---
abstract: 'This work is devoted to improving empirical mass-luminosity relations (MLR) and mass-metallicity-luminosity relation (MMLR) for low mass stars. For these stars, observational data in the mass-luminosity plane or the mass-metallicity-luminosity space subject to non-negligible errors in all coordinates with different dimensions. Thus a reasonable weight assigning scheme is needed for obtaining more reliable results. Such a scheme is developed, with which each data point can have its own due contribution. Previous studies have shown that there exists a plateau feature in the MLR. Taking into account the constraints from the observational luminosity function, we find by fitting the observational data using our weight assigning scheme that the plateau spans from 0.28 [[$M_\odot$]{}]{}to 0.50 [[$M_\odot$]{}]{}. Three-piecewise continuous improved MLRs in K, J, H and V bands, respectively, are obtained. The visual MMLR is also improved based on our K band MLR and the available observational metallicity data.'
author:
- |
    Fang Xia$^{1,2}$[^1], Shulin Ren$^{1,2}$ and Yanning Fu$^{1}$\
    $^{1}$Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China\
    $^{2}$Graduate School of Chinese Academy of Sciences, Beijing 100039, China
title: 'The Empirical Mass-Luminosity Relation for Low Mass Stars'
---

[*Keywords:*]{} [ stars: low-mass–stars: fundamental parameters–methods: numerical]{}

Introduction
============

Mass is one of the most fundamental parameters of a star. Unfortunately, stellar mass is difficult to be determined directly. Therefore, it is often estimated by mass-luminosity relation (MLR).

Since the pioneering papers of Hertzsprung (1923) and Russell et al. (1923) there have been many studies devoted to improving and understanding MLR. To date, the relation has been well constrained for solar-type and intermediate stars [@Del00]. But the empirical MLR remains poorly defined for very low mass stars due to scarcity of data. On the other hand, however, the empirical MLR for very low mass stars is crucial in many aspects of astronomy and astrophysics. For example, without an accurate MLR in this mass interval, the total luminous mass of the Galaxy will never be well determined [@Henry98]. This is because very low mass stars account for at least 70% of all stars, and they make up more than 40% of the total stellar mass of the Galaxy [@Henry99].

Recently, the MLR for very low mass stars has been improved significantly, thanks to the painstaking observations and the required orbital analysis. Henry et al. (1999) obtained or improved some dynamical masses of very low mass stars and discussed the MLR for the mass interval 0.2 [[$M_\odot$]{}]{}to 0.08 [[$M_\odot$]{}]{}. Delfosse et al. (2000) pointed out that MLR is tight only in near-infrared bands. Together with the improved MLRs in K, J and H bands, they suggested that the large dispersion in V band should be due to the difference in stellar metallicity. In Bonfils et al. (2005), with quantitative metallicity estimations, the authors verified this metallicity dependency and provided visual mass-metallicity-luminosity relation (MMLR) for the first time.

Despite of these progresses, the empirical MLR for low mass stars obtained so far is still not satisfying. The existence of a transitional mass interval, where the derivative of MLR presents a plateau feature, is an intrinsic obstacle to a satisfying result. This is because a continuous yet segmented model is needed for MLR. While more data are often necessary for fitting to a segmented model than a non-segmented one, some data with relatively low accuracy were discarded in the previous work. This is not without reason, as many of the data are prone to systematics. Obviously, to remove such data, it should be more reasonable to require that the remaining data set is well behaved. And at the same time, one should avoid discarding data more than necessary, such that the original data set can be fully utilized. Based on a newly developed weight assigning scheme, we are able to achieve this by an iterative process and, as thus, give a more confident MLR. In comparison with previous work [@Del00; @HM93], the improvement of MLR in (0.7 [[$M_\odot$]{}]{}, 1.0[[$M_\odot$]{}]{}) also comes from incorporating more observational data, and that in (0.1[[$M_\odot$]{}]{}, 0.7[[$M_\odot$]{}]{}) from taking into account the underlying physics.

In section 2 and section 3, respectively, we describe our sample data and develop a fitting method with a reasonable weight assigning scheme. In section 4, the best-fitting three-piecewise empirical MLRs in K, J, H and V bands, the MMLR in V band are provided. Concluding remarks are given in the last section.

