Tests of Disk-locking in T Tauri Stars of the Orion Nebula Cluster

We look for specific correlations predicted by magnetospheric accretion models for young stars that assume disk-locking using stellar and accretion parameters derived from low-resolution HST-STIS spectra of 33 T Tauri stars in the Orion Nebula Cluster. Our results provide strong support for the correlation predicted by a model that does not constrain the stellar magnetic field to a specific geometry, while little support is found for the correlation that assumes a dipolar stellar field geometry. These results support the conclusions from similar studies of older T Tauri stars in Taurus and NGC 2264 and underscore the role that trapped flux plays in regulating a young star’s angular momentum as detailed by Ostriker & Shu. While our targets were all selected to be accreting young stars based on photometric indices, approximately half of the observed stars show no significant signs of accretion in our spectra, illustrating the difficulty in using photometric indices to find accreting stars and the possible role that variability has in the appearance of spectra in young stars. Although values of accretion luminosity derived from our models agree well with those derived from Hβ luminosity for strongly accreting stars, we find that accretion luminosity derived from Hβ luminosity is not a reliable parameter for discriminating between weakly accreting and nonaccreting T Tauri stars due to chromospheric emission that is likely present in all T Tauri stars.


Introduction
T Tauri stars (TTSs; Joy 1945) represent a pre-mainsequence stage in stellar evolution where once deeply embedded protostars have shed their natal molecular cloud cores to become optically visible (Shu & Adams 1987).These young (age 10 Myr), low-mass (2.5 M e ) stars are commonly referred to as classical T Tauri stars (CTTSs) if they are actively accreting from a circumstellar disk, or weak emission-line T Tauri stars (WTTSs) if they show little to no accretion (Bertout 1989).
One of the most intriguing properties of CTTSs is their surprisingly long rotation periods.As CTTSs evolve toward the main sequence, gravitational contraction and accretion from their Keplerian circumstellar disks should cause the stars to spin-up to break-up velocity (∼300 km s −1 ) in order to conserve angular momentum (Rebull et al. 2002).However, the observed rotation rates are typically an order of magnitude or more below break-up velocity (Mestel 1965;Bodenheimer 1995).Understanding the mechanisms that regulate the angular momentum of a CTTS remains one of the greatest challenges related to understanding the early stages of stellar evolution (Bouvier 2014;Gehrig & Vorobyov 2023).
Magnetospheric accretion-the funneling of accreting material from a star's disk to its surface by the stellar magnetic field lines-accounts for many of the observed characteristics of CTTSs and is thought to play a role in the regulation of angular momentum.First described by Ghosh & Lamb (1979) for magnetized neutron stars, the magnetospheric accretion model was extended to CTTSs by Camenzind (1990) and Königl (1991).In short, the model states that the stellar magnetosphere truncates the star's disk at a few stellar radii and channels partially ionized disk material to the stellar surface.Shocks formed when the accreting material impacts the photosphere at nearly freefall velocities produce excess UV and optical continuum emission, which contributes to veiling.The high temperature and velocity of the infalling gas generate the broadened emission line profiles and high-velocity redshifted absorption lines of CTTSs (Hartmann et al. 1994(Hartmann et al. , 2016)).Large IR continuum excesses are produced by hot dust in the inner disk reradiating optical radiation absorbed from the star (Eugenio & Mendoza 1968;Cohen & Kuhi 1979;Adams et al. 1987;Kenyon & Hartmann 1987;Muzerolle et al. 2003) and from viscous heating due to accretion processes in the disk (Meyer et al. 1997).The lack of an actively accreting disk around WTTSs explains their more typical stellar and chromospheric emissions and suggests that they have evolved beyond the CTTS stage (Bertout et al. 1988;Williams & Cieza 2011;Alexander et al. 2014).Shu et al. (1994) proposed a variant of magnetic star-disk coupling known as the "X-wind" after recognizing that the field configuration proposed by Ghosh & Lamb (1979) would eventually destroy the magnetic field beyond corotation due to differential twisting of the field lines.In a detailed analysis of the Shu et al. (1994) X-wind model, Ostriker & Shu (1995, hereafter OS95) introduced the concept of trapped magnetic flux produced by the pinching of magnetic field lines toward the corotation radius, R co .In this scenario, one-third of the trapped magnetic field lines produce an ionized disk wind, the X-wind, just outside R co that ejects material with excess angular momentum away from the disk.Another one-third of the field constitutes dead-zone fields.The final third of the trapped magnetic flux funnels accreting material along field lines that bend inward to the star, which creates torques that transport angular momentum back to the disk.In this way, the trapped flux regulates the star's angular momentum and influences the size and geometry of the accretion hot spots on the stellar surface (Shu et al. 1994;Mohanty & Shu 2008).
Although magnetospheric accretion is widely accepted to explain the transfer of disk material to the star, disk-locking as a regulator of angular momentum remains an open question.Photometric and spectroscopic observations that reveal a bimodality of rotation periods within a given cluster of TTSs offer strong support for disk-locking.T Tauri stars with disk indicators, such as infrared excesses, generally have longer rotation periods and narrower period distributions than stars without disk signatures (e.g., Attridge & Herbst 1992;Edwards et al. 1993;Cieza & Baliber 2007;Orcajo et al. 2019).In the Orion Nebula Cluster (ONC), Cieza & Baliber (2007) found that in a sample restricted to stars with masses 0.25 M ☉ , stars with disks dominate the peak of the bimodal distribution at rotation periods of ∼8 days, while most of the stars with rotation periods of ∼2 days are diskless.
Although the results of studies of disk rotation period generally indicate that the circumstellar disk plays a role in the regulation of a star's rotation period, they do not offer conclusive evidence of the disk-locking mechanism.Theoretical studies (Matt & Pudritz 2005, 2008;Gallet et al. 2019) suggest that magnetospheric ejections are more effective at spinning down a star than magnetic coupling (disk-locking) between the star and disk.Clearly, the disk-locking theory remains unresolved.
Noting that disk-locking relies on more than just the the presence of a circumstellar disk, Johns-Krull & Gafford (2002, hereafter JG02) used stellar and accretion parameters for TTSs in Taurus (mean age ∼2 Myr; Küçük & Akkaya 2010) to look for correlations predicted by magnetospheric accretion models that invoke disk-locking (see Section 3).They found poor support for disk-locking models that require a dipole field geometry, while they reported statistically significant correlations for the relationship predicted by the OS95 model after modifying it to allow for a nondipole field topology at the stellar surface.Cauley et al. (2012, hereafter C12) analyzed the same models as JG02 using accretion and stellar parameters derived from model fits to low-resolution spectra of 36 TTSs (25 CTTSs) in NGC 2264 (age ∼2.5-5 Myr; Sung et al. 2004;Dahm et al. 2007;Venuti et al. 2018).Like JG02, they found strong support for the modified OS95 disk-locking model, which does not require a specific magnetic field geometry at the stellar surface, but not for the models that assume a dipole field geometry.Furthermore, C12 note that since both analyses support the modified OS95 picture, they also indirectly support the idea that magnetically powered X-winds also remove angular momentum from the star-disk system.Venuti et al. (2017) also used stellar and accretion parameters from CTTSs in NGC 2264 to test the modified OS95 model and the model predicted by Königl (1991) and Shu et al. (1994).Like JG02 and C12, Venuti et al. (2017) did not find a significant correlation in the disk-locking model that assumes a stellar dipole field geometry.Unlike JG02 and C12, they did not find a significant correlation in the model that relaxes the requirement for a dipole field.Differences in the estimates for accretion luminosity (and hence mass accretion rate) between the studies may explain the different outcomes between the C12 and Venuti et al. (2017) studies.The results of Venuti et al. (2017) are based almost entirely on photometry, including for visual extinction, A v , and for accretion rate through a measure of u-band excess.The u-band emission does not provide information on the shape of the excess in the UV or optical spectra, which is important for accurate estimates of accretion rate.In addition, their assumption that accretion spots and the unspotted stellar surface emit as blackbodies (Venuti et al. 2014) is inaccurate and may have contributed to the conflicting results (Bouvier et al. 1993;Pecaut & Mamajek 2013;Somers et al. 2020).
In this paper, the work of JG02 and C12 is extended to a younger group of TTSs in the ONC (mean age ∼1 Myr; Hillenbrand 1997, hereafter H97;Tobin et al. 2009).We apply an analysis similar to C12 to derive stellar and accretion parameters from optical spectra of 33 TTSs in the ONC, and we test the same models of disk-locking as JG02 and C12 in some of the youngest observable TTSs.Using data from clusters of various ages may allow us to better understand the evolution of angular momentum as TTSs evolve to the main sequence.The ONC provides an excellent region for this study since it is one of the densest (Hillenbrand et al. 2013), best-studied (Muench et al. 2008) star-forming regions in close proximity (∼414 ± 7 pc; Menten et al. 2007) to the Earth.
A drawback of the ONC as a study area is that its hot (∼10,000 K) H II region produces background emissions at UV, visible, and IR wavelengths, making it somewhat challenging to separate the stellar and nebular emission.Here we take advantage of the narrow slit width (0 2) and tight point-spread function of the Hubble Space Telescope (HST)'s Space Telescope Imaging Spectrograph (STIS), which makes it possible to eliminate the nebular emission very close to the star.
We discuss measures taken to minimize the contaminating effects of the H II emission as part of the sample selection and observation details in Section 2. In Section 3, we introduce the two models of magnetospheric accretion for CTTSs that are tested in this study.An explanation of the modeling procedure, derivation of stellar and accretion parameters, and choice of photospheric templates are described in Section 4. Results from the tests of the magnetospheric accretion models and comparisons of our results with the results from other studies are discussed in Section 5. A summary of our findings and their implications are discussed in Section 6.

