the effective temperature response from a spotted-star spectrum. First we measured a single unique effective temperature to each spectral order in the optical (\S \ref{sec:ESP_starfish}) and near-IR (\S \ref{sec:IGR_starfish}). Second we fit a two-temperature mixture model (\S \ref{sec:methods}) to a near-IR echelle spectrum (\S \ref{sec:two_tempIGRINS}). The near-IR was preferred over the optical, since the cool starspots emit most of their flux in the near-IR, enhancing the likelihood of direct detection of emission from the starspots. +We took two approaches to characterizing the effective temperature response from a spotted-star spectrum. + +Using the spectral fitting approach introduced in Section 3.1 \ref{sec:methods}, we first first separately fit the optical (ESPaDOnS) and near-IR (IGRINS) spectra with distinct, single $\teff$ models; these fits are described in Sections \ref{sec:ESP_starfish} and \ref{sec:IGR_starfish} respectively. We then use a two temperature mixture model to perform a fit to the near-IR spectra in Section \ref{sec:two_tempIGRINS}. + + +First we measured a single unique effective temperature to each spectral order in the optical (\S \ref{sec:ESP_starfish}) and near-IR (\S \ref{sec:IGR_starfish}). Second we fit a two-temperature mixture model (\S \ref{sec:methods}) to a near-IR echelle spectrum (\S \ref{sec:two_tempIGRINS}). The near-IR was preferred over the optical, since the cool starspots emit most of their flux in the near-IR, enhancing the likelihood of direct detection of emission from the starspots. \subsection{Methodology}\label{sec:methods} \citet[hereafter \iancze]{czekala15} developed a modular framework\footnote{The open source codebase and its full revision history is available at \url{https://github.com/iancze/Starfish}. The experimental fork discussed in this paper is at \url{https://github.com/gully/Starfish}} to infer stellar properties from high resolution spectra. The \iancze\ technique forward models observed spectra with synthetic spectra from pre-computed model grids. The intra-grid-point spectra are ``emulated'' in a process similar but superior to interpolation, since it seemlessly quantifies the uncertainty attributable to the coarsely sampled stellar intrinsic parameters (see Appendix of \iancze). The forward model includes calibration parameters, line spread functions, and a Gaussian process noise model to account for correlations in the residual spectrum. We employed the \PHOENIX\ grid of pre-computed synthetic stellar spectra, which span a wide range of wavelengths at high spectral resolution with a sampling of 100 K in $\teff$ in our range of interest \citep{husser13}. The modular framework was altered to accommodate starspot measurements in two ways. First, the single photospheric component was updated to include a starspot spectrum. Second, the MCMC sampling strategy was altered to accommodate the additional free parameters added by the starspot model. -The stellar photosphere is characterized as two photospheric components with a starspot temperature $\teffb$ and ambient photosphere temperature $\teffa$, with scalar solid angular coverages $\Omega_{\mathrm{spot}}$ and $\Omega_{\mathrm{amb}}$, respectively\footnote{For flux calibrated spectra, the total solid angle $\Omega$ can be constrained, but for typical echelle spectrographs only relative values of $\Omega_{\mathrm{spot}}$ and $\Omega_{\mathrm{amb}}$ can be inferred.}. Starspots (or spot groups) are assumed to be cooler than the ambient photosphere, but otherwise share the same average intrinsic and extrinsic stellar parameters $(\vsini, \logg, \Z, v_z)$. The composite mixture model for observed flux density is: +The stellar photosphere is characterized as two photospheric components with a starspot temperature $\teffb$ and ambient photosphere temperature $\teffa$, with scalar solid angular coverages $\Omega_{\mathrm{spot}}$ and $\Omega_{\mathrm{amb}}$, respectively\footnote{For flux calibrated spectra, the total solid angle $\Omega$ can be constrained, but for typical echelle spectrographs only relative values of $\Omega_{\mathrm{spot}}$ and $\Omega_{\mathrm{amb}}$ can be inferred.}. Starspots (or spot groups) are assumed to be cooler than the ambient photosphere, but otherwise share the same average intrinsic and extrinsic stellar parameters $\vsini, \logg, \Z, v_z$. The composite mixture model for observed flux density is: \begin{eqnarray} \label{eqn:mix_M} S_{\mathrm{mix}} = \Omega_{\mathrm{amb}} B(\teffa) + \Omega_{\mathrm{spot}} B(\teffb) \end{eqnarray} @@ -207,7 +212,7 @@ \subsection{Methodology}\label{sec:methods} The term $f_{\Omega}$ represents an instantaneous, observational fill factor seen on one projected hemisphere, not $f_{\rm spot}$, the ``ratio of the spotted surface to the total surface areas'' \citep{somers15}. In the limit of homogenously distributed spots, $f_{\Omega}$ tends to $f_{\rm spot}$, but in general $f_{\rm spot}$ is not a direct observable since some circumpolar regions of inclined stars will always face away from Earth; $f_{\Omega}$ can vary cyclically through rotational modulation, whereas $f_{\rm spot}$ changes secularly through starspot evolution that is not yet understood. -The starspot model includes two more free parameters than the standard \iancze\ model, namely $\teffb$ and $f_{\Omega}$. The addition of these two parameters makes the MCMC sampling much more correlated than it was before because the relative contribution of the starspot and ambient photosphere are nearly degenerate over small changes in $\teffa$, $\teffb$, and $f_{\Omega}$. This challenge motivated the second important change to the spectral inference framework, which involves technical aspects of switching from sampling the nuisance and stellar parameters separately in a blocked Gibbs framework to ensemble sampling using \texttt{emcee} \citep{foreman13}. The affine-invariant \texttt{emcee} ensemble sampler is more resilient to correlations among stellar temperatures and fill factor than the Metropolis-Hastings sampler used in Gibbs sampling. The practical effect of this switch is that all 14 stellar and nuisance parameters are fit simultaneously in a single spectral order, making stellar parameter estimates $\vt_{o}$ unique for each spectral order $o$, whereas the \iancze\ strategy had the power to provide a single set of stellar parameters $\vt$ that was based on all $N_{\rm ord}$ spectral orders. The $N_{\rm ord}$ sets of inferences on $\vt_{o}$ are combined by weighted averaging point estimates of results from reliable orders, which offers resilience to flagrant calibration artifacts or conspicuous model mismatches. This process can be thought of as a coarse approximation to the much more sophisticated spectral line outlier rejection described in \iancze. No attempt was made to downweight spectral-line outliers using local kernels. +The starspot model includes two more free parameters than the standard \iancze\ model, namely $\teffb$ and $f_{\Omega}$. The addition of these two parameters makes the MCMC sampling much more correlated than it was before because the relative contribution of the starspot and ambient photosphere are nearly degenerate over small changes in $\teffa$, $\teffb$, and $f_{\Omega}$. This challenge motivated the second important change to the spectral inference framework, which involves technical aspects of switching from sampling the nuisance and stellar parameters separately in a blocked Gibbs framework to ensemble sampling using \texttt{emcee} \citep{foreman13}. The affine-invariant \texttt{emcee} ensemble sampler is more resilient to correlations among stellar temperatures and fill factor than the Metropolis-Hastings sampler used in Gibbs sampling \citep{}. The practical effect of this switch is that all 14 stellar and nuisance parameters are fit simultaneously in a single spectral order, making stellar parameter estimates $\vt_{o}$ unique for each spectral order $o$, whereas the \iancze\ strategy had the power to provide a single set of stellar parameters $\vt$ that was based on all $N_{\rm ord}$ spectral orders. The $N_{\rm ord}$ sets of inferences on $\vt_{o}$ are combined by weighted averaging point estimates of results from reliable orders, which offers resilience to flagrant calibration artifacts or conspicuous model mismatches. This process can be thought of as a coarse approximation to the much more sophisticated spectral line outlier rejection described in \iancze. No attempt was made to downweight spectral-line outliers using local kernels. This two-temperature mixture model for starspots requires single stars, or at least double stars with extremely large (>100) luminosity ratios or wide separations. Unresolved single-lined or double-lined binaries could mimic a signal that would be misattributed to starspots. As pointed out earlier, LkCa 4 has no detected companion from direct imaging, and its RV variations are consistent with starspot modulation, so any binary companion would have to have exceptionally large luminosity or mass ratios, rendering it undetectable with our spectral inference methodology with the finite signal to noise available in the IGRINS data. LkCa 4's absense of mid-IR excess emission attributable to a disk also means that the spectral modeling requires no further parameters like veiling or accretion. @@ -218,27 +223,27 @@ \subsection{Single temperature fitting to the ESPaDOnS spectrum}\label{sec:ESP_s We performed spectral fitting on an ESPaDOnS spectrum\footnote{The fourth ESPaDOnS spectrum from Table \ref{tbl_estimated_V} was arbitrarily chosen.} acquired on 2014 January 11. The spectrum was apportioned into subsets of $N=35$ chunks, corresponding approximately to spectral order boundaries. Full-spectrum fitting was performed separately on each of 35 spectral orders following the Metropolis-Hasting MCMC sampling procedure described in \iancze. The spectral emulator was trained on stellar parameters in the range $\logg \in [3.5, 4.0]$, $\Z \in [-0.5, 0.5]$, and $\teff \in [3500, 4200]$. -The Metropolis-Hastings step sizes were tuned with several iterations of burn-in procedure and the final chains were visually checked for convergence. Nine of the 35 available spectral orders failed to converge, due to numerical artifacts arising from poor model fits. For the other 26 spectral orders, the number of samples was generally much longer than the estimated integrated autocorrelation length. Overlays of models with observed spectra show modest agreement, with the exception of conspicuous spectral line outliers. +The Metropolis-Hastings step sizes were tuned with several iterations of burn-in procedure and the final chains were visually checked for convergence. Nine of the 35 available spectral orders failed to converge, due to numerical artifacts arising from poor model fits. For the other 26 spectral orders, the number of samples was generally much longer than the estimated integrated autocorrelation length. Overlays of models with observed spectra showed conspicuous spectral line outliers, but the bulk of spectral lines were replicated in the synethic spectral model fits. We computed the median value and 5$^{th}$ and $95^{th}$ percentiles of burned-in subsets of the MCMC samples described above. Overall the 26 sets of point estimates for stellar parameters show relatively good agreement, with exceptions. The best-fitting spectral orders show effective temperature point estimates in the range $\teff=4000\pm130$ K. Figure \ref{fig:SingleTeffvsOrder} displays the $\teff$ point estimates with 5$^{th}$ and $95^{th}$ percentile error bars placed at the central wavelength of each spectral order. \begin{figure} \centering \includegraphics[width=0.48\textwidth]{figures/single_Teff_v_order} - \caption{Effective temperature as derived from unique full spectrum fitting to each of 58 spectral orders in the optical through infrared portions of the spectrum and assuming a single component photosphere. The effective temperature derived in the $K-$band is about 800 K lower than that derived in optical. The increasing flux density from starspots compared to the flux from ambient photosphere can explain the observed trend in derived $\teff$.