DATA COLLECTION
===============

In order to map out MLR for low mass main sequence stars, we need a sample of dynamical masses derived from orbital analysis [@Henry04] and absolute magnitudes. By searching the literatures, the resulting sample of 48 main sequence stars is listed in Table 1. The columns of Table 1 are, respectively, name of these stars, the dynamical mass ($M\pm \Delta M$) (spanning from 0.07 [[$M_\odot$]{}]{}$\sim$1.086 [[$M_\odot$]{}]{}), absolute V magnitude ($M_V\pm\Delta
M_{V}$), absolute K magnitude ($M_K\pm \Delta M_{K}$), absolute J magnitude ($M_J\pm\Delta M_{J}$) and absolute H magnitude ($M_H\pm\Delta M_{H}$). Most of the data can be found directly from the literatures except that some values of $M_V, M_J$ and $M_H$ are obtained from apparent magnitudes (color index if necessary) and parallaxes. The types of spectrum as well as references are also indicated.

  --------- --------------------- --------------------- --------- --------------- -------- --------------- --------- --------------- -------- --------------- ---------- --------------
  Name      $M $                  $\Delta{M}$               $M_V$ $\Delta{M_V}$   $M_K$    $\Delta{M_K}$       $M_J$ $\Delta{M_J}$      $M_H$ $\Delta{M_H}$   Spectrum   Reference
            ([[$M_\odot$]{}]{})   ([[$M_\odot$]{}]{})       (mag) (mag)           (mag)    (mag)               (mag) (mag)              (mag) (mag)                      
  GL22A     0.43                  0.039                     10.56 0.07            6.44     0.10                                          6.74 0.08            M2V        1&2&9
  GL22C     0.14                  0.014                     13.64 0.12            8.43     0.13                                          8.85 0.10            M2V        1&2&9
  GL25A     0.94                  0.088                      4.67 0.088           3.82     0.27               *3.18* *0.08*              3.88 0.26            G8V        6&9&10$^{a}$
  GL25B     0.7                   0.079                      4.99 0.088           3.98     0.27                                          4.13 0.26            G8V        6&9&10$^{a}$
  GL65A     0.102                 0.01                      15.41 0.05            8.76     0.07               *9.68* *0.05*            *9.15* *0.03*          M5.5V      8
  GL65B     0.100                 0.01                      15.87 0.06            9.16     0.07                10.06 0.05                9.45 0.03            M6V        8
  GL67A     0.933                 0.231                      4.45 0.03            *2.87*   *0.12*             *3.14* *0.12*              2.90 0.12            M4V        9
  GL67B     0.28                  0.071                     12.07 0.5             7.30     0.13               *7.52* *0.27*              7.40 0.17            M4V        9
  GL166C    0.177                 0.029                     12.68 0.03            7.58     0.07                 8.49 0.07              *7.87* *0.07*          M4.5V      1&9
  GL234A    0.2027                0.0106                    13.07 0.05            7.64     0.04                 8.52 0.06                7.93 0.04            M4.5V      8
  GL234B    0.1034                0.0035                    16.16 0.07            9.26     0.04                10.31 0.25                9.56 0.10            M4.5V      8
  GL340A    0.657                 0.101                                           4.25     0.12               *4.73* *0.12*            *4.23* *0.12*          K3V        9
  GL340B    *0.590*               *0.090*                                         *4.33*   *0.12*             *5.08* *0.12*            *4.65* *0.12*          K3V        9
  GL352A    *0.195*               *0.060*                 *10.80* *0.20*          *6.38*   *0.19*             *7.13* *0.19*            *6.59* *0.19*          M3V        9
  GL352B    *0.203*               *0.062*                 *11.20* *0.18*          *6.61*   *0.19*             *7.46* *0.19*            *6.85* *0.19*          M3V        9
  GL473A    0.143                 0.011                     15.01 0.07            8.40     0.06                 9.44 0.06                8.84 0.06            M5.5V      8
  GL473B    0.131                 0.01                      15.00 0.07            8.84     0.08                 9.57 0.06                9.04 0.07            M5.5V      8
  GL508A    0.709                 0.179                      8.43 0.17            4.95     0.17                 5.72 0.17                5.12 0.17            M0.5V      9
  GL508B    0.497                 0.126                     10.14 0.19            5.87     0.17                 6.80 0.18                6.10 0.17            M1V        9
  GL559A    1.086                 0.025                      4.71 0.05            2.90     0.05                 3.24 0.05                3.00 0.05            G2V        9
  GL559B    0.903                 0.021                      6.08 0.05            3.78     0.05                 4.37 0.05                3.89 0.05            K1V        9