Sample Selection
We use blue optical spectra to measure stellar and accretion properties of our target stars.An initial group of ∼400 premain-sequence stars in the ONC was selected based on the availability of rotation periods, stellar masses, and radii.From these candidates, we attempted to identify actively accreting stars that reside in areas of low nebular emission.
The strong hydrogen emission from the H II region and spatial variability of the nebulosity of the ONC can reduce the quality of the extracted spectra, especially at blue optical wavelengths (H97).In order to minimize these contaminating effects, we used flux-calibrated images obtained at the Wisconsin-Indiana-Yale-NOAO (WIYN) 0.9 m telescope to measure the stellar-to-nebular contrast ratio in the immediate vicinity of the ∼400 candidates.Only those stars whose local nebular brightness in 1 arcsec 2 is at least 1 mag fainter than the stellar brightness were retained.This metric ensures that the nebular flux is at least ∼20 times below that of the stellar continuum in a spectrum observed with the 0 2 slit of the HST-STIS.
We relied on near-infrared (NIR) excesses to identify stars with circumstellar disks.To improve the likelihood that the targeted stars are actively accreting, only stars with at least a 0.35 mag excess I C − K color as determined by Hillenbrand et al. (1998) were selected.In the end, this criterion proved insufficient for distinguishing between accreting and negligibly accreting stars in our sample: an analysis of the acquired spectra revealed that approximately half of the stars selected for the study lacked the excess UV and optical emission that characterize accreting stars (see Tables 3 and 4).
Finally, to take advantage of the brief gaps in the HST observation schedule that can be used in a Snapshot (SNAP) Program, the targeted stars had to be bright enough to observe in less than one half-orbit of the HST (<48 minutes).A total of 93 stars met the selection criteria and were included as targets in a SNAP program executed on HST (GO-12995; PI: Johns-Krull).Of these 93 stars, 35 were eventually observed by the HST.The spectra of JW73 and JW853 were eliminated from the sample due to their poor quality.In the end, the spectra of 33 TTSs in the ONC are used in this study.An additional nine spectra from stars outside the ONC are used as templates.The targeted ONC stars are listed in Table 1 by their Jones & Walker (1988) designations.The templates are listed in Table 2.

Observations
Spectra of the ONC stars used in this study were recorded beginning in 2012 October and ending in 2014 March.Observations were made with the G430L grating using the 52″ × 0 2 slit.The exposure times and observation dates of each star are listed in Table 1.
Following the generation of 2D calibrated spectra of the observed stars by the HST data reduction pipeline (calstis), we extract 1D spectra using custom IDL routines.To derive accurate mass accretion rates from the excess optical emission of each star, it is imperative that the extraction procedure minimizes the effects of other sources of emission, particularly the nebular emission around each star.Using a custom IDL extraction program, we narrow the extracted region to each spectrum's unique width on the detector in order to exclude as much of the background emission as possible without loss of the stellar signal.We then select a tight offset value for each spectrum to extract and subtract flux from background sources such as line radiation from the H II region.After extracting the spectra, we interactively delete bad pixels and cosmic-ray hits.
The majority of TTSs in this study appear as single stars.However, six of the 33 observed stars are reported to have at least one companion.Only the binary star JW867 is close enough (0 29; Reipurth et al. 2007) for the two stars to show a slight overlap of their spectra.Nevertheless, the two stars were well enough separated on the detector so that the primary star's spectrum was extracted such that the vast majority of the flux from the secondary star was excluded.We were also able to extract the portion of the secondary star's spectrum that falls within the slit.The strong Balmer lines and Balmer jump in the extracted spectrum show the secondary to be a strongly accreting star (see Figure 14).

