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Distances of Galactic Supernova Remnants Using Red Clump Stars

We carry out a project to independently measure the distances of supernova remnants (SNRs) in the first quadrant of the Galaxy. In this project, red clump (RC) stars are used as standard candles and extinction probes to build the optical extinction (A$_V$) - distance(D) relation in each direction of extinction-known SNRs. 15 SNRs' distances are well determined. Among them, the distances of G65.8-0.5, G66.0-0.0 and G67.6+0.9 are given for the first time. We also obtain 32 upper/lower limits of distances, and the distances to G5.7-0.1, G15.1-1.6, G28.8+1.5 and G78.2+2.1 are constrained. Most of the distances measured by the RC method are consistent with previous results. The RC method provides an independent access to the distances of SNRs.


INTRODUCTION
Supernova remnants (SNRs) play key roles in the final evolution of stars, reshaping and heating the interstellar medium, and the birth of the high-energy cosmic rays. Reliable distances to SNRs are essential to constrain their physical parameters such as age, physical size, expansion velocity and explosion energy of the progenitor supernovae, which reveal the evolutionary process of SNRs. However, obtaining reliable distances of SNRs is a really challenging job. About 20% of Galactic SNRs have distance measurements (Green 2014a).
There are several popular methods to measure the distances to Galactic SNRs. Firstly, the kinematic method is based on the flat rotation curve of the Milky Way. By combining 21 cm HI absorption with CO emission, Tian et al. (2007) developed an improved way to measure the distances of the extended radio sources by minimizing the possibility of a false absorption spectrum. Their methods have been applied to several SNRs, e.g., SNRs Kes 69 &75, Tycho's SNR , 2011. Secondly, distance determinations to the shell-type SNRs can be inferred by the relation between the mean surface brightness (Σ) at a specific radio frequency and physical diameter (D) of an SNR, Σ=aD β . Distance is the ratio of physical diameter and the angular diameter (e.g. Clark & Caswell 1976;Milne 1979;Case & Bhattacharya 1998). Σ-D relation is frequently used since Σ is easy to be observed in radio bands for most radio SNRs. Thirdly, the distances can be accessible when SNRs are associated with the objects with known distances like OB associations (e.g. Cha et al. 1999, Vela remnant) or pulsars (e.g. Cordes & Lazio 2002). Additionally, the proper motion and the shock velocity can be used to calculate the distance (e.g. Vink 2008; Katsuda et al. 2008, Kepler's SNR). For the shell-type SNRs in the adiabatic phase, distances can be calculated by the X-ray flux and thermal temperature of X-ray -emitting gas (Kassim et al. 1994). Finally, the extinction measurements can also indicate distances (e.g. Chen et al. 2017;Zhao et al. 2018), which this paper focuses on.
Red clump (RC) stars are characterized by an obvious concentration region in the colour-magnitude diagram (CMD). They are usually low-mass stars in the early stage of core He-burning. Their helium cores almost have the same mass. Meanwhile their absolute magnitude weakly depends on metal abundance and ages in the K band (Alves 2000). Hence, RC stars are good enough to be standard candles in the infrared band. Assuming that intrinsic colour of RC stars is homogeneous, then their CMD spread along the colour is just caused by interstellar extinction traced by RC stars. Zhu et al. (2015) applied a similar measurement to determine the distance of SNR G332.5-5.6.
We closely follow the RC method and systematically measure the distances to 47 SNRs with known extinction in the first Galactic quadrant, with the aim of enlarging the reliable distance sample of SNRs. In Section 2, the method is described in detail. We summarize methods of measuring the optical extinction, the hydrogen column density and the distances of SNRs compiled from the literature in Section 3. Figure 1. Colour-magnitude diagram for 21032 stars within 0.5 deg 2 of G29.7-0.3, the grey colours denote stellar densities in the logarithmic scale. The red dot and lines show the fitted location of the RC peak density and its extent with 1σ. Figure 2. Histogram of the J − K values of the selected stars in the 11.1 < K < 11.4. The black curve is the best fit to this histogram to obtain the value of J − K for the peak density of RC stars. The red dotted curves are the Gaussian and power-law components, respectively.
The uncertainties of this method are analyzed in Section 4. In Section 5, we discuss our results and make a comparison with distances measured by other methods. Finally, a brief summary is given.