  GL570B    0.5656                0.0029                     9.28 0.09            5.39     0.03                 6.21 0.03                5.61 0.03            M1V        6&8$^{b}$
  GL570C    0.377                 0.0018                    11.09 0.17            6.57     0.04                 7.40 0.04                6.76 0.04            M3V        6&8$^{b}$
  GL623A    0.3432                0.0301                    10.74 0.05            6.46     0.04                 7.19 0.04                6.70 0.04            M2.5V      8
  GL623B    0.1142                0.0083                    16.02 0.11            9.33     0.14                10.47 0.29                9.35 0.05            M2.5V      8
  GL644A    0.4155                0.0057                    10.76 0.06            6.35     0.04                                          6.61 0.05            M3V        8
  GL661A    0.379                 0.035                     11.10 0.06            6.36     0.05                 7.10 0.05                6.56 0.04            M4.5V      8
  GL661B    0.369                 0.035                     11.15 0.06            6.78     0.05                 7.51 0.04                7.02 0.04            M4.5V      8
  GL677A    *0.285*               *0.092*                  *8.37* *0.22*          *4.94*   *0.22*             *5.65* *0.21*            *5.10* *0.21*          K3V        9
  GL677B    *0.251*               *0.081*                  *8.93* *0.22*          *5.22*   *0.22*             *5.95* *0.21*            *5.41* *0.21*          K3V        9
  GL702A    0.9                   0.074                      5.63 0.047           3.293    0.048                4.15 0.06                3.39 0.05            K0V        4&6&9$^{b}$
  GL702B    0.78                  0.04                       7.43 0.047           4.53     0.04                 5.63 0.05                                     K0V        6&8$^{b}$
  GL704A    0.932                 0.187                      3.95 0.15            *2.38*   *0.17*             *2.62* *0.16*            *2.47* *0.15*          F6V        9
  GL704B    0.616                 0.124                      7.31 0.15            4.44     0.18               *5.15* *0.16*              4.87 0.16            K4V        9
  GL725A    0.37                  0.061                     11.18 0.05            6.72     0.03                 7.48 0.03                6.95 0.03            M3V        9
  GL725B    0.316                 0.052                     11.96 0.05            7.25     0.03                 8.00 0.03                7.48 0.03            M3V        9
  GL747A    0.2137                0.0009                    12.30 0.06            7.53     0.04                                                               M3V        8
  GL747B    0.1997                0.0008                    12.52 0.06            7.63     0.04                                                               M3V        8
  GL820A    0.562                 0.08                       7.58 0.04            4.598    0.015                5.32 0.02                4.71 0.02            K5V        3&4
  GL820B    0.562                 0.08                       8.41 0.04            4.958    0.018                5.75 0.02                5.11 0.02            K5V        3&4
  GL831A    0.2913                0.0125                    12.52 0.06            7.08     0.05                                          7.36 0.05            M4.5V      8
  GL831B    0.1621                0.0065                    14.62 0.08            8.36     0.05                                          8.62 0.05            M4.5V      8
  GL860A    0.2711                0.0100                    11.76 0.05            6.95     0.04                 7.84 0.04                7.26 0.04            M3V        8
  GL860B    0.1762                0.0066                    13.46 0.09            8.32     0.07                 9.03 0.08                8.40 0.05            M5.5V      8
  GL866A    0.1187                0.0011                    15.39 0.07            9.11     0.32                                                               M5.5V      5&7
  GL866B    0.1145                0.0012                    15.64 0.08            8.98     0.05                                          9.29 0.04            M5.5V      5&7
  GL1245A   0.11                  0.021                     15.18 0.03            8.96     0.03                 9.79 0.03                9.32 0.03            M5.5V      1&2&9
  GL1245C   0.07                  0.013                   *18.47* *0.06*          *9.99*   *0.04*            *11.05* *0.04*             10.40 0.04            M5.5V      1&2&9
  --------- --------------------- --------------------- --------- --------------- -------- --------------- --------- --------------- -------- --------------- ---------- --------------

1.[@Henry99]; 2. [@Henry04]; 3. [@Cester83]; 4. [@Alonso94]; 5. [@Woitas03]; 6.[@Pourbaix00]; 7. [@Woitas00] 8. [@Del00]; 9. [@HM93]; 10. [@Soderhjelm99].\
$^a$ parallax from 6 and color index from 10;    $^b$ parallax and apparent magnitude from 6.