Magnetospheric Accretion Models
Johns-Krull & Gafford (2002) derived specific relationships among stellar and accretion parameters from models of magnetospheric accretion by Königl (1991), Collier Cameron & Campbell (1993), 3 Shu et al. (1994), and a more detailed analysis of the Shu et al. (1994) model by OS95.These relations depend on stellar mass (M å ), stellar radius (R å ), period of rotation (P rot ), strength of the magnetic field (B å ), mass accretion rate (M acc  ), and accretion filling factor ( f acc ).We look for correlations based on these relationships using values of M å , R å , f acc , and M acc  derived from fits to the observed spectra of CTTSs as described in Section 4. We rely on P rot from studies by Herbst et al. (2000Herbst et al. ( , 2002) ) and Stassun et al. (1999).
We attempted to extract TESS light curves to update these rotation periods; however, the field surrounding the observed Orion stars proved to be too crowded for the large TESS pixel scale (21″ pixel −1 ).In addition to the substantial blending due to crowding, strong nebular emission contaminates the field.
Since magnetic field measurements are unavailable for most of our CTTSs, we follow JG02 and C12 and assume that the mean magnetic field strength remains relatively constant from star to star, an assumption supported by Guenther et al. (1999), Johns-Krull et al. (1999, 2001), Johns-Krull (2007), and Yang & Johns-Krull (2011).
Assuming an aligned stellar dipole field geometry, a consistent magnetic field strength from star to star, and a magnetized accretion flow that locks the star to its disk, the correlation between M å , R å , P rot , and M acc  from models by Königl (1991) and Shu et al. (1994) is  Like most magnetospheric accretion models, the above correlation depends on an assumed stellar dipole field geometry.However, observed surface magnetic field topologies of CTTSs are often more complex (Gregory et al. 2006;Johns-Krull 2007;Donati et al. 2008;Yang & Johns-Krull 2011).To mitigate this discrepancy between theory and observation, JG02 add a term that represents the fraction of the stellar surface participating in the accretion flows, f acc , to the correlation predicted from the OS95 model.The resulting correlation eliminates the dependence of the model on a specific field geometry at the star's surface (Mohanty & Shu 2008).
Assuming disk-locking and a constant magnetic field strength for the stars, the relationship derived by JG02 from the OS95 model becomes  where f acc is defined by where A acc is the surface area of the star covered by accreting material.It is the correlations in Equations ( 1) and (2) that we seek to test.

Spectral Analysis
To derive the parameters needed to test Equations ( 1) and (2), the observed spectrum of each Orion CTTS is fitted with the spectrum of a WTTS of similar spectral type (the photospheric template), to which is added a model of the spectrum produced by the accreting material (Valenti et al. 1993, hereafter VBJ) as described in Section 4.1.The templates are dereddened using extinctions from Herczeg & Hillenbrand (2014).An example fit is shown in Figure 1.This figure shows the observed spectrum of Orion M1.2 CTTS JW123 (solid black line) overplotted with the fitted model (solid red line).The dashed blue line is the spectrum of the flux emitted by accretion processes.The scaled photospheric template is shown as a solid green line.
Table 2 lists the photospheric templates used in this study along with their distances derived from parallaxes given by Gaia Early Data Release 3 (Gaia EDR3; Gaia Collaboration et al. 2016Collaboration et al. , 2021) ) and radii from Herczeg & Hillenbrand (2014, hereafter HH14), which are scaled to Gaia EDR3 distances.The photospheric templates used in this study are shown in Figure 14, while all 33 spectra of the observed Orion TTSs overplotted with their fitted models are found in Figures 15 and  16.Stellar and accretion parameters derived from the models are then used to test the relationships predicted by the magnetospheric accretion models (Section 5).

Modeling the Observed Spectra
To model emission from accretion processes, we follow VBJ (see also, e.g., C12, Manara et al. 2021) and treat the emission region as a slab of hot, isothermal hydrogen in local thermodynamic equilibrium.The fraction of the stellar surface covered by the slab is expressed as the filling factor, f acc .Although accretion processes contribute to both continuum and line emission, like VBJ we only include the Hδ line in fitting the slab's model.The strength of the Hβ line is often affected by winds in CTTSs (Alencar et al. 2005;Donati et al. 2008), and lines blueward of Hδ are not well resolved by the lowresolution spectra.Emission lines not related to hydrogen emission are omitted from the fitting procedure since our slab model only includes hydrogen.
The modeling procedure begins with an initial set of nine parameters.The distance to the star and turbulent velocity are held fixed.Distances were calculated using parallax measurements from Gaia EDR3 (Gaia Collaboration et al. 2016, 2021).The one star (JW867) that did not have a parallax measurement reported was assigned a distance of 414 pc (Menten et al. 2007).
As described in VBJ, the model for the accretion emission includes a turbulent velocity parameter, V T .Although V T does not reflect an intrinsic property of the accretion emission, it serves to match the Balmer line widths of the model with those of the observed spectrum, which are subject to substantial instrumental broadening in these low-resolution spectra.A value of V T = 120 km s −1 provides a good fit for all of the observed stars in this study and is used for all model fits.
The remainder of the parameters-temperature, number density of hydrogen atoms, depth of the emission region, filling factor of the emission region on the stellar surface, stellar radius, visual extinction, and spectral type (T, n, l, f acc , R å , A v , and SpT)-are varied in order to optimize the fit of the model to the observed spectrum.At the start of the fitting routine, the free parameters are set as follows: T = 9000 K, n = 10 14 cm -3 , l = 10 7 cm, f acc = 0.01, and R å = 1.5 R ☉ , while the spectral type and A v for each star are customized for each source.However, if a star's spectrum shows no signs of accretion and the addition of a slab is not necessary for the photospheric template to provide the best fit, we perform a new fit to minimize the impact of the accretion slab but still fit for the stellar radius and extinction.This fit to the model is made by fixing f acc to 1 × 10 −6 and allowing only  R and A v to vary for the model fit.In all cases, a nonlinear least-squares curve-fitting algorithm based on the Marquardt method (Bevington & Robinson 1992) varies the stellar and accretion parameters to find the suite of parameters that provides the best-fit model to the observed spectrum.For the remainder of this paper, we use CTTSs to refer to the stars for which we fit an accretion slab and measure an accretion rate, and we call the stars for which we do not require an accretion slab negligible accretors (NAs).These stars may still be accreting at a low level but we are unable to reliably detect this accretion.
To identify the optimal SpT of each observed TTS, we fit four to five templates of different SpTs to each spectrum.The SpT of the template that produces the best-fit model (i.e., lowest reduced χ 2 ) is adopted for that star.In most cases, the best-fit SpT is within two subtypes of the one reported in Hillenbrand et al. (2013, see Table 1).
A good estimate of A v toward each star is important in order to appropriately deredden the flux measured in the accretion continuum before converting it to an accretion luminosity.An overestimation of an A v value will result in an overestimation of M acc  .We initially fix A v for each star to those values derived using optical photometry by H97 and fit for the other parameters.After this initial fit, we allow A v , along with the other parameters, to vary.In all cases, A v parameters from the models fit to the observed spectra provide better fits than those using values from H97.The relationship between the A v values given by H97 and those derived from our model fits is shown in Figure 2. H97 relies on the longer wavelengths of V − I C colors to minimize the effects of veiling.While V magnitudes are not strongly affected by veiling, our analysis indicates that there is some effect.Using the ratio of fluxes at 5500 Å, we find the contribution from accretion luminosity to be as high as ∼40% of the total luminosity.By not accounting for the accretion continuum, the A v values from H97 may lead to misestimated accretion luminosities and mass accretion rates.Additionally, the A v values in this study were derived from models fit to HST-STIS spectra, which have a higher spatial resolution than the relatively coarse resolution of the ground-based photometry used by H97, which they note may contribute to less reliable extinctions, especially in areas of high nebulosity.Therefore, we choose to rely on our estimates of A v , which are fitted simultaneously with the accretion emission.