BUILD A V -D RELATION
To better illustrate this method, we start with G29.7-0.3 as an example. We extract the stars from the 2MASS Allsky Point Source Catalog in the J and K S (hereafter K) bands (Skrutskie et al. 2006) centered on the SNR in 1 • × 0.5 • ( l × b ) area. The reason why we choose the size of 0.5 deg 2 will be discussed later in this section. Their magnitudes in J and K bands are used to construct the CMD (K vs. J-K) since the RC stars are easy to be identified on CMDs (e.g. Gao et al. 2009). In Figure 1, RC stars are concentrated in the middle of the CMD. The bulk of stars in the left region of the CMD are predominantly main-sequence stars; those in the right are mainly dwarfs and red-giant-branch stars.
In principle, there is a maximal density of RC stars in each range of K apparent magnitude. We divide the stars sample into a number of horizontal strips in K through the CMD. The locations of the RC stars in different strips indicate different distances and reddening. The width of each strip is usually 0.3 mag and it will be extended to 0.5 mag or 0.7 mag when the counts of the RC's peak density are less than 10. The length of each strip in J − K is fixed by the RC's distribution in order to include most of the RC stars and minimize the contamination of the stars of other types. For each strip, we apply an empirical function to fit the histogram of star counts (Durant & van Kerkwijk 2006): Where (J−K) represents the stellar colour, A RCs and A C stand for the normalizations of the RC stars and the contaminant stars, respectively. The first term is a Gaussian of (J − K) peak and the width σ to fit the RC stars distribution; the second term is a power law to fit the contaminant stars. For instance, Figure 2 shows the best fit for the 11.1 < K < 11.4 strip: the stellar colour (J − K) at the peak density of the RC stars (J − K) peak is 1.56 mag and the σ is 0.19 mag. The (J − K) peak value is applied to calculate the average extinction of this field as equation (2). We assume that the intrinsic colour (J − K) 0 is 0.63 mag and the mean absolute magnitude of RC stars in K band is −1.61 mag. We discuss further in section 4. Then the extinction and the corresponding distance are derived from the following functions (Indebetouw et al. 2005): Where A V and A K are extinction in V and K bands, R V = 3.1 ± 0.18 (we discuss the value of R V in Section 3.1 ). In Equation 3, the conversion from A K to A V follows the empirical relation of Cardelli et al. (1989), which contributes an uncertainty of ∼3% to the optical extinction. This process is repeated for all strips until the 2MASS observational limit is reached. Since the extinction grows with the increasing distance in a given line of sight, there is a oneto-one correspondence between the extinction and distance. Hence, the distance of G29.7-0.3 is obtained by overlapping its extinction value on the A V − D relation in its direction. We tempt to find an optimal bin size for each SNR. On the one hand, a larger bin size can help increase the accuracy in determining the extinction. This would enlarge the amount of the RC stars, then decrease the uncertainty of RC's colour (J − K) peak . Meanwhile it would allow a narrow horizontal strip in m K that is used to perform the fits of Equation 1, then yield a better sampling of the pairs of extinction and distance. On the other hand, a smaller bin sizes can decrease the dispersion of extinction in the line of sight.
We present two typical examples (one with high stellar density, the other one with low stellar density ) to illustrate how to select the bin sizes. For the SNR with high stellar density in the sightlines, the A V -D relations are constructed within the bin sizes of 0.5 deg 2 (1.0 • × 0.5 • ), 0.125 deg 2 (0.5 • × 0.25 • ), 0.0625 deg 2 (0.25 • × 0.25 • , the smallest possible area around the target), respectively (See Figure 3 (a) ). It is found that the three A V -D relations are fully consistent with each other. In this case, the effect of bin sizes can be neglected when deriving the distance of SNRs. For the one with low stellar density , the A V -D relations are constructed within the bin sizes of 1.5 deg 2 ( l 1.5 • × b 1 • ), 0.5 deg 2 (1.0 • × 0.5 • ), 0.125 deg 2 (0.5 • × 0.25 • ,the smallest possible area around the target), respectively (See Figure 3 (b) ). The numbers of sampling points are 9, 7, 6, respectively. The A V -D relations of 0.5 deg 2 and 0.125 deg 2 are almost the same while the data points of 1.5 deg 2 are systematically lower. As equation (2) shows, A V is determined by the stellar colours of RC stars located in the peak density (J − K) peak . The lower A V for 1.5 deg 2 is a result of 1.5 deg 2 concentrating in stellar colour(J − K) peak lower than those of 0.5 deg 2 and 0.125 deg 2 . The A V -D relations indicate that extending the area to 1.5 deg 2 might enlarge extinction dispersion. In principle, we choose a larger size of bin when the dispersion of extinctions is at the same level. The of 0.5 deg 2 is a good balance for the two examples. Then, we test the bin sizes of 0.5 deg 2 and 0.125 deg 2 for the objects almost at an interval of 15 • in the longitude. In addition to Figure 3, Figure 4 presents the results in the four typical directions. In each of the panels, the two curves are consistent with each other, except for the number of sampling points within 0.125 deg 2 is less than that of 0.5 deg 2 for some objects. Hence, we usually build the A V -D curve within 0.5 deg 2 when deriving the distance of an SNR.
In recent years, both theory and observations have shown the fine structure of RC, which includes two subclass stars, the main RC stars and the secondary RC stars (SRCs) . The main RC stars with low mass that we usually take as standard candles have almost the same luminosity and electronic-degenerate core; while the SRCs whose luminosity are greatly changed contain non-degenerate He-cores (Girardi 1999). The largest sample of RC stars identified based on LAMOST survey DR3 (Wan et al. 2015(Wan et al. , 2017 shows the ratio of the main and secondary RC stars is about 3:1. To investigate the effects of the SRCs for this method, we test to use a double Gaussian function to fit the distribution of RC stars of each strip. No apparent secondary Gaussian component can be found. Hence, the effects of the SRCs can be neglected in this method.