In this context, it should be mentioned that accuracy is sometimes used as a cutoff principle in deciding which star can be used for defining MLR. For example, 10% mass uncertainty is used as the cutoff accuracy in Delfosse et al. (2000). To do so, one can not only avoid the difficulty of assigning weight but also reduce the influence of observational systematic errors. But in order to make the best use of the present observations, we will not discard data just for their relatively low accuracy (for details, see Section 4).

Weight assigning scheme
=======================

MLR is often modeled as $$\log M = f(M_P),$$ where $M_P$ is absolute magnitude, $f$ is a linear function or polynomial. In this section, $f$ is considered as a fourth order polynomial [@DK91; @Del00]. As both $M$ and $M_P$ are subject to non-negligible errors, the weight of each data point is not easy to be assigned. Because the error of magnitude is relatively small, there is an easy fitting method (hereafter Method 1) in which the weight is assigned only considering the error of mass. But we found it is not acceptable for quantitatively describing the MLR for low mass stars. For our purpose, a new method (hereafter Method 2) in which errors in both coordinates are used to assign appropriate weight is developed.

To introduce Method 2, let’s write $$y(x)=f(M_P)=a_1+a_2x+a_3x^{2}+a_4x^{3}+a_5x^{4}$$ and suppose ($x_i\pm\sigma_{x_i},y_i\pm\sigma_{y_i}$) (i=1,…,N) to be the observational data with respective errors. The merit function is, as usual, $$\chi^{2}=\sum_{i=1}^N[\frac{y_i-y(x_i)}{\sigma_i}]^{2},$$ where the reciprocal of $\sigma_i$ is the assigned weight $w_i$ of ($x_i,y_i$).

For linear model $$y=a+bx,$$ the weight of each data point can be quantified as [@Press92]: $$w_i\equiv\frac{1}{\sigma_i}=\frac{1}{\sqrt{b^{2}\sigma_{x_i}^{2}+\sigma_{y_i}^{2}}}.$$ In nonlinear case, the weight of each data point depends on the local property of the fitting result, so an iteration process is unavoidable. The main steps of the iteration process are described as follows.

\(1) A linear fitting with errors in both coordinates is performed with the help of Press et al. (1992).

\(2) With the slope “b” of the linear model obtained in the first step, the weight is approximately quantified by using equation (5) and then, the data are fitted to the given model, $y=y(x)$.

\(3) By using the preliminary fitting result, the weight of ($x_i,y_i$) can be re-quantified according to the local slope. In terms of normalized local tangent vector, the weight writes: $$w_i\equiv\frac{1}{\sigma
_i/\sqrt{b^{2}_i+1}}=\sqrt{\frac{b_i^{2}+1}{b^{2}_i\sigma_{x_i}^{2}+\sigma_{y_i}^{2}}},$$ where $b_i=f'(x_i)$. And the model parameters are updated accordingly.

\(4) The third step is iterated until each model parameter changes less than, say 1%.

In order to compare the above mentioned two methods, we do the following tests. We start with a fourth order polynomial $Log
M=f(M_V)$, named as “true MLR”, qualitatively resembling the MLR for very low mass stars (M $< 0.25$ [[$M_\odot$]{}]{}). We then generate “observational data” from the “true MLR” by adding Gaussian errors at a level of the relevant observations.

There are three kinds of fitted MLRs. For each kind, 100 MLRs are obtained with different sets of generated observational data. The MLRs of the first and the second kinds are obtained by fitting 20 data points (approximately the number of very low mass sample stars) by Method 1 and Method 2, respectively. And the MLRs of the third kind are obtained by fitting different sets of 40 data points, using Method 2. The third kind is included because comparing results obtained from different number of data points can shed some light on the role of future observations in improving MLR.