Deriving M acc  from the Shock Region
Equations (1) and (2) require accurate estimates of the emission released from the accretion shock region in order to calculate M acc  .A key aspect of the modeling is the template spectra used to represent the star.We rely on WTTSs from HH14 to model our CTTS spectra since the surface gravity and emission from the magnetically active WTTS chromospheres provide a good match to the overall spectra of CTTSs (Houdebine et al. 1996;Ingleby et al. 2011).While the optical spectra of our nine templates do not show signs of accretion, Ingleby et al. (2009) found that chromospheric emission in TTSs can mask accretion emission in weakly accreting stars.The morphology of the He I λ10830 line offers a more dependable accretion indicator for stars that have low levels of accretion since it is highly sensitive to stellar and disk winds (Edwards et al. 2006;Ingleby et al. 2011).A literature search of the stars in this study revealed only one star, TWA 3B, with a spectrum showing the He I λ10830 line.The spectrum shown in Edwards et al. (2006) does not have the blueshifted or redshifted absorption below its continuum that often characterizes CTTSs.In addition, spectral energy distributions (SEDs) constructed by Andrews et al. (2010) and Kellogg et al. (2017) show no evidence of a disk surrounding TWA 3B.Similarly, Hill et al. (2019) found TWA 6 to be diskless since there is no IR excess in its SED.
The remaining photospheric templates are also diskless based on SEDs constructed from Spitzer (Luhman et al. 2010).While we cannot be 100% certain that the photospheric templates in Table 2 are not weakly accreting, the fact that our templates are diskless is a strong indication that these stars are not accreting.Once we have fit the observed spectrum, the total accretion flux at the stellar surface, F acc , is calculated by integrating the model slab emission over 100 Å λ 40000 Å.Using R å and f acc from model fits to the observed spectra and F acc from the slab's emission, the accretion luminosity is calculated as In order to calculate the stellar luminosity, L å , of each star, we use the model-derived R å with the effective temperature from the spectral type-to-effective temperature conversions determined by Herczeg & Hillenbrand (2014, see their Table 5).We interpolate to find the effective temperatures for those spectral types that fall between spectral types listed in the table.The stellar luminosity is then calculated using the following equation: where σ is the Stefan-Boltzmann constant and T eff is the effective temperature.M å of each star is estimated using our T eff and L å values with the evolutionary tracks of Baraffe et al. (2015).Finally, M acc  of each CTTS is calculated using where G is the universal gravitational constant, and the truncation radius, R T , is the distance from the star to where the stellar magnetic field lines truncate the accretion disk.Following Calvet & Gullbring (1998), we assume R T = 5R å .
Assuming that the truncation radius is the corotation radius provides an alternative method of calculating a value of R T unique to each star: We calculate two sets of M acc  for the 15 Orion CTTSs, one set from each of the two equations for R T .Figure 3 illustrates the good agreement between the M acc  values derived from each R T equation.Next, we test the relationships described by Equations (1) and (2).The resulting best-fit lines to the relationships are essentially the same for both choices of R T .Therefore, for the rest of the paper, we assume R T = 5R å .

Results
Tables 3 and 4 present the stellar and accretion parameters derived from the best-fit models for the accreting and negligibly accreting Orion TTSs, respectively.Parameters related to accretion ( f acc , L acc , and M acc  ) are not shown for the stars in Table 4 since the photospheric template of the appropriate spectral type is all that is necessary to match their spectra well.The spectra of negligible accretors and CTTSs overplotted with their fitted models are shown in Figures 15 and 16 in the Appendix.

Tests of Magnetospheric Accretion Models
Using the CTTS stellar and accretion values listed in Table 3, we test correlations described by Equation (1), which assume disk-locking and a dipole stellar magnetic field.We look for signs of a correlation between the left and right sides of the equation by plotting * M M P Using the logged values of the x-and y-axis variables, we find that the Pearson's correlation coefficient (r = 0.22) is low, and the associated false-alarm probability is high (P f = 0.438), all indicating no true correlation exists.
Similarly, we create a log-log graph of the proportionality derived from the magnetospheric accretion model described by Equation (2), which also assumes disk-locking but allows for a more complex stellar field topology.To identify the best-fit line to the correlation between the logarithmic values of  (y-axis values), we use a linear least-squares fitting procedure that derives the best straight-line fit when uncertainties exist in both the x-and yaxes (Press & Teukolsky 1992).It is difficult to determine true uncertainties for these quantities so we rely on the scatter of the data as a measure of that uncertainty.Initially, the standard deviations of the logarithmic values of  serve as the uncertainties for the x-and y-data.Once new x-and y-values for each star are predicted from the initial fit, the standard deviations of the residuals from the actual and predicted x-and y-values serve as the uncertainties to derive an improved fit.We continue to optimize the fits to the x-and y-values until the fit converges.The slope and intercept from the final fit define the line of best fit for the correlation as shown in Figure 4(b).The dashed line represents the best-fit line for Equation (2) with a slope of 1.0 as predicted by the equation.For comparison, we recomputed accretion parameters using masses derived from evolutionary tracks of Feiden (2016) and found negligible difference in the final correlations.
When comparing the scatter plots in Figures 4(a) and (b), it is apparent that the relationship in Figure 4(b)-the model that allows for nondipole field geometries-shows a much stronger correlation.The Pearson's correlation coefficient (r = 0.52) for the nondipole model provides quantitative support for this interpretation.On the other hand, the false-alarm probability for the nondipole model is >0.01 (P f = 0.046), which weakens our confidence in the correlation.We find that the low Pearson's correlation coefficient and high false-alarm probability are driven primarily by one star, JW1000, which has a particularly noisy spectrum at wavelengths <3800 Å (see Figure 16).If JW1000 is removed from the correlation calculation, the strength of the correlation improves (r = 0.67) and the falsealarm probability (P f = 0.009) suggests that this is a significant correlation.