} + \caption{Effective temperature as derived from unique full spectrum fitting to each of 58 spectral orders in the optical through infrared portions of the spectrum and assuming a single component photosphere. The effective temperature derived in the $K-$band is about 800 K lower than that derived in optical. The increasing flux density from starspots compared to the flux from ambient photosphere can explain the observed trend in derived $\teff$. The cluster of $H-$band orders at 1.7 $\mu$m correspond to a local peak in flux density of cool starspots.} \label{fig:SingleTeffvsOrder} \end{figure} \subsection{Single temperature fitting to the IGRINS spectrum}\label{sec:IGR_starfish} -We performed full-spectrum fitting on 32 of the 54 available IGRINS spectral orders, again fitting unique stellar parameters $\vt = (\teff, \logg, \Z)$ for each spectral order $o$. We used the same analysis procedure as described for the ESPaDOnS spectra, with one exception. For the IGRINS $K-$band, we employed an expanded search range for the effective temperature, $\teff \in [3000, 4200]$ K, since the IGRINS $H-$ band demonstrated some saturation at the $\teff=3500$ K lower bound. +We selected a subset of 32 of the 54 available IGRINS spectral orders\footnote{The finite computational cost limited running all the spectral orders.} with the low telluric absorption artifacts. We performed full-spectrum fitting, again fitting unique stellar parameters $\vt = (\teff, \logg, \Z)$ for each spectral order $o$. We used the same analysis procedure as described for the ESPaDOnS spectra, with one exception. For the IGRINS $K-$band, we employed an expanded search range for the effective temperature, $\teff \in [3000, 4200]$ K, since the IGRINS $H-$ band demonstrated some saturation at the $\teff=3500$ K lower bound. We find a larger dispersion in the point estimates for the stellar parameters derived from the IGRINS data than those derived in optical. The most conspicuous trend is in the derived effective temperature as a function of wavelength shown in Figure \ref{fig:SingleTeffvsOrder}. The effective temperature peaks at values of $\sim4200$K in the short wavelength end of $H-$band and saturates at $<3500$ K at the long wavelength end of $H-$band. The $K-$band shows even lower derived effective temperatures of $\sim3300$ K, or 700 K cooler than estimated from optical. No single temperature can describe all the spectral lines present in the high resolution optical and IR spectra. Sources with such discrepancies have been seen previously, for example in Figures 4 and 5 in \citet{bouvier92}. These discrepancies are circumstantial evidence for the detection of spectral features attributable to starspots. \subsection{Heightened sensitivity to starspot spectral lines in the infrared}\label{sec:whyNearIR} -Some care should be taken when directly comparing results between the ESPaDOnS and IGRINS spectra since they were not taken at the same time. The particular ESPaDOnS spectrum used in Section \ref{sec:ESP_starfish} has an estimated $V-$band magnitude of 12.90 while the IGRINS spectrum has 12.83. The fainter magnitude during the ESPaDOnS spectrum acquisition implies a greater coverage fraction than during the IGRINS spectrum. The $\teff$ derived in the optical bands and short-wavelength end of $H$-band yield similar values of $\sim 4100$ K, suggesting the \emph{bulk}\footnote{The bulk appearance and disappearance of spectral lines is mostly controlled by the temperature of the emitting region of the local photosphere. In other words, most of the variance in a spectrum is attributable to temperature, assuming near-solar metallicity. Starspots imbue dearths of flux in the line profiles of optical spectra, but these are secondary to the mere presence of temperature-sensitive lines.} spectral features are broadly consistent with a emission from a single temperature component. A single ESPaDOnS order surrounding the TiO bands shows an exceptionally low estimated $\teff$. The long wavelength portion of $H-$band and all of $K-$band are more sensitive to starspot spectral signatures than the shorter wavelength portions. +Some care should be taken when directly comparing results between the ESPaDOnS and IGRINS spectra since they were not taken at the same time. The particular ESPaDOnS spectrum used in Section \ref{sec:ESP_starfish} has an estimated $V-$band magnitude of 12.90 while the IGRINS spectrum has 12.83. The fainter magnitude during the ESPaDOnS spectrum acquisition implies a greater coverage fraction than during the IGRINS spectrum. The $\teff$ derived in the optical bands and short-wavelength end of $H$-band yield similar values of $\sim 4100$ K, suggesting the \emph{bulk}\footnote{The bulk appearance and disappearance of spectral lines is mostly controlled by the temperature of the emitting region of the local photosphere. In other words, most of the variance in a spectrum is attributable to temperature, assuming near-solar metallicity. Starspots imbue dearths of flux in the line profiles of optical spectra, but these are secondary to the mere presence of temperature-sensitive lines, despite carrying useful information about the longitudinal distribution of the spots.