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The Distance Measurement of NGC 1313 With Cepheids

We present the detection of Cepheids in the barred spiral galaxy NGC 1313, using the Wide Field and Planetary Camera 2 on the Hubble Space Telescpoe. Twenty B (F450W) and V (F555W) epochs of observations spanning over three weeks were obtained, on which the profile-fitting photometry of all stars in the monitored field was performed using the package HSTphot. A sample of 26 variable stars have been identified to be Cepheids, with periods between 3 and 14 days. Based on the derived period-luminosity relations in B and V bands, we obtain an extinction-corrected distance modulus of mu = 28.32 +- 0.08 (random) +- 0.06 (systematic), employing the Large Magellanic Cloud as the distance zero point calibrator. The above moduli correspond to a distance of 4.61 +- 0.17 (random) +- 0.13 (systematic) Mpc, consistent with previous measurements reported in the literature with uncertainties. In addition, the reddening to NGC 1313 is found to be small.


INTRODUCTION
Cepheids are luminous variable stars with radial pulsations, which have been proven to be reliable distance indicators with the well-established period-luminosity relation in both observational and theoretical aspects. Given their intrinsic visual brightness range from −2 to −6 mag, Cepheids are ideal standard candles at Galactic and extragalactic distances, and provide useful calibrations for secondary extragalactic distance indicators including for example Tully-Fisher relation and planetary nebular luminosity functions.
Thanks to the improved resolution of the Hubble Space Telescope (HST), many Cepheids have been discovered in the galaxies in the Local Supercluster and the distances to these galaxies have therefore been determined. This campaign extends the distance scale that can be determined reliably by a factor of 10 as compared to the pre-HST era, when Cepheids were mostly detected and studied from the ground. For a more detailed understanding of Cepheids distance scale and its implications to the local universe, the readers are referred to the classic review by Madore & Freedman (1991).
The nearby starburst galaxy NGC 1313 is a spiral galaxy with morphological type SB(s) (de Vaucouleurs et al. 1991). It is often described as a galaxy in transition between the Magellanic type galaxies and normal disk galaxies such as M33, given its high rotation velocity and very little or no radial metallicity gradient (Mollá & Roy 1999). Being very active in star formation, NGC 1313 and its super shell nebula have been extensively studied (e.g. Ryder 1995;Pellerin et al. 2007), as well as the X-ray sources NGC1313 X-1 and X-2 (Feng & Kaaret 2005, 2006Liu et al. 2007Liu et al. , 2012 and typeII supernova SN1978K (Ryder et al. 1993). The distance to this galaxy has been determined previously using the methods including for example supernova radio flux density and H II region size distribution, which are listed in Table 1 and show modest diversity. Securing an independent distance measurement using Cepheids in NGC 1313 is necessary in this context.

Observation and data analysis
The sky region around NGC 1313 X-2 was monitored 20 times over three weeks in 2008 with HST WFPC2 in the F450W and F555W filters for the program GO-11227 (PI: Jifeng Liu), which was aimed to search and identify periodicity of NGC 1313 X-2 in the optical. For each observation, two 500 s exposures in F450W were taken, followed by two shots of 400 s and 700 s in F555W. The observations were calibrated with the best calibration files on 2008 August 26, and standard photometric analysis were performed on each frame using the photometry package HSTphot (Dolphin 2000), which is designed to handle the undersampled point-spread functions (PSF) in WFPC2 images. The package conducts procedures for cleaning cosmic rays, removing hot pixels, fitting PSF, and applying aperture corrections to secure precise photometry in the VEGAmag photometric system.
In total, 32482 sources are detected and identified as well-photometered stars in the 5.7 arcmin 2 of NGC 1313 covered by the WFPC2 field-of-view. Further investigations will be based on them.

Variable Star Search and Cepheids Identification
Only well-photometered point sources are included for the search of variable stars. Following a similar approach as Dolphin et al. (2003) in their study of variables in Sextans, we define a variable star with magnitude variation σ mag 0.15 mag and chi square χ 2 ν 2, where Periods of these variable star candidates are determined using a modified Lafler-Kinman algorithm (Lafler & Kinman 1965), but with a relatively loose criterion of minimum Θ 0.85 in either B or V light curves for in order not to miss any periodic variables. Among the total 32, 482 stars, 1616 are found to be variables, and ∼ 65% of them show periodic variations.
The light curves of these variable stars with periodic variations were visually examined, and 26 of them are found to be Cepheid on the basis of their distinctively rapid brightening, followed by a long, linear decay phase. They are presented in the color-magnitude diagram (CMD) as in Fig. 1 as red filled circles, along with all 32, 482 stars in black dots.