Combining Data from Previous Studies
The above correlation analysis is repeated after combining data from JG02, C12, and this study.We select the 17 CTTSs from the VBJ data set in JG02 (see their Table 1) for our calculations since it is the only data set in their study derived from detailed fits to spectra similar to that done here.From C12, we use the 25 CTTSs reported in their Tables 5  and 6 for strongly and weakly accreting stars.Figures 5(a 5. Again, a strong correlation is seen for Equation (2) while no correlation is seen for Equation (1).
The slope, 0.71 ± 0.10, of the best-fit line to the relationship described by the nondipole model (Equation (2)) is somewhat less than the predicted value of 1.0.The difference in the slope is marginally significant at the 2.9σ level; therefore, we investigated potential implications of this.A possible explanation for this discrepancy is that the fundamental assumption of our test-that magnetic field strengths of TTSs do not vary significantly from star to star-is not correct.If a star's magnetic field, B * , varies with a stellar parameter such as mass or radius, this would imply that magnetic fields are not constant between stars.
Here, we look for a relationship between B * and M * in order to explain the slope we observe in Figure 5(b).To begin, we reintroduce B * to Equation (2) in its appropriate place (JG02), which gives


After systematically varying β in Equation (11) and testing the correlations that result, we find that when β = 1.6, the best-fit line to the proportionality described by the equation has the expected slope of 1.0 ± 0.12.This would imply that B * ∝ * -M 1.1 and would explain why the correlation for Equation (2) is not the expected value of 1.0.Figure 6 shows the correlation given by Equation (11) when β = 1.6.The correlation statistics, r = 0.84 and P f = 6.49× 10 −16 , are slightly better than the original statistics derived using Equation (2) (see Figure 5(b)).While this is quite promising, it implies a specific relationship between B * and M * that we can look for.
In principle, B * ∝ * -M 1.1 can produce a correlation between the explored accretion parameters with a more consistent slope than given by Equation (2); however, Johns-Krull (2007) did not find any significant correlations between mean magnetic field measurements of 15 CTTSs in Taurus and seven stellar parameters (M * , - P rot 1 , L * , T eff , age, convective turnover time (τ c ), and t - P c rot 1 ).Today, magnetic field measurements are available for many more stars.Here, we use stellar parameters for 38 TTSs from Sokal et al. (2020) to look for a correlation between B * and M å (Figure 7).Field strengths are from Sokal et al. (2020), while stellar masses are derived using T eff and L * /L ☉ from Sokal et al. (2020) with tracks from Baraffe et al. (2015).Although Figure 7 provides some evidence that field strengths are proportional to stellar mass, the correlation statistics for the best-fit line to all TTSs on the graph,  r = −0.29 and P f = 0.077, are much too weak to conclude that the assumption of consistent stellar field strengths between TTSs is not valid.In addition, Figure 7 clearly shows that the slope of the best-fit line for the combined sample, −0.25, is very different from the predicted slope of −1.1.This further confirms that the observed magnetic fields of these stars do not vary with stellar mass in the way predicted by the correlation seen in Figure 7.Following the same prescription as above, we use an analogous version of Equation (11) to search for a value of β such that B * ∝ * b R .We are unable to find a value for β that satisfies this proportionality, and so conclude that field strengths varying with stellar radius do not explain the lowerthan-expected slope when fitting the data to Equation (2).The results of these tests strongly suggest that something other than our assumption that field strengths are roughly constant from star to star is influencing the slope of the line of best fit to Equation (2).
Notwithstanding the lower than expected slope, the results of the three studies clearly show that the equilibrium relationship given by the nondipole model (Equation ( 2)) provides strong correlations between the stellar and accretion data from each study, while none of the studies find significant correlations for the pure dipole model (Equation ( 1)).The overwhelming support given to the nondipole model clearly strengthens suggestions by JG02 and C12 that the assumption of a purely dipole magnetic field geometry for these accreting TTSs is too stringent and underscores the importance of including a term representing the stellar surface area affected by accretion ( f acc ).

Estimating L acc from L Hβ
The strength and shape of the Balmer emission lines of CTTSs are thought to originate from accreting gas as it freefalls along stellar magnetic field lines from the disk to the star's photosphere, from winds driven by accretion that are magnetically launched from the disk, and, to a lesser extent, from the shock produced as accreting gas impacts the stellar surface (Alencar & Basri 2000;Lima et al. 2010;Günther 2013;Hartmann et al. 2016).Here we measure Hβ line luminosities and their associated uncertainties from HST-STIS spectra as an alternative method for estimating L acc .For several of the spectra, identifying the precise endpoints that distinguish the Hβ emission line from the continuum is difficult.In these cases, we select several possible endpoints and average the results.
After using A v values from our models to correct the Hβ flux for extinction, the Hβ luminosity, L Hβ , is calculated from this flux and the Gaia distance to the ONC stars.L Hβ is then used in the relationship derived by Fang et al. (2009) to estimate L acc .Figure 8 shows the strong agreement between L acc estimated from L Hβ and L acc calculated using parameters from full model fits to the observed CTTS spectra.The uncertainties for L acc derived from Hβ emission are too small to show.
While there is a strong correlation between L acc derived from L Hβ and L acc derived from model fits to the spectra of the observed CTTSs (r = 0.85, P f = 7.15 × 10 −5 ), the magnetically active chromospheres of TTSs contribute a small amount of emission to the Hβ lines, which may bias the estimated L Hβ (Manara et al. 2013).In order to estimate the contribution from chromospheric emission to the Hβ lines, we derive L acc from L Hβ for each observed negligibly accreting TTS by applying the same prescription used for the CTTSs.Figure 8 compares L acc estimated from L Hβ with the 3σ upper limits to L acc derived from fits to spectra of the negligible accretors that have had accretion emission in the form of a hot hydrogen slab added to their spectra.The average temperature, density, thickness, and turbulent velocity of the observed CTTS slabs are used to  a The 3σ upper limits to mass accretion rates estimated after adding accretion emission to each spectrum of a negligibly accreting T Tauri star.b Mass accretion rates using L Hβ in the translation from Fang et al. (2009) to estimate L acc .These rates assume L Hβ is entirely from accretion.However, it is likely that a substantial amount of the emission is a result of strong chromospheric activity.The upper limit to the accretion rate listed for JW365 is uncertain due to difficulty in discerning the extent of the Hβ emission line.c P rot from Herbst et al. (2000).
represent the accretion slab for the negligibly accreting T Tauri stars.The slab parameters are combined with f acc , M å , R å , A v , and SpT from the original model fits to the negligible accretors.
All parameters are fixed at their original values except f acc , which is varied manually in the modeling program until χ 2 of the fit is at the 3σ level of detection.The total accretion flux at the stellar surface, F acc , is found by integrating the emission from the slab over all wavelengths.f acc and F acc are used in Equation (4) to derive L acc , which represents the upper limit to the accretion luminosity that would escape detection for each negligible accretor.While a majority of negligible accretors fall below the CTTSs in Figure 8, six negligible accretors have accretion luminosities derived from L Hβ that are greater than the CTTS with the lowest L Hβ accretion luminosity, suggesting that the strong chromospheric emission of those negligible accretors could mask accretion emission.