COMPILATION OF A V , N H AND D
Drawing from the catalogs of Green (2014b) and Ferrand & Safi-Harb (2012), we investigate each of 161 SNRs in the first Galactic quadrant. Among them, 47 SNRs have access to the optical extinction or hydrogen column density data in the literatures. We collect their parameters on optical extinction A V , hydrogen column density N H and the distance D, then discuss the methods for determing determine the three parameters. The three parameters and the corresponding methods are listed in Tables 1 and 2 . 3.1. Optical extinction A V      The interstellar extinction is the absorption and scattering of electromagnetic radiation by dust and gas. The most common method to obtain optical extinction A V is measuring the reddening via the intensity ratios between the two emission lines and converting the reddening into the colour excess E B−V . Then we gain the extinction values A V = R V × E B−V . The total to selective extinction ratio R V is ∼ 3.1 for the diffuse interstellar medium in the Milky Way, which is widely used (e.g. Fitzpatrick 1999;Draine 2003;Schlafly & Finkbeiner 2011). Schlafly et al. (2016) measured the reddening of 37,000 stars in the Galactic disk based on APOGEE, PS1, 2MASS and WISE data, then determined the uncertainty of the R V , ∼ 0.18. Although R V has significant variance in some regions, there is a wide wedge of intermediate R V in the first quadrant (see the Figure 3 of the Schlafly et al. (2017)). Hence it is robust to adopt R V = 3.1 ± 0.18. T he uncertainties of A V are approximately The frequently used line ratio we present in Table 1  . The group lines from the transition with the same upper level weakly depend on the physical conditions such as temperature and density of the gas. Hence, they are feasible to estimate the extinction to the extended sources (e.g. Oliva et al. 1989, SNRs and pulsar wind nebulae). And another approach is to measure the extinction of the individual stars with known distances that are associated with an SNR.

N H
Hydrogen column density N H is usually used to approximately denote X-ray extinction which is caused by any element not fully ionized, especially the abundant heavy elements when energy is above 0.25 keV. The dust grains of the same abundant heavy elements also contribute to the optical extinction A V (Güver & Özel 2009). Both theoretical and observational studies for decades have indicated that there should be a reasonable correlation of A V and N H . We adopt the latest value N H /A V = (2.04 ± 0.05) × 10 21 H cm −2 mag −1 for the first and fourth Galactic quadrants (Zhu et al. 2017).The conversion error is approximately esti- More than half of SNRs have been detected in X-ray band (Ferrand & Safi-Harb 2012). About 30% of SNRs are associated with optical emission (Green 2014b). Therefore, we can obtain more A V values transformed from the N H . N H is usually derived from the best fitting of X-ray spectrum. Here we only collect the N H derived from solar abundances (Anders & Grevesse 1989) to keep interstellar abundances consistent in the whole transition. If the uncertainty of N H has not been given in the literature, we use the average errors of the N H , 20%, which are derived from our sample with known uncertainties.

Distance
The distance measurements of SNRs mainly include the Σ-D relation, the kinematic method, the proper motion measurements, extinction measurements, Sedov estimates, and associated objects with known distance. Due to different distance measurements with varying uncertainties, we select distances of SNRs in the literature with the following priority: kinematic method, proper motion estimates, associated objects with known distance, Sedov estimates and Σ-D relation.