} spectral features are broadly consistent with a emission from a single temperature component. A single ESPaDOnS order surrounding the TiO bands shows an exceptionally low estimated $\teff$. The long wavelength portion of $H-$band and all of $K-$band are more sensitive to starspot spectral signatures than the shorter wavelength portions. The heightened sensitivity to starspot spectral lines as a function of wavelength can be understood in the following way. Starspots are cooler than their surrounding photosphere and will, therefore, have a longer wavelength of peak emission. The average ratio of flux density between starspot and bulk photosphere will increase with wavelength until asymptoting to a fixed value in the Rayleigh-Jeans tail \citep{wolk96}. The bottom panel of Figure \ref{fig:SingleTeffvsOrder} shows the flux density ratio for patches of photosphere with equal areas (50\% filling factor) but different temperatures-- 2800 K and 4100 K for the starspot and ambient photosphere respectively. The black body ratios predict smooth flux ratio increases with wavelength, while the ratio of \PHOENIX\ models shows wavelength regions with heightened sensitivity to starspot fluxes, for example in $J-$band. @@ -247,11 +252,11 @@ \subsection{Heightened sensitivity to starspot spectral lines in the infrared}\l \subsection{Two-temperature fitting to IGRINS spectra}\label{sec:two_tempIGRINS} -Full-spectrum fitting was performed for 48 of the 54 available IGRINS spectral orders as described in Section \ref{sec:methods} and Appendix \ref{methods-details}. The MCMC samples generally had 5000 steps with 40 walkers. The stellar parameter ranges were $\logg \in [3.5, 4.0]$ and $\Z \in [-0.5, 0.5]$; the \PHOENIX\ model spectrum temperature range\footnote{The \PHOENIX\ model spectra are indexed by $\teff$, but this term carries a different meaning for spotted stars than non-spotted stars.} was $\teffa, \teffb \in [2700, 4500]$. $H-$band fits had normal distribution priors in place: solar metallicity to $\pm0.05$ dex, $logg=3.8\pm0.1$, and $\vsini=29\pm5$ km/s. The MCMC chains all appeared to converge after about 1500 steps. +Full-spectrum fitting was performed for 48 of the 54 available IGRINS spectral orders, omitting only the orders with the most pathological telluric spectral artifacts. We applied the mixture model as described in Section \ref{sec:methods} and Appendix \ref{methods-details}. The MCMC samples generally had 5000 steps with 40 walkers. The stellar parameter ranges were $\logg \in [3.5, 4.0]$ and $\Z \in [-0.5, 0.5]$; the \PHOENIX\ model spectrum temperature range\footnote{The \PHOENIX\ model spectra are indexed by $\teff$, but this term carries a different meaning for spotted stars than non-spotted stars.} was $\teffa, \teffb \in [2700, 4500]$. $H-$band fits had normal distribution priors in place: solar metallicity to $\pm0.05$ dex, $logg=3.8\pm0.1$, and $\vsini=29\pm5$ km/s. The MCMC chains all appeared to converge after about 1500 steps. Marginalized distributions were computed for all 14 stellar and nuisance parameters by selecting the final 200 $\times$ 40 walkers = 8000 samples, and point estimates were obtained by computing the median, $5^{th}$ and $95^{th}$ percentiles of the marginal distributions. The fit quality was first assessed by examining the consistency of the point estimates of $v_z$ and $\vsini$ across the spectral orders. The distribution of $v_z$ and $\vsini$ exposed extremely poor fits in two orders ($o=91$ and $94$), with all other orders demonstrating $v_z = 12.4 \pm 2.6$ km/s and $\vsini = 28.8 \pm 2.0$ km/s. -Overplotting forward-modeled spectra with the observed IGRINS spectra yielded insights on why some spectral orders performed better than others in assessing stellar properties. This model comparison is performed by sorting the 8000 samples by their fill factor estimates and examining random draws of the cool spectrum, hot spectrum, and composite spectrum, excluding the Gaussian process coveriance and white noise steps. Fitting defects were conspicuous. Orders with extremely poor telluric correction residuals, large spectral line outliers, and uncorrected H line residuals from A0V standard division, were all excluded from our final stellar parameter compilation. Notably, several orders in $K-$band were rejected due to large biases in metallicity, since $K-$band had uninformative priors on $\Z$, resulting in overfitting of orders possessing deep metal lines. For example, orders $o=$ 79 and 81 overfit Ca~I and Na~I lines and orders $o=74-76$ overfit the CO lines, which are known for their gravity sensitivity \citep{rayner09}. These and other metal lines can also be biased by Zeeman broadening, which could explain the heightened effect size with wavelength \citep{deen13}. Additionally, some orders without these conspicuous faults, but simply possessing mediocre fits, or relatively uninformative fits, were also removed from downstream estimates. What's left were the most informative orders relatively devoid of spectral line outliers, referred to hereafter as the Reliable Order subset. Figure \ref{fig:TwoTempResults} shows the distribution of $\teffa$ (blue shading), $\teffb$ (red shading), and $f_{\Omega}$ (yellow shading) for all of the spectral orders, with the rejected orders grayed out, and the Reliable Orders shown in bold. The point estimates for each spectral order are listed in Appendix Table \ref{tbl_order_results}. +Overplotting forward-modeled spectra with the observed IGRINS spectra yielded insights on why some spectral orders performed better than others in assessing stellar properties. This model comparison is performed by sorting the 8000 samples by their fill factor estimates and examining random draws of the cool spectrum, hot spectrum, and composite spectrum, excluding the Gaussian process coveriance and white noise steps. Fitting defects were conspicuous. Orders with extremely poor telluric correction residuals, large spectral line outliers, and uncorrected H line residuals from A0V standard division, were all excluded from our final stellar parameter compilation. Notably, several orders in $K-$band were rejected due to large biases in metallicity, since $K-$band had uninformative priors on $\Z$, resulting in overfitting of orders possessing deep metal lines. For example, orders $o=$ 79 and 81 overfit Ca~I and Na~I lines and orders $o=74-76$ overfit the CO lines, which are known for their gravity sensitivity \citep{rayner09}. These and other metal lines can also be biased by Zeeman broadening, which could explain the heightened effect size with wavelength \citep{deen13}. Additionally, some orders without these conspicuous faults, but simply possessing mediocre fits, or relatively uninformative fits, were also removed from downstream estimates. The remaining orders are rerelatively devoid of spectral line outliers and include the most information rich portion of the spectrum. Figure \ref{fig:TwoTempResults} shows the distribution of $\teffa$ (blue shading), $\teffb$ (red shading), and $f_{\Omega}$ (yellow shading) for all of the spectral orders, with the rejected orders grayed out, and the reliable order subset shown in bold. The point estimates for each spectral order are listed in Appendix Table \ref{tbl_order_results}. \begin{figure*} \centering @@ -260,7 +265,7 @@ \subsection{Two-temperature fitting to IGRINS spectra}\label{sec:two_tempIGRINS} \label{fig:TwoTempResults} \end{figure*} -The IGRINS spectrum demonstrates some features that are present only in the ambient photosphere model, and some features that are present only in the starspot model. Figure \ref{fig:specPostageStamp} shows a selection of six such spectral features for a range of plausible fill factors. In some cases, featureless starspot spectra veil isolated spectral lines predicted in the ambient photosphere models. In other cases, the ambient photosphere model veils sequences of shallow spectral features predicted in the starspot model. The starspot spectral models predict shallow features because line blanketing from multiple indistinct molecular bands overlap from rotational broadening. This line blanketing has probably hampered efforts to identify isolated spectral features suitable for line-depth-ratio analysis in the spectra of rotationally broadened young stars. One such feature has been previously identified in the near-IR: the OH 1.563 $\mu$m line noted by \citet{oneal01} shows a clear pattern of 3 lines in our data, with the central line exceeding the depth of the adjacent two lines, although the \PHOENIX\ models predict a non-negligible ambient photosphere contribution to the middle line for our range of ambient and spot temperatures. In comparison, the forward-modeling technique described in this work thrives in the presence of long sequences of indistinct yet predictably correlated spectral features. Similar forward-modelling strategies have successfully identified patterns of metal lines in the line-blanketed spectra of low metallicity stars \citep{kirby11,kirby15}. Figures \ref{fig:Hband3x7} and \ref{fig:Kband3x7} in the Appendix show 42 of the $H-$ and $K-$band spectra on a log scale with a single random composite model spectrum overplotted. +The IGRINS spectrum demonstrates some features that are present only in the ambient photosphere model, and some features that are present only in the starspot model. Figure \ref{fig:specPostageStamp} shows a selection of six such spectral features for a range of plausible fill factors. In some cases, featureless starspot spectra veil isolated spectral lines predicted in the ambient photosphere models. In other cases, the ambient photosphere model veils sequences of shallow spectral features predicted in the starspot model. The starspot spectral models predict shallow features because line blanketing from multiple indistinct molecular bands overlap from rotational broadening. Any feature of interest will be biased to non-zero veiling. This line blanketing has probably hampered efforts to identify isolated spectral features suitable for line-depth-ratio analysis in the spectra of rotationally broadened young stars. One such feature has been previously identified in the near-IR: the OH 1.563 $\mu$m line noted by \citet{oneal01} shows a clear pattern of 3 lines in our data, with the central line exceeding the depth of the adjacent two lines, although the \PHOENIX\ models predict a non-negligible ambient photosphere contribution to the middle line for our range of ambient and spot temperatures. In comparison, the forward-modeling technique described in this work thrives in the presence of long sequences of indistinct yet predictably correlated spectral features. The level of veiling of ambient photospheric lines is set by the starspot filling factor and temperature, while the level of veiling of the starspot lines is set by the filling factor and temperature of the ambient photosphere lines. Similar forward-modelling strategies have successfully identified patterns of weak metal lines in the line-blanketed spectra of low metallicity stars \citep{kirby11,kirby15}. Figures \ref{fig:Hband3x7} and \ref{fig:Kband3x7} in the Appendix show 42 of the $H-$ and $K-$band spectra on a log scale with a single random composite model spectrum overplotted. \begin{figure*} \centering @@ -270,7 +275,7 @@ \subsection{Two-temperature fitting to IGRINS spectra}\label{sec:two_tempIGRINS} \includegraphics[width=0.45\textwidth]{figures/spectral_postage_stamp_06} \includegraphics[width=0.45\textwidth]{figures/spectral_postage_stamp_02} \includegraphics[width=0.45\textwidth]{figures/spectral_postage_stamp_03} - \caption{Examples of spectral features in the observed IGRINS spectrum (light gray line). The composite spectrum model (purple thin line) is consistent with the observed spectrum for a range of fill factors, with examples of the the median fill factor (middle panel of triptych) and $\pm2\sigma$ fill factors demarcated on the spectral postage stamps. The upper right triptych shows a Zeeman-sensitive Mg I line that is modeled with no special attention to magnetic field, but is still coarsely reproduced in its gross appearance as a function of temperature, but potentially biasing estimates of $\teffa$ and/or $f_{\Omega}$ for individual spectral orders.