Distinguishing between CTTSs and Negligible Accretors Based on the Estimated Level of L Hβ
That several negligibly accreting TTSs fall among the CTTSs in Figure 8 clearly illustrates the challenge of distinguishing negligible accretors with strong chromospheric emission from CTTSs with low accretion rates.Here we estimate the level of Hβ line luminosities derived from photospheric templates in Table 2 that have measurable Hβ emission to define an upper envelope to the chromospheric emission of the negligible accretors.As before, we translate Hβ line luminosities to L acc using the relation in Fang et al. (2009).
The dashed lines in Figure 8 correspond to the templates' chromospheric emission derived from L Hβ .Only five of the templates have measurable Hβ emission lines; however, we derived two values of L acc for V830 Tau: one from the spectrum observed by HH14 (denoted V830 Tau (HH14)) and the second from the HST-STIS spectrum observed as part of GO-9790; PI: Johns-Krull.L acc derived from the V830 Tau (HH14) spectrum has the highest L Hβ emission of the templates (0.003 L acc /L ☉ ), which is twice the value derived from the HST-STIS spectrum (0.0015 L acc /L ☉ ).The difference between the two values of L acc from the V830 Tau spectra is attributed to variability in the activity of the star's chromosphere.
Figure 8 shows that V830 Tau (HH14) has the highest chromospheric emission of the templates with discernible Hβ emission lines.To better estimate the relative strength of its chromospheric emission, we compare its Hβ and C IV λ1549 line luminosities with line luminosities from the 15 WTTSs shown in Figure 5 of Yang et al. (2012).After using extinctions from Yang et al. (2012) to deredden their flux values, we compute V830 Tau's Hβ and C IV line luminosities using L = F Hβ 4πd 2 , where d = 140 pc, the distance used by Yang et al. (2012).Our results show V830 Tau to be the strongest chromospheric emitter of the WTTSs in Figure 5 of Yang et al. (2012).
Given that the dashed lines in Figure 8 represent Hβ emission that is unrelated to accretion, stars that fall above these lines should show signs of accretion.Indeed, all of the observed Orion TTSs identified as CTTSs in this study fall above the dashed lines.However, seven negligibly accreting T Tauri stars also fall above the topmost dashed line, which suggests that the chromospheric emission of these stars could mask low accretion rates.For those stars, the Balmer emission lines blueward of the Hβ line are either missing or poorly developed, and the photospheric templates provide good fits to the spectra without the addition of a slab.Although Balmer jumps are not apparent in the spectra, the poor quality of the spectra blueward of 3600 Å for JW378 makes it difficult to conclude that a Balmer jump is not present.Our results show that L Hβ does not provide a reliable threshold for distinguishing between accreting and weakly accreting TTSs, especially for those negligibly accreting T Tauri stars with active chromospheres.1) that require a stellar dipole magnetic field.The dashed line shows the best-fit line predicted by Equation (1) (slope = 1).A best-fit line to the data is not shown due to the apparent lack of correlation.(b) Log-log plot of parameter values derived from models of Orion CTTSs applied to Equation (2) that allows for a nondipole field geometry.The dashed line shows the best-fit line predicted by Equation (2) (slope = 1).The solid line is the best-fit line for the parameters derived from the full model fits (slope = 0.78).

Estimating Surface Flux and M acc
 from L acc The luminosity of any diagnostic, including chromospheric emission, is affected by the intrinsic strength of that emission and the total surface area of the star.That is, weak chromospheric Hβ emission could still have substantial Hβ luminosity per unit surface area for a relatively large star.Thus, Hβ surface flux may be a better way to define the boundary between chromospheric and accretion emission.We estimate the Hβ surface flux using the Hβ luminosities given in Section 5.3.1 and the stellar radii from model fits in the equation F surf = L Hβ /4πR 2 .Figure 9   The red, blue, and black lines are the best-fit lines for the CTTSs, WTTSs, and the combined sample, respectively.We can now compute accretion rates for the observed negligible accretors and CTTSs using surface flux values from this section, L acc from Section 5.3.1, and R * from model fits in Equation (6).Six negligibly accreting TTSs fall above the lowest CTTS accretion rate derived using L Hβ .It is not surprising that these six stars also have the highest chromospheric emission based on where they fall in Figure 8.In Figures 8 and 10, the 15 CTTSs are evenly spread above and below the dashed line, which has a slope of 1, whereas the negligibly accreting TTSs systematically fall above the dashed line, further indicating that at low accretion rates, Hβ is not a good diagnostic.Table 4 lists the 3σ upper limits to the accretion rates derived using accretion luminosities from L Hβ and from the negligibly accreting TTSs that have had a slab added to their spectra.

Final Estimates of M acc  from Hβ Line Emission
Estimating the accretion rate of individual sources depends on measuring an accretion diagnostic such as L Hβ as well as other stellar parameters of the individual source.Here, we use L Hβ derived from full model fits to the STIS spectra in this study with stellar parameters from H97, Manara et al. (2012, hereafter MRD12), and this study to estimate M acc  .H97 derived A v , R å , and M å for ∼900 TTSs in the ONC.Values of A v were estimated from V − I C photometry and spectral type.Stellar radii were calculated using L å derived from extinctioncorrected I C -band photometry and effective temperatures   translated from temperature scales of Cohen & Kuhi (1979) according to each star's spectral type.Stellar masses were interpolated from tracks of D' Antona & Mazzitelli (1994).
Accretion luminosities were estimated using the relationship in Fang et al. (2009), where F Hβ derived in this study are dereddened with extinctions from H97.Values of L acc were then used to calculate M acc  for the 15 CTTSs from H97 in common with this study.Figure 11 shows that M acc  derived using H97 stellar parameters and M acc  from this study are in good agreement (r = 0.71, P f = 0.005).MRD12 derived A v , R å , and M å for ∼700 TTSs in the ONC.Values of A v were estimated according to their displacement from a theoretical isochrone on a (B − I) versus (U − B) diagram.If this method did not provide an A v value, MRD12 adopted the A v value from Da Rio et al. (2012).Stellar radii were calculated using T eff from Da Rio et al. (2012) and L å from I-magnitudes corrected for extinction and accretion.Stellar masses were estimated from three different evolutionary tracks.For simplicity, we limit our evaluation to M å derived from tracks of Siess et al. (2000).We calculate M acc  using our Hβ measurements and stellar parameters from MRD12.
In Figure 11, stars dereddened with A v derived from the (B − I) versus (U − B) diagram are labeled MRD12-1.The remaining stars that were dereddened with A v values from Da Rio et al. (2012) are labeled MRD12-2.The correlation statistics (r = 0.15, P f = 0.68) suggest a much weaker correlation between the results from this study and the MRD12 study when compared to the statistics with H97.These results show how sensitive accretion rate measurements can be to the determined stellar parameters (mass and radius).In this section, we compare M acc  derived from photometric measurements by MRD12 with those derived from full model fits to the STIS spectra.MRD12 use two methods to derive L acc .For each star in the group that we label MRD12-1, L acc /L tot (where

Comparisons to Previous Studies
For the remaining stars in MRD12-2, Hα excess is estimated by comparing the flux measured using the Hα narrowband filter (F656N) to the photospheric (Hα -I) color for each star.The excess is then converted to W λ (Hα).Sources that fall within 3 Å < W λ (Hα) < 1000 Å are considered accretors.For these stars, L acc /L ☉ is derived from L Hα using the relation derived by MRD12, which is then used to calculate M acc  .
Figure 12 shows the relationship between log M acc  from MRD12 and this study for stars in common to both studies.Although the sample size is small, the M acc  rates from the two studies appear to be poorly correlated.The correlation statistics (r = 0.26, P f = 0.465) confirm the weakness of the correlation.One star, JW750, appears as an outlier on the graph.M acc  derived by MRD12 for JW750 is the second highest of the accreting stars in common to both studies, whereas our analysis shows M acc  for JW750 to be among the lowest.A comparison of the spectra from JW123, the star with the highest accretion rate for MRD12, and JW750 underscores why spectral analysis is important for accurate measurements of M acc  .The spectra, shown in Figure 16 of Appendix C, show that while both JW750 and JW123 are accreting stars, the shapes of their spectra are very different.The spectrum of JW123 has greater UV emission and a stronger Balmer jump than the spectrum of JW750, which has a spectrum more similar to spectra of stars with lower accretion rates (e.g., JW248).The higher accretion rate derived by MRD12 is explained by the higher extinction (A v = 2.26 mag) used by MRD12 compared to the extinction (A v = 0.46 mag) from our model fits.