UNCERTAINTY ANALYSIS
The uncertainties of the derived distances to SNRs are mainly attributed to the errors of the SNRs' A V and the RC's distances. The errors of SNRs' A V are calculated by a standard deviation formula. Here, we focus on the discussion on the errors caused by RC stars. The errors of RC's distances mainly include the dispersion of absolute magnitude and extinction traced by RC stars.
The absolute magnitude M K of RC stars is extensively studied from the perspective of observations, especially in the K band (e.g.  Groenewegen (2008) based on the revised Hipparcos parallaxes. Taking all of these studies into consideration, the variance of the absolute magnitude M K leads to a systematic uncertainty of 0.1 mag that contributes about 5% error in the mean distance calculated by equation (3).
The uncertainties of extinction are caused by the dispersion of intrinsic colour and the random errors of the Gaussian fitting. The absolute magnitudes of RC stars are more sensitive to [Fe/H] and age in the J band than the K band, which leads to the variation of the intrinsic colour (J − K) 0 (Güver et al. 2010). The values of intrinsic colour (J − K) 0 for RC stars concentrate in the range from 0.5 to 0.75 mag (e.g. Yaz Gökçe et al. 2013;Grocholski & Sarajedini 2002). We adopt (J − K) 0 as 0.63 mag, and assume that its dispersion is 0.1 mag which leads to about 3% uncertainty in the distance, according to equations (2) and (3). Meanwhile we estimate the uncertainty of mean colour on the peak density location of RC stars : σ J−K = σN −1/2 RC , where N RC = A RC σ √ 2π is the sum of RC stars in each magnitude strip (Durant & van Kerkwijk 2006). It is a good estimate if the Gaussian fit is valid and the contamination is not significant. The typical error of the mean colour is about 0.05 mag, which brings ∼ 2% uncertainties in distance estimation.
In summary, the systematic uncertainties of the distances traced by the RC stars are about 10% in total.

RESULTS AND DISCUSSION
We measure the run of reddening along distance using the RC method in each line of sight of 47 SNRs in the first Galactic quadrant. Among them, 32 SNRs' extinctions are beyond the range of A V traced by the RC method, hence the upper/lower limits of distances are obtained. Fortunately, there are 15 SNRs' extinction bands overlapping with the extinction measured by the RC stars, which provides an opportunity to estimate the distance accurately and with precision. Figure 5 presents the CMDs with the locations of RC's peak density for each of the 15 SNRs. Figure 6 shows the corresponding A V -D relations and the probability distribution over distance to the SNRs.

Derive the Distances of SNR
To determine the distances of SNRs, we calculate the probability distribution of distance using the product of two probability distributions and marginalizing over the extinction (Güver et al. 2010): where P SNR (A V ) presents the probability distribution of an SNR's extinction. We assume P RC (D|A V ) = P RC (A V |D). P RC (A V |D) presents the distribution of the extinction traced by RC at each distance bin. Both distributions are denoted as Gaussian functions.
In Figure 6 (right column), the panels show the probability distributions over distance calculated by equation 5. Then, we fit these distributions with a Gaussian function, yielding the distance with the highest probability. For the objects with good Gaussian fitting, the uncertainty of the distance is equal to the standard deviation of the Gaussian. However, for some objects there are apparent and sudden decreases in the distance probability. The red lines mark such decreases (see Figure 6). In this case, the uncertainty of distance reflects the cut-off distance. The results are listed in Tables 1 and 2.

Summarize the results
We obtain 15 new distances, three of which are given for the first time. G65.8-0.5 1 , G66.0+0.0 and G67.6+0.9 are identified as SNRs by Sabin et al. (2013). We estimated their distances as 2.4 kpc, 2.3 kpc, 2.0 kpc, respectively. Note that G66.0+0.0 is not detected in the most sensitive Galactic Plane surveys (Anderson et al. 2017). Maybe more observations are needed for its classification.
We have also given 20 lower distance limits and 12 upper distance limits. Among them, the distances of 4 SNRs have been further constrained by combining the lower or upper limits inferred by the RC method and the previous results.
The distance of G5.7-0.1 is ambiguous, at either 3.1 or 13.7 kpc, as inferred by the OH maser velocity (Hewitt & Yusef-Zadeh 2009). The RC method's distance is about 2.9 kpc. Therefore, we predict its distance is around 3 kpc. Boumis et al. (2008) suggested a lower limit distance of 2.2 kpc for G15.1-1.6. The RC method gives an upper limit distance of 2.1 kpc. Hence, we conclude that the distance of G15.1-1.6 is around 2.2 kpc.