} + \caption{Examples of spectral features in the observed IGRINS spectrum (light gray line). The composite spectrum model (purple thin line) is consistent with the observed spectrum for a range of fill factors, with examples of the median fill factor (middle panel of triptych) and $\pm2\sigma$ fill factors demarcated on the spectral postage stamps. The upper right triptych shows a Zeeman-sensitive Mg I line that is modeled with no special attention to magnetic field, but is still coarsely reproduced in its gross appearance as a function of temperature, but potentially biasing estimates of $\teffa$ and/or $f_{\Omega}$ for individual spectral orders.} \label{fig:specPostageStamp} \end{figure*} @@ -292,13 +297,13 @@ \section{The Two Temperature Fit to the SED and Stellar Rotation}\label{sec:GJHs \begin{figure*} \centering \includegraphics[trim=2.1cm 3.0cm 1cm 7.7cm, clip=true, width=0.70\textwidth]{figures/lores_panels_2750k} - \caption{Top: The low-resolution optical/near-IR spectrum of LkCa 4 obtained from Palomar/DBSP and APO/Triplespec on 30 December 2008 (black), compared to a synthetic spectrum of a two temperature photosphere (purple). The inset shows that the 2750 K (red, 70\% fill factor) and 4100 K (blue, 30\% fill factor) components contribute equally to the near-IR spectrum, but the 4100 K component dominates the blue emission. The synthetic spectrum is reddened by $A_V=0.4$ mag and scaled to the observed $J$-band spectrum. Bottom: The low-resolution optical (left) and near-IR (right) spectrum of LkCa 4, compared with a 3900 K photosphere (blue), a 3500 K photosphere (red), and the two temperature photosphere (pink) that best fits the IGRINS spectrum. The synthetic spectra are scaled separately to the optical spectrum at 0.75 $\mu$m and to the near-IR spectrum at 1.5 $\mu$m. Warm photospheres accurately reproduce molecular bands at $0.7$ $\mu$m but fail to fit the spectral features at longer wavelengths. Cooler photospheres predict molecular bands at $<0.7$ $\mu$m that are much deeper than observed. The two temperature photosphere accurately fits spectral features in the optical and near-IR.} + \caption{Top: The low-resolution optical/near-IR spectrum of LkCa 4 obtained from Palomar/DBSP and APO/Triplespec on 30 December 2008 (black), compared to a synthetic spectrum of a two temperature photosphere (purple). The inset shows that the 2750 K (red, 80\% fill factor) and 4100 K (blue, 20\% fill factor) components contribute equally to the near-IR spectrum, but the 4100 K component dominates the blue emission. The synthetic spectrum is reddened by $A_V=0.4$ mag and scaled to the observed $J$-band spectrum. Bottom: The low-resolution optical (left) and near-IR (right) spectrum of LkCa 4, compared with a 3900 K photosphere (blue), a 3500 K photosphere (red), and the two temperature photosphere (pink) that best fits the IGRINS spectrum. The synthetic spectra are scaled separately to the optical spectrum at 0.75 $\mu$m and to the near-IR spectrum at 1.5 $\mu$m. Warm photospheres accurately reproduce molecular bands at $0.7$ $\mu$m but fail to fit the spectral features at longer wavelengths. Cooler photospheres predict molecular bands at $<0.7$ $\mu$m that are much deeper than observed. The two temperature photosphere accurately fits spectral features in the optical and near-IR.} \label{fig:lores} \end{figure*} -In the previous section, we established that the high resolution optical and near-IR spectra of LkCa 4 may be explained by a two-temperature photosphere. In this section, we use testable predictions from this fit to demonstrate that the fit reasonably matches spectral features and their rotational modulation. We adopt the best-fit two temperature model to the IGRINS spectrum, with components of 2750 K covering 80\% of the visible stellar surface and 4100 K covering 20\% of the stellar surface. The IGRINS spectrum occurred when LkCa 4 had an estimated brightness of $V=12.84$ mag. These parameters are selected without any adjustments to attempt to improve fits to the SED or broadband spectra. +In the previous section, we established that the high resolution optical and near-IR spectra of LkCa 4 may be explained by a two-temperature photosphere. In this section, we use testable predictions from this fit to demonstrate that the fit reasonably matches spectral features and their rotational modulation. We adopt the best-fit two temperature model to the IGRINS spectrum, with components of 2750 K covering 80\% of the visible stellar surface and 4100 K covering 20\% of the stellar surface. The IGRINS spectrum was obtained when LkCa 4 had an estimated brightness of $V=12.84$ mag. The starspot and ambient temperatures and the filling factor are adopted without any adjustments to attempt to improve fits to the SED or broadband spectra. Figures~\ref{fig:sed}-\ref{fig:lores} compares synthetic spectra from the two-component photosphere to the SED and flux-calibrated spectra of LkCa 4. The only free parameters in this comparison are the luminosity and extinction, which are both scaled to match the spectrum (see \S 5.1-5.2). The optical-IR SED is obtained from estimating photometry from \citet{grankin08} during the 2MASS epoch, with $V\sim12.61$ mag. The SED also includes mid-IR photometry from Spitzer/IRAC \citep{hartmann05}, without adjusting for epoch. In the spectroscopic comparison, some minor discrepancies between the optical and near-IR spectra may be introduced because the spot coverage changed in the $\sim 5$ hrs between observations ($\Delta V=0.08$ mag). The synthetic spectrum is obtained from the Phoenix models, as in \S 3, and is extended beyond the longest wavelength (5 $\mu$m) for calculating the