Comparing TTS Classifications between Studies
Here, we compare our TTS classifications (see Tables 3 and  4 Diskless stars that emit only intrinsic photospheric flux levels are identified as Class III sources.Excess MIR emission is likely produced from an optically thick circumstellar disk that reradiates energy from the central star.Although the presence of a disk does not ensure that the central star is accreting, a star without a disk is clearly not accreting. In addition, D14 identified accretion emission from W λ (Ca II) as measured from spectra observed by Hillenbrand et al. (1998).Following guidelines suggested by Hillenbrand et al. (1998) and Flaccomio et al. (2003), they interpreted stars having W λ (Ca II) < −1 Å as accreting TTSs and stars with W λ (Ca II) > 1 Å as nonaccreting.
Overall, we find that TTS classifications by SA05 generally agree with the classifications from this study.Excluding stars that SA05 flag as having questionable interpretations, 82% (9/11) of the stars in common to both studies have the same classifications.By comparison, 40% (10/25) of the stars from the MRD12 study in common with this study have the same classification.Of those 25 stars, 63% (5/8) of the MRD12 star classifications based on M acc  derived from L Hα match those from this study, whereas only 29% (5/17) of the classifications based on U-band excess are in agreement.
While MIR excess suggests the presence of a circumstellar disk, it does not predict with certainty that the star is accreting.However, of the stars common to both studies, all TTSs that showed MIR excess according to D14 were identified as CTTSs in this study.On the other hand, four TTSs that we identified as negligibly accreting TTSs should have disks based on MIR excess (JW25, which has an uncertain classification, JW243, JW636, and JW836).Detections of disks around negligible accretors are of interest since they are somewhat uncommon and may provide insight into the early stages of disk disbursement.Evidence of IR excess might indicate that the negligible accretor is surrounded by a passive disk that is undergoing significant disk evolution or that it is actually an accreting star that was observed during a nonaccreting state (McCabe et al. 2006).Table 7 provides a summary of the classifications for the T Tauri stars common to MRD12, SA05, and this study.Additional information regarding the accretion classifications from MRD12, SA05, and this study can be found in Appendix E. open cluster NGC 2264 (C12) to ∼1 Myr old stars in the ONC.Like JG02 and C12, we find evidence in support of the disklocking hypothesis in the form of the modified OS95 model, which assumes a consistent magnetic field strength among T Tauri stars but does not rely on a dipolar stellar magnetic field geometry.On the other hand, we do not find a statistically significant correlation for the disk-locking model derived from Königl (1991) and Shu et al. (1994), which requires a dipole magnetic field at the stellar surface.
Using stellar and accretion parameters from JG02, C12, and this study, we have for the first time computed slopes for the correlations derived from the modified OS95 model using uncertainties in both the x-and y-axes.Within admittedly large uncertainties, all three studies give the same slopes (Table 5), and the combined samples overlie each other well (Figure 5(b)).The combined slope is approximately equal to the value of 1.0 that is predicted from the X-wind model as extended by Johns-Krull & Gafford (2002).However, the slope is formally lower than 1.0 with ∼3σ confidence.
The lower-than-expected slope led us to question the fundamental assumption of our test-that the magnetic field strength is consistent from star to star.We explored the possibility that field strength may vary with a star's mass or radius, which would indicate that the field strength is not consistent between TTSs.Initially, we found that if B * ∝ * -M 1.1 we could explain why the slope for the correlation seen in Equation (2) is less than 1.0.However, using data from Sokal et al. (2020), we found no such correlation between B * and M * (Figure 7).Using the same approach to determine whether B * varies with R * , we again found no indication of a correlation.The reason for the lower-than-expected slope remains unsettled.Nevertheless, the strong correlation statistics for the combined sample lend further support to the premise that the dipole magnetic field geometry assumed by many magnetospheric accretion models is too stringent.The X-wind model as extended by Johns-Krull & Gafford (2002) allows a natural means to compare observations to model predictions, producing good correlations with the observed parameters.More sophisticated numerical simulations do not currently afford such a straightforward comparison with observed samples of stars.
To improve the results for future tests of the accretion models in the ONC, we recommend longer observation times and acquiring near-contemporaneous HST-STIS spectra from accreting T Tauri stars over a broader range of wavelengths (near-UV to optical).Observations at these shorter wavelengths have been shown to provide improved estimates of accretion rate and filling factor (Herczeg et al. 2004;Ingleby et al. 2013Ingleby et al. , 2015;;Robinson & Espaillat 2019).The spectra acquired for the ULLYSES, ODYSSEYS, and Penellope programs will provide an excellent resource for the type of analysis used in this study and for more detailed future work related to improving our understanding of the evolution of T Tauri stars.
We find that limiting the targeted stars to those with at least a 0.35 mag excess I C − K color did not limit our sample to accreting T Tauri stars given that over half of the observed stars have negligible accretion signatures in the HST-STIS spectra.This may be due to the variability of young stars or a limitation in the use of photometric indices.
Accretion luminosities estimated from Hβ line fluxes correlated well with accretion luminosities derived from model fits to the STIS spectra.However, our attempt to use L Hβ derived from the template spectra to define a threshold between accretion and chromospheric luminosity showed that L Hβ is not always a reliable indicator for distinguishing between weakly accreting and nonaccreting T Tauri stars.
Lastly, results from our spectral analyses were not always in agreement with the results from studies that relied primarily on photometry.For example, the HST-STIS spectra of several of the T Tauri stars classified as CTTSs by MRD12 do not show signs of accretion.While photometric studies provide an efficient way to determine physical parameters for a statistically large sample of stars, a model spectrum fit to the shape of the UV and optical continuum provides a more direct measure of accretion luminosity and mass accretion rate unique to each star since the shape of the spectrum is a consequence of the properties of the accretion flow.
Figure 14 shows the photospheric templates that were used to determine if a star's spectrum shows signs of accretion.If the star was indeed found to be accreting, then the derived stellar and accretion parameters were used to test the magnetospheric accretion models described in Section 3.

Figure 1 .
Figure 1.Example fit to the spectrum of a T Tauri star.The observed spectrum of Orion CTTS JW123 is shown as a solid black line.The solid red line is the fitted model spectrum, while the dashed blue line represents emission from accretion processes.The solid green line is the scaled M1.2 photospheric template.