Discussion
We have estimated distances for 15 SNRs and upper or lower limits for 32 SNRs using the RC method.
To further understand the precision of the distances indicated from the RC method, we compare these results with the previous studies by two steps. Firstly, we compare 8 new SNRs' RC distances with their kinematic distances. The kinematic distances are denoted as D Kinematic . When the corresponding uncertainties are not given by the literature, we will empirically assume 20% of their distances as uncertainties. The distances measured by the RC method are denoted as D RC . Assuming that each measurement with no errors for the same source, then regression function will be D RC = D Kinematic . The MPFITXY routine based on the MP-FIT package is used to fit a straight line via data with errors in both coordinates (Markwardt 2009;Williams et al. 2010). In Figure 7(a), the fitted regression equation, D RC = (1.0 ± 0.1) × D Kinematic , means the RC distances of SNRs are highly consistent with their kinematic method within the range of uncertainty. Secondly, we compare 44 distances constrained by the RC method with the corresponding distances measured by other methods. As Figure 7 (b) shows, all distances estimated by the RC method are all in the range of 1.5-8 kpc, which is consistent with our expectations. All 20   lower limits of SNRs' distances coincide with the trend of the fitting lines, while 6 of the 12 upper limits are in agreement with other distance measurements. We conclude that most of RC distances are in agreement with the previous measurements, and the lower limit distances are more reliable. Therefore, SNRs' distances can be independently constrained by the RC method.
We analyse the reasons why the RC method only can trace either upper or lower limits. For 20 relatively distant SNRs, we only draw their lower limits for two reasons. One restricted condition is the 2MASS Survey completeness limit as J=15.8, K S =14.3 mag. The other is that the RC stars are likely mixed with the highly reddening main-sequence stars in CMDs when the apparent magnitudes begin to be fainter than 13 mag (see Figure 5). For 12 relatively closer SNRs, we only obtain their upper limits because the sample of RC stars is not enough for statistics when their apparent magnitudes are brighter than 9 mag in most cases. Hence, the RC method based on 2MASS data can be effective in the range from 1.5-8 kpc and the specific range of distance it can trace depends on the RC stars sample in a given direction.
We next check the seven discrepant measurements. First, the differences between old and new distances for G32.8-0.1, G39.7-2.0, G73.9+0.9 are less than 30%, which may be caused by the uncertainties of the two different methods. Then, the key investigation is conducted on the other four SNRs. The optical extinction values towards SNRs G13.3-1.3, G85.9-0.6 are smaller than 1 mag. We expect that the extinction values are not sensitive to the distance when extinction is extremely slight in the line of sight. For G32.1-0.9, the difference between RC and kinematic distance is greater than 50% likely due to the large error of N H that is up to 40%. Note the distance of G67.7+1.8 listed in Table 1 is not reliable since its probability distribution over distance is not well fitted by a Gaussian function. It might be product of the broad range of the SNR's A V and the slight extinction which is much lower than the average magnitudes of extinction per kiloparsec (c V ∼ 0.7mag kpc −1 ) along the line of sight (Indebetouw et al. 2005). These discrepant results suggest that the slight extinction or large uncertainties of extinction might significantly affect the accuracy of the RC method.

SUMMARY
We have taken advantage of the RC stars from 2MASS data to construct the A V -D relations along the directions of 47 SNRs in the first Galactic quadrant. In total, 15 distances and 32 upper or lower limit distances of SNRs have been obtained by overlapping their extinction values on the A V -D relations. Among them, the distances of SNRs G65.8-0.5, G66.0-0.0 and G67.6+0.9 are estimated as 2.4 kpc, 2.3 kpc, and 2.0 kpc for the first time. Distances of SNRs G5.7-0.1, G15.1-1.6, G28.8+1.5 and G78.2+2.1 have been better constrained as about 3 kpc, 2.2 kpc, 3.4 kpc, and 1.9 kpc, respectively.
By comparison, distances estimated by the RC method are consistent with other measurements within the range of the allowed errors. In addition, the RC method tends to give a reliable lower limit distance. In addition, we analyze the possible reasons why six upper limits are incompatible with the previous results. Finally, we highlight the RC method can independently constrain distances of SNRs in the range of 1.5 kpc to 8 kpc and the distances can be well determined by this method when the samples of RC stars are relatively abundant along the line of sight.
We all acknowledge supports from NSFC program (11473038, 11603039, U1831128). We appreciate it that the referee provided constructive comments and useful suggestions. We also thank Dr. Liu Chao, Dr. Xiang Mao-Sheng, Dr. Chen Bing-Qiu and Dr. Wan Jun-Chen for helpful discussion. This publication makes use of data products from the Two Micron All Sky Survey and Software: MPFIT (Markwardt 2009)