bolometric luminosity. @@ -335,7 +340,7 @@ \section{The Two Temperature Fit to the SED and Stellar Rotation}\label{sec:GJHs \subsection{The $V$-band lightcurve and spot coverage}\label{sec:rotSpot1} -The $V$-band brightness corresponds to the filling factor of the cool and hot components. In the two-temperature photosphere, the $V$-band emission is dominated by the hotter component and is therefore a good proxy for the visible surface area of the hot component. Figure~\ref{fig:vband_spot} shows the 2015--2016 ASAS-SN lightcurve, converted into a cool spot filling factor. In this epoch, the brightest and coolest phase corresponds to cool component fill factors of $74\%$ and $86\%$, respectively. Since 1992 the $V$ band photometry has been as bright as $12.3$ mag, which corresponds to a cool component fill factor of $65\%$, and as faint as $13.2$ mag, or a cool component fill factor of $87\%$. This drastic change in visible area of the hot spot (35\% to 13\%) is needed to explain the full $\sim 1$ mag range in brightness, assuming no temperature change in either component. +The $V$-band brightness reflects the instantaneous filling factor of the cool and hot components. In the two-temperature photosphere, the $V$-band emission is dominated by the hotter component and is therefore a good proxy for the visible surface area of the hot component. Figure~\ref{fig:vband_spot} shows the 2015--2016 ASAS-SN lightcurve, converted into a cool spot filling factor. During this period, the brightest and coolest phase corresponds to cool component fill factors of $74\%$ and $86\%$, respectively. Since 1992 the $V$ band photometry has been as bright as $12.3$ mag, which corresponds to a cool component fill factor of $65\%$, and as faint as $13.2$ mag, or a cool component fill factor of $87\%$. This drastic change in visible area of the hot spot (35\% to 13\%) is needed to explain the full $\sim 1$ mag range in brightness, assuming no temperature change in either component. \subsection{Rotational Modulation of Colors}\label{sec:rotSpot} @@ -346,7 +351,7 @@ \subsection{Rotational Modulation of Colors}\label{sec:rotSpot} The optical emission is dominated by the hotter component while both components contribute equally to the infrared emission (see inset in top panel of Figure \ref{fig:lores}). Figure~\ref{fig:colors} demonstrates that the $B-V$ color from \citet{grankin08} does not depend on $V$, which is consistent with expectations for a single hot component with no contributions from the cooler component. The standard deviation of 0.03 mag in $B-V$ color, which includes a $\sim 0.01$ mag uncertainty in both $V$ and $B$, indicates that the spot temperature is stable to $\lesssim 50$ K. The stability of this temperature implies that this hotter component may be the ambient temperature of the photosphere. -The correlation between $V$ and $V-R$ indicates that the star becomes redder when the cool spot has a higher filling factor. Our simple model predicts that the correlation should be much weaker than is measured. Most likely, our two temperature model for the photosphere is overly simplistic, since the $V-R$ could could be reproduced with some contribution from intermediate temperature. +The correlation between $V$ and $V-R$ indicates that the star becomes redder when the cool spot has a higher filling factor. Our simple model predicts that the correlation should be much weaker than is measured. Most likely, our two temperature model for the photosphere is overly simplistic, since the $V-R$ could be reproduced with some contribution from intermediate temperature. Rotational modulation is expected to lead to much smaller brightness changes at near-IR wavelengths than at optical wavelengths (see Table~\ref{tab:photrange}). The smaller amplitude of near-IR brightness is also seen in a few spotted stars in the optical/IR monitoring of NGC 2264 \citep{cody14}, although those stars have a much smaller $V$ band amplitude than LkCa 4. @@ -447,11 +452,11 @@ \subsection{Spot-corrected HR diagram placement} The discrepancies in temperature are even larger when spectral type/temperature estimates focus on features in either blue-optical or $H$ or $K$-band spectra, and may explain the severe differences in effective temperature between APOGEE and other (optical) methods \citep{cottaar14}. On the other hand, GAIA-ESO spectra \citep{frasca15} cover only a narrow wavelength around H$\alpha$ and would yield an effective temperature of 4000 K. Spectral types and effective temperatures measured from TiO depths \citep{herczeg14} are lower than those from blue spectra but still overestimate the effective temperature. Temperature uncertainties for young K and M dwarfs are larger than any formal uncertainties in individual measurements. -The improved characterization of the photospheric emission and radius of LkCa 4 is supposed to lead to more accurate estimates of mass and age from pre-main sequence tracks. However, our effective temperature and luminosity yield a mass of 0.33 $M_\odot$ and an age of $\sim 0.5$ Myr in the \citet{baraffe15} evolutionary models\footnote{The \citet{baraffe15} evolutionary tracks were calculated using BT-Settl photospheric spectra \citep{allard14}, which may introduce some uncertainties when compared to results from the PHOENIX models used here.