Figure 1 shows typical cases of the three kinds of fitting. The “true MLR” and the “observational data” are shown as solid curve and points with error bars, respectively. Comparing the upper two panels, we see clearly that Method 2 is better than Method 1. And, as is evident in the top panel, Method 1 is apt to bring spurious characteristics to the obtained MLR. This suggests that an improper way of weight assigning tends to lead not only quantitative but also qualitative difference between a fitted and a realistic MLR. From the middle panel, we know that, by using Method 2, 20 points already have the opportunity to be used to recover rather satisfactorily the “true MLR”. As expected, the improvement brought by doubling observational data is evident, as can be seen by comparing the lower two panels. This shows the importance of future observations in improving further the empirical MLR.

![Typical results of the three different kinds of fitted mass-luminosity relation for very low mass stars. In each panel, the dots with error bars are the “observational” data, the dashed and the solid lines, respectively, the fitted and the “true” mass-luminosity relations.[]{data-label="fig_1"}](xf_fig1.eps){width="45.00000%"}

For comparing the two methods statistically, we calculate the absolute errors ($\Delta M$) of the masses estimated from the above MLRs obtained using 20 data points. For each fitted MLR, $\Delta M$ at 100 randomly chosen magnitudes are calculated. In total, there are 20000 such errors {$\Delta M_{kij},k=1,2; i=1,\ldots,100;
j=1,\ldots,100$}, where the subscripts $k$, $i$ and $j$ indicate the used methods, the generated data sets and the chosen magnitudes, respectively. The maximal (mean) $\Delta M_{2ij}$ over both $i$ and $j$ is about 40% (76%) of that of $\Delta M_{1ij}$. These results clearly show that Method 2 is better than Method 1.

The introduced weight assigning scheme is still effective in multiple dimension fittings. For fitting functions like $z=z(x,y)$ the weight writes $$w_i=\sqrt{\frac{a_i^{2}+b_i^{2}+1}{a^{2}_i\sigma_{x_i}^{2}+b^{2}_i\sigma_{y_i}^{2}+\sigma_{z_i}^{2}}},$$ where $a_i=z'(x_i),b_i=z'(y_i)$. Tests show that much better results can be derived when the weight of each data point is reasonably assigned.

Piecewise MLR
=============

Early empirical MLRs were often given in the form of a power law. Thereafter, in order to more accord with observations, three-piecewise continuous model was used [@HM93]. This is also physically justified. In fact, as the decreasing stellar mass reaches about 0.5 [[$M_\odot$]{}]{}, convection becomes more and more efficient and more H$_2$ forms in the stellar atmosphere [@Cester83; @Laughlin97]. And when stellar mass reaches about 0.3[[$M_\odot$]{}]{}, stars should be fully convective [@Copeland70; @Grossman74]. As a result, the derivative of MLR should display a plateau feature between $\sim 0.3$ [[$M_\odot$]{}]{}and $\sim
0.5$ [[$M_\odot$]{}]{}. This feature is evident from observations. However, the observations show that the plateau feature in the MLR spans slightly larger mass interval than the theoretically estimated one.

In fact, there is a well-known relation between mass function (MF), luminosity function (LF) and MLR, written in V band as $$LF_V= MF \times |\frac{dM}{dM_{V}}|.$$ The term $|dM/dM_{V}|$ is the derivative of V band MLR. The golden rule about the three quantities is:“ if an LF presents features like peaks or dips, they should not be automatically attributed to features of the MF, before one has excluded the possibility that they correspond (even if not exactly) to expected features of the MLR” [@Antona98]. Synthetically considering the available observational LFs [@DM83; @Wielen83; @Scalo86; @Gould96; @Reid02], we find that it is safe to assume $0.17\leq a\leq0.30$ and $0.50\leq
b\leq0.60$, where $(a,b)$ is the mass interval spanned by the plateau feature. Let $D$ be the above-mentioned area in the $(a,b)$-plane. We note that Henry & McCarthy (1993) claimed according to their sampling mass and luminosity data $(a,b)=(0.18, 0.5)$ in D. In the following, we will improve this result, as well as the associated MLRs in various bands, in a more sophisticated and reliable way.