Light curves and average magnitudes for Cepheids
In Table 2 and 3, V -and B-bands time-series photometry for the Cepheids are presented for each epoch, along with photometric uncertainties, respectively. We determined the pulsation period and the average magnitudes in V -and Bbands for each Cepheid by using template fitting procedures. The template fitting method based on Fourier analysis and Principal component analysis (PCA) was firstly introduced by Stetson (1996) and Kaluzny, J. and Stanek, K. Z. and Krocken (1998). It can obtained more robust periods and precisely measure average magnitudes even in few observational epochs and bad-sampled light curves. Yoachim, P. and McCommas, L. P. and Dalcanto (2009) and Pejcha, O. and Kochanek, C. S. (2012) expanded on these methods to build new templates for analysis of HST Cepheids data. The templates of Yoachim, P. and McCommas, L. P. and Dalca (2009) include BV I bands and short-period Cepheids (< 10 days) in the analysis. It was ideal for characterizing our discovered Cepheids.

The training set included the light curves of fundamental (FU) type Cepheids stars comes from OGLE database for the LMC (Udalski et al. 1999a), SMC (Udalski et al. 1999b)
and VizieR database for the Milky Way (Ochsenbein, F. and Bauer, P. and Marcout, J. 2000). We put in a single fake Iband observation as there are no I band photometry. Considering there have fewer data points in Bband and deblending problems can cause outlier points in light curves, we fixed all additional PCA parameters in fitting procedure. As a result, the magnitudes and periods were determined through template fitting. The best-fit light curve templates also generated from obtained parameters.
The light curves of photometry data along with their best-fit templates (solid lines) in V -and Bare plotted in Figure 2 with their Ids, periods, V and B magnitudes. The average V -and Bmagnitudes and their measurement uncertainties for each Cepheid are given in Table 4. The χ 2 /dof for the fitting are given in the last column.

PL relations and the distance to NGC 1313
The phase-weighted apparent To determine the distance to the galaxy NGC 1313, the apparent PL relations need to be calibrated with the absolute PL relations in LMC galaxies, which contain a large number of Cepheids and have a well determined distance modulus of 18.493 ± 0.008 (statistical) In order to avoid incompleteness bias in fitting a slope to the NCG 1313 data, only the intercept of the linear regression was fitted, while keeping the slope fixed to the LMC values. We shift the NGC 1313 magnitudes to match those of LMC, and determine the relative distance moduli by minimizing the best-fit chi square of the linear fitting of the combined data set. As a result, the apparent relative distance moduli between NGC 1313 and the LMC are measured to be ∆µ V = 9.893 ± 0.049 mag and ∆µ B = 9.872 ± 0.057 mag.

The true distance modulus and its uncertainties
The above derived distance moduli need to be corrected for foreground extinction and reddening effect. The true absolute wavelength-independent distance modulus to NGC 1313 can be expressed as a function of the relative apparent modulus ∆µ between NGC 1313 and LMC, the LMC distance modulus, and the extinction for NGC 1313, as: Where the reddening law of for NGC 1313 and the Magellanic Clouds (Ferrarese et al. 1996). The true modulus to µ LM C = 18.493 mag (Pietrzyński et al. 2013). By subtracting Equation 3 to Equation 2, we yield the extinction of NGC 1313:

Based on Equation 3 and Equation 2
, we found the apparent distance moduli for NGC 1313 are then µ V = 28.386 ± 0.049 mag and µ B = 28.365 ± 0.057 mag. Those two modulus agree to each other within errors. That implied zero reddening. We computed an errorweighted average value as the true distance modulus to NGC 1313 µ = 28.377 ± 0.038 mag instead of applying dereddening precedure of Equation 4.
In the above results,errors for the distance modulus consist of two independent factors: random errors and systematic errors. The random errors are calculated using Equation (18) of Saha et al. (2000), propagated mainly from the errors on stellar photometry and intrinsic widths of the PL relation in V and B: The first item in the right side is the estimated photometric errors for the mean V and B magnitudes, as propagated through the dereddening procedure, where σ V = 0.029 and σ B = 0.070 mag respectively. The second item is associated with the intrinsic width of the PL relation (i.e., a given Cepheid may not be on the mean ridge line of the PL relation), which is σ PL,V = 0.046 mag, and σ PL,B = 0.050 mag.
Several systematic errors also come into play: uncertainties due to the distance modulus of zero point galaxy, which is 0.008 (statistical)±0.047 (systematic) mag for LMC (Pietrzyński et al. 2013), errors derived from the observed scatter in PL relation of LMC and (all lower than 0.02 mag), uncertainties due the metallicity dependency of the PL relation. Walsh & Roy (1997) performed multifiber emission-line spectrophometry of 33 H II regions in NGC 1313, and derived a flat disc metallicity of log 10 (O/H) = −3.6 ± 0.1, which is very similar to that for LMC with log 10 ( We list all our known sources of error in table 5. The distance modulus based on our 24 Cepheids is 28.32 ± 0.10 (random)±0.06 (systematic), giving a corresponding distance of 4.61 ± 0.21 (random)±0.13 (systematic) Mpc.

Discussion
We have derived a Cepheid distance to NGC 1313 based on V -and B-bands HST WFPC2 observations. A total of 26 Cepheids have been found with periods ranging from 3.12 to 14 days, with their phase-weighted average magnitudes in both bands and periods in V determined. Apparent V and B distance moduli to NGC 1313 have been derived, employing the LMC Cepheids samples to be zero point reference galaxies. After applying extinction corrections, the true distance modulus to NGC 1313 is 28.32±0.10 (random)±0.06 (systematic) using LMC for the calibration of distance. These moduli correspond to distance of 4.61 ± 0.21 (random)±0.13 (systematic) Mpc. The Cepheid distance found in this work is slightly larger than previous measurements, but are consistent within errors with most results, especially those obtained using tip of RGB stars (summarized in Table 1).
An important factor on distance measurement is that Cepheids are oscillating in two modes -fundamental (FU) and first over-tone (FO). Cepheids with periods shorter than 8 days are likely to be oscillating as a first over-tone pulsator that appears brighter and bluer than fundamental Cepheids. To confuse this two type Cepheids would make the distance underestimated. Since the PL relations are calculated by only using FU Cepheids from OGLE database, we must make sure that our 26 Cepheids are all belong to this type. Beaulieu et al. (1995) studied large sample of Cepheids light-curve in 490 nm in LMC, and found that the first over-tone (FO) Cepheids have a strong tendency to be more symmetric, and having lower amplitudes in light curves than the FU Cepheids. Fig. 4 presents the amplitudes against their periods of 26 Cepheids in V and 6 in B with detections in all 20 observations, along with the 94 LMC Cepheids from Beaulieu et al. (1995). It is obvious that their LMC FU and FO Cepheids occupy quite different regions in the Amplitude-Period (AP) plane. All of our Cepheids are within the area where LMC FU Cepheids are located, and they should be of FU type.
The extinction of NGC 1313 is also obtained when deriving the distance modulus. Based on Equation 5, we found the extinction to NGC 1313 is E(B − V ) = −0.01 ± 0.07. The value for the estimated extinction is low compare with previous studies around the region of NGC 1313 X-2. An extinction of E(B-V) = 0.11 mag can be obtained from the Schlegel et al. (1998) dust map as the Galactic average value within 0.6 • of NGC1313 X-2. An extinction of E(B-V) = 0.44 mag (corresponding to n H = 2.7 × 10 21 cm −2 ) is inferred from the power-law fit to the X-ray spectrum of NGC 1313 X-2 (Miller et al. 2003), which also includes contribution from circumbinary materials. The extinction value we obtained (< 0.03) through Cepheids suggests that the average extinction within 2 ′ of NGC1313 X-2 is slightly smaller than the Galactic average value within 0.6 • of NGC1313 X-2.      Table 4:: The final sample of 26 Cepheids, with their IDs, coordinates and periods presented in Columns 1-4. Final photometry, photometric error, and number of observations in V are given in Columns 5-7, while those in B in Columns 8-10. The χ 2 /dof for the fitting are given in the last column.    Beaulieu et al. (1995). The NGC 1313 Cepheids clearly belong to the FU type.