}. This age is uncomfortably young and this mass is uncomfortably low. LkCa 4 is not located deeply within a molecular cloud and is not associated with any nearby Class 0/1 stars, which would be expected for a 0.5 Myr star. Dynamical measurements of masses of young stars with similar optical properties suggest that LkCa 4 should have a mass of $\sim 0.6-1.0$ M$_\odot$ \citep[e.g.][]{guilloteau14,czekala16,rizzuto16}. Even though LkCa 4 is only a single data point, the inferred age and low mass may be evidence that strong magnetic fields are inhibiting convection in LkCa 4. As a result, the surface is cooler and releases less radiation, slowing the contraction rate. LkCa 4 then appears more luminous and cooler than expected from evolutionary tracks that do not consider magnetic fields. This shift to lower temperature may also help to resolve some of the age discrepancies between intermediate mass and low mass young stars \citep[e.g.][]{herczeg15}. The corrected placement of spotted stars on the HR diagram reflects the accurate luminosity, radius, and effective temperature, but evolutionary models are not yet equipped to interpret such heavily spotted stars \citep{somers15}. +The improved characterization of the photospheric emission and radius of LkCa 4 would ideally lead to more accurate estimates of mass and age from pre-main sequence tracks. However, our effective temperature and luminosity yield a mass of 0.33 $M_\odot$ and an age of $\sim 0.5$ Myr in the \citet{baraffe15} evolutionary models\footnote{The \citet{baraffe15} evolutionary tracks were calculated using BT-Settl photospheric spectra \citep{allard14}, which may introduce some uncertainties when compared to results from the PHOENIX models used here.}. This age is uncomfortably young and this mass is uncomfortably low. LkCa 4 is not located deeply within a molecular cloud and is not associated with any nearby Class 0/1 stars, which would be expected for a 0.5 Myr star. Dynamical measurements of masses of young stars with similar optical properties suggest that LkCa 4 should have a mass of $\sim 0.6-1.0$ M$_\odot$ \citep[e.g.][]{guilloteau14,czekala16,rizzuto16}. Even though LkCa 4 is only a single data point, the inferred age and low mass may be evidence that strong magnetic fields are inhibiting convection in LkCa 4. As a result, the surface is cooler and releases less radiation, slowing the contraction rate. LkCa 4 then appears more luminous and cooler than expected for a star of its genuine age from evolutionary tracks that do not consider magnetic fields. This shift to lower temperature may also help to resolve some of the age discrepancies between intermediate mass and low mass young stars \citep[e.g.][]{herczeg15}. The corrected placement of spotted stars on the HR diagram reflects the accurate luminosity, radius, and effective temperature, but evolutionary models are not yet equipped to interpret such heavily spotted stars \citep{somers15}. The IGRINS and ESPaDOnS spectra indicate indicate $\vsini\sim$ 28 km~s$^{-1}$ and a rotation period of 3.375 days. Combining these numbers gives $R\sin{i} \sim 1.87 R_{\odot}$. Our HR diagram analysis gives $R \sim 2.3R_{\odot}$. These numbers are consistent for $\sin{i} \sim 0.813$, or an inclination of about 35$^{\circ}$ from edge on. These values show broad consistency between rotational properties, spectral fitting values, and our interpretation of a tilted star revealing a circumpolar region with large areal coverage of starspots. -As seen in Figures \ref{fig:PhotTime} and \ref{fig:PhotPhase}, the areal coverage of spots appears to wax and wane secularly through the observing seasons of the last 31 years. It is conceivable that LkCa 4 happens to be going through a short-lived episode of relatively high coverage of spots analogous to solar maximum. The intensity and duration of such episodes would further confound the interpretation of its instantaneous position on the HR diagram. +As seen in Figures \ref{fig:PhotTime} and \ref{fig:PhotPhase}, the areal coverage of spots appears to wax and wane secularly through the observing seasons of the last 31 years. Notably, it appears to be in a relatively faint phase currently, 0.2 mag fainter in $V$ than it was through much of the late 1980's and early 1990's. It is conceivable that LkCa 4 happens to be going through a short-lived episode of relatively high coverage of spots analogous to solar maximum. The intensity and duration of such episodes would further confound the interpretation of its instantaneous position on the HR diagram. \section{Discussion} @@ -527,26 +532,6 @@ \section{Conclusions} \appendix -\section{Examples of full spectrum fitting} - -Figures \ref{fig:Hband3x7} and \ref{fig:Kband3x7} show spectra from 42 IGRINS orders. - - -\begin{figure*} - \centering - \includegraphics[width=0.95\textwidth]{figures/H_band_spectra_3x7} - \caption{IGRINS Orders 94 and $99-119$. Note that the $y-$axis is on a logarithmic scale. The red line is the cool photosphere while the blue line is the hot photosphere. The purple line is the composite mixture model.} - \label{fig:Hband3x7} -\end{figure*} - -\begin{figure*} - \centering - \includegraphics[width=0.95\textwidth]{figures/K_band_spectra_3x7} - \caption{IGRINS Orders $73-93$. Note that the $y-$axis is on a logarithmic scale. Cools are the same as Figure \ref{fig:Hband3x7}} - \label{fig:Kband3x7} -\end{figure*} - - \section{Methodological details} \label{methods-details} @@ -583,6 +568,28 @@ \subsection{Addressing absolute flux calibration} %\todo{Verify that the final term possesses a $q$ and not a $q^2$ term.} + +\section{Examples of full spectrum fitting} + +Figures \ref{fig:Hband3x7} and \ref{fig:Kband3x7} show spectra from 42 IGRINS orders. + + +\begin{figure*} + \centering + \includegraphics[width=0.95\textwidth]{figures/H_band_spectra_3x7} + \caption{IGRINS Orders 94 and $99-119$. Note that the $y-$axis is on a logarithmic scale. The red line is the cool photosphere while the blue line is the hot photosphere. The purple line is the composite mixture model.} + \label{fig:Hband3x7} +\end{figure*} + +\begin{figure*} + \centering + \includegraphics[width=0.95\textwidth]{figures/K_band_spectra_3x7} + \caption{IGRINS Orders $73-93$. Note that the $y-$axis is on a logarithmic scale. Cools are the same as Figure \ref{fig:Hband3x7}} + \label{fig:Kband3x7} +\end{figure*} + + + \section{Full table of IGRINS best fits} Table \ref{tbl_order_results} lists all the IGRINS spectral order results.