We choose a three-piecewise continuous model for the MLR, where linear model is used for the mass interval associated to the plateau, and two-order polynomial model for the other two [@HM93; @Henry99]. And, with a step of 0.01 [[$M_\odot$]{}]{}, the attempted precision for mass, we construct a set of two-dimensional grid points on the region $D$. Each grid point corresponds to a possible mass interval associated to the plateau. By fitting K, H, J and V band data, respectively, there are four $\chi^{2}$ of best fitting at each grid point. The searched $a,b$ then correspond to the grid point with the smallest sum of these four $\chi^{2}$.

In order to remove the data point prone to systematics, an iterative process is used. Firstly, using all the collected data, we map out MLRs in K, J, H and V bands. Though there are some data prone to systematics, the resulting MLRs are good first approximation. This is because the fact that most of these data have low accuracy and our weight assigning scheme effectively reduces their negative influence. Therefore, as a second step, the deviation of observational data to an MLR, approximated by the ones obtained in the first step, can be reasonably estimated. If the absolute expectation of this deviation is lager than 0.01, the data point corresponding to the maximum of absolute deviation is discarded. And the reduced data set is processed again through the two steps described above until the absolute expectation is less than 0.01. The discarded data are shown in italic in Table 1 and as open circles in Fig 2. It is evident that these data are indeed deviate from the final MLR.

The resulting K, H, J and V band MLRs are, respectively, $$\log M =\left \{\begin{array}{ll}
0.427-0.123M_K-0.000327M_K^{2} & \textrm{if $0.50<M<1.086$}\\
0.825-0.192M_K & \textrm{if $0.28\leq M\leq0.50$}\\
3.60\:\:-0.893M_K+0.0433\:M_K^{2} & \textrm{if $0.1<M<0.28$}
\end{array}\right.$$ $$\log M =\left \{\begin{array}{ll}
0.451-0.132M_H+0.00142M_H^{2} &  \textrm{if $0.50<M<1.086$}\\
0.865-0.191M_H & \textrm{if $0.28\leq M\leq0.50$}\\
1.87\:-0.430M_H+0.0139M_H^{2} &  \textrm{if $0.1<M<0.28$}
\end{array}\right.$$ $$\log M =\left \{\begin{array}{ll}
0.184-0.000028M_J-0.0111M_J^{2} &  \textrm{if $0.50<M<1.086$}\\
0.958-0.189M_J & \textrm{if $0.28\leq M\leq0.50$}\\
3.76\:-0.834M_J\:\:\:\:+0.0367M_J^{2} & \textrm{if $0.1<M<0.28$}
\end{array}\right.$$ $$\log M =\left \{\begin{array}{ll}
0.213-0.0250M_V-0.00275\:M_V^{2} & \textrm{if $0.50<M<1.086$}\\
0.982-0.128M_V & \textrm{if $0.28\leq M\leq0.50$}\\
4.77\:\:-0.714M_V\:\:+0.0224M_V^{2} & \textrm{if $0.1<M<0.28$}
\end{array}\right.$$

We show these MLRs in Fig 2 as solid curves. For comparison, we also plot the results of Henry & McCarthy (1993) (hereafter HM93) and Delfosse et al. (2000) (hereafter Del00) as dashed lines and dotted lines, respectively. Because Del00 improved the MLR in (0.1[[$M_\odot$]{}]{}, 0.7[[$M_\odot$]{}]{}), only the MLR in (0.7[[$M_\odot$]{}]{}, 1.0[[$M_\odot$]{}]{}) of HM93 is shown. The gray points with error bars are the data used in fitting MLR.

![Quantitative descriptions of K, H , J and V band mass-luminosity relations for low mass stars. The gray points with error bars are the data used in fitting MLR while the open circles are the discarded data, the solid curves show our fitted three-piecewise mass-luminosity relations, the dashed and the dotted lines are the results of Henry & McCarthy (1993) and Delfosse et al.(2000), respectively.[]{data-label="fig_3"}](ff3.eps){width="45.00000%"}

In comparison with HM93, we notice that our plateau region is more accord with the theoretical result. And From Fig 2, it’s easy to see our K, J and H band MLRs are lower than HM93. Because one more parameter is added to our MLRs, no definitive conclusion can be made through comparing $\chi^{2}$ directly. Therefore, F-test is used to evaluate whether adding parameters is statistically significant.