Figure 2 .
Figure 2. Comparison of A v from H97 with A v values from this study.The dashed line represents a line with slope = 1.

Figure 3 .
Figure 3.Comparison of M acc  (10 −8 M ☉ yr −1 ) from two methods for estimating the truncation radius.The dashed line has a slope = 1.0.
) and (b) show the relationships for Equations (1) and (2), respectively, for the combined data.The correlation statistics and slopes of the best-fit lines determined from the log of the values are shown in Table

Figure 4 .
Figure 4. (a) Log-log plot of parameter values derived from models of Orion CTTSs applied to Equation (1) that require a stellar dipole magnetic field.The dashed line shows the best-fit line predicted by Equation (1) (slope = 1).A best-fit line to the data is not shown due to the apparent lack of correlation.(b) Log-log plot of parameter values derived from models of Orion CTTSs applied to Equation (2) that allows for a nondipole field geometry.The dashed line shows the best-fit line predicted by Equation (2) (slope = 1).The solid line is the best-fit line for the parameters derived from the full model fits (slope = 0.78).

Figure 5 .
Figure 5.The combined data from C12 (open diamonds), JG02 (open triangles), and this study (black circles).(a) Log-log plot of the correlation given by Equation (1) that requires an aligned stellar dipole magnetic field.The dashed line is the line with slope = 1.0.(b) Log-log plot of the correlation given by Equation (2) that relaxes the requirement for a dipole field.The solid line represents the best-fit line for the three studies.The dashed line is the line of best fit with slope = 1.0.

Figure 6 .
Figure 6.Log-log plot of the correlation given by Equation (11) when β = 1.6.The solid line is the line of best fit, which has a slope = 1.Figure 7. Log-log plot of magnetic field data from Sokal et al. (2020) vs. stellar mass derived from tracks of Baraffe et al. (2015) for a sample of TTSs.The dashed line has a slope of −1.1, the slope predicted from Equation (11).The red, blue, and black lines are the best-fit lines for the CTTSs, WTTSs, and the combined sample, respectively.Table5Correlation Results for Three Studies

Figure 7 .
Figure 6.Log-log plot of the correlation given by Equation (11) when β = 1.6.The solid line is the line of best fit, which has a slope = 1.Figure 7. Log-log plot of magnetic field data from Sokal et al. (2020) vs. stellar mass derived from tracks of Baraffe et al. (2015) for a sample of TTSs.The dashed line has a slope of −1.1, the slope predicted from Equation (11).The red, blue, and black lines are the best-fit lines for the CTTSs, WTTSs, and the combined sample, respectively.Table5Correlation Results for Three Studies

Figure 8 .
Figure 8.Comparison of excess emission derived from Hβ line strengths with excess emission estimated from fits to the observed spectra of CTTSs and negligibly accreting (NA) T Tauri stars.The symbols of the negligible accretors (NA) show the 3σ upper limit to each star's accretion luminosity.Horizontal dashed lines represent the chromospheric emission based on L Hβ from each of six photospheric templates that have measurable Hβ line widths.From top to bottom, the lines represent measurements from the following spectra: V830 Tau (HH14), LkCa 7, TWA 6, V830 Tau (from STIS-HST spectrum), LkCa 5, and TWA 3B.The diagonal dashed line has slope = 1.Detection of the Hβ line is uncertain for JW365 so the upper limits to both values of L acc are shown.

Figure 9 .
Figure 9. Distribution of T Tauri stars according to each star's Hβ surface flux.Detection of the Hβ line is uncertain for JW365; however, we estimate a surface flux that is less than 1.0 × 10 6 erg cm -2 s −1 .

Figure 10 .
Figure 10.Comparison of M acc  derived from L Hβ with M acc  derived from model fits.The diagonal dashed line has slope = 1.The symbols of the negligibly accreting (NA) TTSs show the 3σ upper limit to each star's M acc  .Detection of the Hβ line is uncertain for JW365 so the upper limits to both values of M acc  are shown.

Figure 12 .
Figure 12.Log M acc  from photometric measurements from MRD12-1 (+ signs) and MRD12-2 (× signs) compared to log M acc  from full model fits to STIS spectra derived in this study.The dashed line has a slope = 1.
) to classifications by MRD12, Sicilia-Aguilar et al. (2005), and Davies et al. (2014).As described in Section 5.4.1,MRD12 identified accreting stars in the ONC from (U − B) and Hα excess.Sicilia-Aguilar et al. (2005, hereafter SA05) classified CTTSs and WTTSs in the ONC based on Hα profiles from high-resolution spectra from the Hectochelle multiobject echelle spectrograph on the 6.5 m MMT.Stars with Hα lines that are unambiguously broader than Hα lines from nebular emission are identified as CTTSs, while stars with weak nebular emission, strong stellar photospheric continua, and narrow stellar Hα lines are classified as WTTSs.T Tauri stars with weak or noisy signals are referred to as possible WTTSs, while those with broad lines that are not clearly recognizable as stellar emission or wings of nebular emission are referred to as possible CTTSs.Davies et al. (2014, hereafter D14) relied on high-spatialresolution mid-infrared (MIR) measurements from published Spitzer Infrared Array Camera (IRAC) fluxes to classify TTSs with excess MIR (relative to radiation) as Class II sources.

Figure 14 .
Figure 14.Spectra of weak T Tauri star templates from Herczeg & Hillenbrand (2014) used to derive stellar and accretion parameters.

Table 3
Stellar and Accretion Parameters for CTTSs

Table 5
Megeath et al. (2012)or Three Studies the distribution of the Hβ surface flux.The two CTTSs that have the lowest Hβ surface flux values, JW36 and JW248, have small but observable Balmer jumps, Balmer emission lines, and excess UV and optical continuum emission.On the other hand, while we do not detect accretion in JW320, a negligibly accreting TTS, it has the highest Hβ surface flux of the stars that do not show obvious signs of accretion.AlthoughMegeath et al. (2012)identify a disk surrounding JW320 from Spitzer mid-IR colors, Balmer emission lines blueward of the Hβ line are weak or indiscernible; a Balmer jump is not apparent; and the star's spectrum does not require a slab to fit its K0 spectrum.Since L acc is directly proportional to L Hβ , Figure8shows that six negligible accretors have Hβ luminosities greater than the Hβ luminosity of the weakest CTTS.However, only two negligible accretors have Hβ surface flux values greater than the lowest Hβ surface flux of the CTTSs.These results suggest that Hβ surface flux is a better indicator for distinguishing between negligibly accreting TTSs with strong chromospheric emission and weakly accreting CTTSs.
a Equation (1)-assumes stellar dipole magnetic field configurations.bEquation (2)-allows for more complex magnetic field configurations.cSlopes are given for nondipole models only due to poor correlations for the dipole models.shows Hβ with stellar parameters from MRD12, H97, and this study, and log M Table 6 lists log M acc  derived from our model fits to the observed spectra, log(L Hβ /L ☉ ) from this study, log M acc  derived using L acc  estimated by MRD12.

Table 7 A
Comparison of T Tauri Star Classifications