Following Lucy & Sweeney (1971) and Pourbaix & Jorissen (2000), comparing the old and new models, let $L$ be the number of parameters, $\chi^{2}$ be the weighted sum of the squares of the residuals, $N$ be the number of observational data, the efficacy of adding parameters can be measured by the ratio: $$F =
\frac{N-L_{new}}{L_{new}-L_{old}}\cdot\frac{\chi^{2}_{old}-\chi^{2}_{new}}{\chi^{2}_{new}}.$$ If the hypothesis that the old model is acceptable, $F$ is distributed as $F(L_{new}-L_{old},
 N-L_{new}$) [@BR1992] and the probability density function is: $$\Phi(F)=(1+\frac{F}{\beta})^{-(1+\beta)},$$ where $\beta=(N-L_{new})/(L_{new}-L_{old})$. If equation (9) gives $F=\overline{F}$, then the probability of $F$ could have exceeded $\overline{F}$ is: $$p=(1+\overline{F}/\beta)^{-\beta}.$$ In fact, $p$ is the first risk error of rejecting the null hypothesis while it is actually true. Usually 5% level of significance is adopted [@pb03], that is, if the calculated $p$ is smaller than 0.05, the new model can be definitely accepted.

Comparing with HM93, one parameter is added in near-infrared MLRs. According to equation (11), the calculated $p_K, p_J$ and $p_H$ (the subscripts K, J and H stand for the corresponding band) are all smaller than 0.05, which show the statistically significant improvement to HM93. Moreover, comparing the $\chi^{2}$ taking into account the different degrees of freedom, our $\chi^{2}$ are less than 50% of the corresponding value of HM93. These comparison results show our appreciable progress in constraining MLR. We list the values of $p$, $\chi^{2}/{(N-L)}$ in the second column of Table 2.

  -------------------------- ----------- ------- ----------- -------
                             this work   HM93    this work   Del00
  $p_K(\%)$                                                  
  $\frac{\chi^{2}_K}{N-L}$   2.76        7.21    2.27        3.52
  $p_H(\%)$                                                  
  $\frac{\chi^{2}_H}{N-L}$   2.59        6.63    1.29        2.13
  $p_J(\%)$                                                  
  $\frac{\chi^{2}_J}{N-L}$   4.56        14.57   1.84        2.17
  $p_V(\%)$                                                  
  $\frac{\chi^{2}_V}{N-L}$   2.23        2.26    4.91        11.63
  -------------------------- ----------- ------- ----------- -------

  : The comparison results

Del00 used a non-segmented fourth-order polynomial model in (0.1[[$M_\odot$]{}]{}, 0.7[[$M_\odot$]{}]{}). So for his MLR, the slope in the plateau is increased and in the other mass regions the slopes are decreased. On the basis of an increase of three parameters, the values of $p$ are calculated and listed in the third column of Table 2. From this table, we can see that our MLRs in K, H and V bands have statistically significant improvement to Del00. And the values of $\chi^{2}/(N-L)$ are reduced about 40%. For J band, due to the number of observational data is the fewest, our improvement is not as obvious as in the other bands and the $\chi^{2}/(N-L)$ is reduced about 15%.

For V band, the difference in stellar metallicity induces a large scatter while the near-infrared relations are much tighter [@Del00]. With observational estimations of metallicity, Bonfils et al. (2005) provided MMLR for the first time. We collect the latest data from Woolf & Wallerstein (2006) and with an improved three-piecewise K band MLR, an improved V-band MMLR is provided in the same way as Bonfils et al. (2005). The resulting formula writes: $$\begin{aligned}
M_V
\mskip-7mu=\mskip-7mu13.4\mskip-7mu+\mskip-7mu1.29M\mskip-7mu-\mskip-7mu22.3M^{2}\mskip-7mu+\mskip-7mu12.8M^{3}\mskip-7mu+\mskip-7mu1.26[Fe/H]\end{aligned}$$ for M $\epsilon[0.13,1.19]$[[$M_\odot$]{}]{}, \[Fe/H\] $\epsilon[-1.5,0.3]$.

Concluding remarks
==================

In this paper, based on a new reasonable weight assigning scheme, we try to make the best use of the present observations to restrict MLRs in K, J, H, V bands and MMLR in V band for low mass stars. Further improvements of this relation would have to come from future observations [@Henry04]. This is especially true for the transitional mass interval. Apart from its importance in estimating stellar masses, the empirical MLR in this mass interval is important in testifying simulations aimed at understanding the underlying physics. Since the precision of observations depends heavily on vastly different situations of individual stars, we believe the weight assigning scheme developed in the present paper will be helpful in the relevant studies. Furthermore, for relations like MLR, only good schemes can demonstrate those not obvious features thus can improve the corresponding physical investigation.

It should be pointed that, while it is very interesting to quantify how empirical MLR in V band depends on metallicity, further improvements to the presently available observations seem to be necessary. Besides, the age-dependency of empirical MLR, which remains to be poorly defined due to the fact that there are few confident observations available [@Henry04], also demands more painstaking observational efforts.

[*acknowledgments:*]{} [ We thank the reviewer whose critical comments help us a lot in improving the manuscript. We are also grateful to Dr. Todd J. Henry and Dr. X. Delfosse for their helps. This work is supported by NSFC 10473025 and NSFC 10233020.]{}

[99]{} Alonso A., Arribas S., Martinez-Roger C., 1994, A&AS, 107, 365A Bevington P.R., Robinson D.K., 1992, Data ruduction and error analysis for the physical sciences, McGraw-Hill, 2nd edition Bonfils X., Delfosse X., Udry S., Santos N. C., Forveille T., Ségransan D., 2005, A&A, 442, 635B Cester B., Ferluga S., Boehm C., 1983, Ap&SS, 96, 125C Copeland H., Jensen J. O., Jorgensen H. E., 1970, A&A, 5, 12C D’Antona F., Mazzitelli I., 1983, A&A, 127, 149D D’Antona F., 1998, ASPC, 142, 157D Delfosse X., Forveille T., Ségransan D., Beuzit J.-L., Udry S., Perrier C., Mayor M., 2000, A&A, 364, 217D Demircan O., Kahraman G., 1991, Ap&SS, 181, 313D Gould A., Bahcall J. N., Flynn C., 1996, ApJ, 465, 759G Grossman A. S., Hays D., Graboske H. C. Jr., 1974, A&A, 30, 95G Henry T. J., McCarthy D. W. Jr., 1993, AJ, 106, 773H Henry T. J., 1998, in ASP Conf. Ser. 134, Brown Dwarfs and extrasolar Planets, ed. R. Rebolo, E. L. Martin, M. R. Zapatero Osorio, 28 Henry T. J. et al., 1999, ApJ, 512, 864H Henry T. J., 2004, in ASP Conf. Ser. 318, The Mass-Luminosity Relation from End to End, ed. R. W. Hilditch, H. Hensberge, K. Pavlovski, 159 Hertzsprung E., 1923, Bulletin of the Astronomical Institutes of the Netherlands, Vol. 2, p.15 Laughlin G., Bodenheimer P., Adams F., 1997, ApJ, 482, 420L Lucy L.B., Sweeney M.A., 1971, AJ, 76, 544L Pourbaix D., Jorissen A., 2000, A&AS, 145, 161P Pourbaix D., 2000, A&AS, 145, 215P Pourbaix D., Boffin H.M., 2003, A&A, 398, 1163 Press W.H., Teukolsky S.A, Vetterling W.T., Flannery B.P., 1992, Numerical Recipes In Fortran, Second Edition, Cambridge University Press, p660 Reid I. N., Gizis J. E., Hawley S. L., 2002, AJ, 124, 2721R Russell H. N., Adams W. S., Joy A. H., 1923, PASP, 35, 189R Scalo J. M., 1986, FCPH, 11, 11S Söderhjelm S., 1999, A&A, 341, 121S Wielen R., Jahreiss H., Krüger R, 1983, in IAU Colloq. 76, The Nearby Stars and the Stellar Luminosity Function, ed. A. G. D. philip & A. R. Upgren, 163 Woitas J., Leinert Ch., Jahrei$\beta$ H., Henry T., Franz O. G., Wasserman L. H., 2000, A&A, 353, 253W Woitas J., Tamazian V. S., Docobo J. A., Leinert Ch., 2003, A&A, 406, 293W Woolf V. M., Wallerstein G., 2006, PASP, 118, 218W

[^1]: E-mail:xf@pmo.ac.cn
