M39 Open Cluster Distance Calculation

Photometric measurements were conducted to determine the distance of the M39 open cluster. For inclusion, field stars were required to have cluster membership probably greater than 50%. Based on main sequence fitting the cluster distance modulus was estimated to be 7.14 (268 pc).


Open clusters are a crucial step in determining astronomical distances. M39 (NGC 7092) is a loose open cluster near the galactic plane. On the basis of proper motion measurements Platais (1984) found 62 stars with M39 cluster membership probability greater than 50%. The aim of this work was to do photometry on these 62 stars and construct a color magnitude diagram to derive cluster distance.

Distance to stars within a few hundred light years can be measured directly by parallax but most star clusters are more distant. One method to derive star cluster distance uses main sequence fitting which relies on stellar physics. This is the method used here.

The color and magnitude of many stars are related. Blue stars have higher luminosity than red stars. A star’s color can be measured as the difference between a star’s apparent magnitude through a blue (B) filter and a star’s apparent magnitude through a yellow-green (V) filter. This quantity is B-V. Since all stars in a star cluster lie at about the same distance, when B-V vs. apparent magnitude in V is plotted for many stars in a cluster, a Main Sequence pattern begins to form. An example Main Sequence is shown in Figure 1.

Main Sequence
Figure 1. Example Main Sequence

A star’s absolute magnitude can be calculated from a star’s apparent magnitude if its distance is known. Nearby stars with known distances and therefore known absolute magnitudes produce a Color Magnitude diagram with a similar shape to the diagram above but with absolute magnitudes brighter than any known star cluster. The vertical difference between the main sequences is due to the star cluster’s distance. The magnitude difference can be used to calculate the star cluster’s distance using the distance modulus equation:

d = 10 ^ (0.2 * (modulus + 5))

where modulus is the magnitude difference and d is in parsecs.


Observations were made in a single night in July 2016 with a Takahashi TOA-150 150mm Refractor, 1095mm FL, Johnsons-Cousins BV filters and an SBIG STL-11000 3 CCD large format camera, 1.69 arc-seconds / pixel. The observing location was AstroCamp, Nerpio, Spain. Two 60-second exposures were taken, one in each filter. The exposure time was selected to prevent stars in M39 from saturating and to minimize slight variations in the timing of the shutter.

The IRAF software package V 2.16.1 was used for analysis of the CCD images. The DAOFIND task was used to locate stars. The PHOT task was used to derive instrumental magnitudes. IRAF tasks FITPARAMS and INVERTFIT were used to transform instrumental magnitudes to standard magnitudes. The equations to transform instrumental magnitudes to standard magnitudes were:

B = b + 4.1427 - 0.0416 * (b - v)
V = v + 4.3746 + 0.0530 * (b - v)

where B and V are standard magnitudes and b and v are instrumental magnitudes.

To integrate the light received from a star an aperture of 10 pixels was used. Changing the aperture a few pixels up or down made little difference. The annulus used was 16 and dannulus was 9. Star images were on average 2.4” FWHM.

Three reference stars in the field were selected based on their separation from other field stars. Magnitudes were obtained from SIMBAD [2]. The three reference stars are shown in Table 1. Using Landolt stars instead of field reference stars would have increased photometry accuracy but would have required additional exposures.

Table 1. Reference stars

Processing with IRAF

IRAF processing followed the steps described in "A User's Guide to Stellar CCD Photometry with IRAF" by Philip Massey and Lindsey Davis. FITS data files used were: M39 B Filter and M39 V Filter. A Python script was written to automate these steps. The Python code requires the PyRAF, numpy and AstroPy packages. The processing steps were:

1. Run hedit to edit the FITS header in each FITS file to add any missing header keywords. Each FITS file must contain: AIRMASS, EXPOSURE, RA, DEC, EPOCH, DATE-OBS, FILTER and ST. Airmass at the start of the exposure was used since it was assummed airmass did not change appreciably during the short exposure.

2. Run daofind to locate target stars for each image (B and V)

a. epadu = 0.8 (from SBIG spec sheet)
b. read noise = 11 (from SBIG spec sheet)
c. sigma = 15
d. fwhmpsf = 4.0
e. threshold = 20
f. datamin = 40
g. datamax = 64000
Input files: M39-B.fit, M39-V.fit
Output files: B-coordinates.txt, V-coordinates.txt

3. Create a daofind output file manually with 3 reference stars only

Output files: B-coordinatesReference.txt, V-coordinatesReference.txt

4. Run phot to determine instrumental magnitudes for each image (B and V)

a. annulus = 16
b. dannulus = 9
Input files: B-coordinates.txt, V-coordinates.txt
Output files: B-phot.txt, V- phot.txt

5. Run phot to determine instrumental magnitudes for each image (B and V) for reference stars

a. annulus = 16
b. dannulus = 9
Input files: B-coordinatesReference.txt, V-coordinatesReference.txt
Output files: B-photReference.txt, V- photReference.txt

6. Create the files imageset, apercors and shifts manually

Output files: imageset.txt, apercors.txt, shifts.txt

7. Create local Landolt catalog. Copy onlandolt.dat file and add reference stars.

8. Run mknobsfile to create a single file with instrument magnitudes combined from all filters for reference stars

a. tolerance = 4
Input files: imagesets.txt, shifts.txt
Output files: references.txt

9. Run mkconfig to edit transformation equations

Output file: ctio.cfg

10. Run fitparams to compute transformation equation coefficients

a. tolerance = 1E-8
b. maxiter = 25
Input files: references.txt, ctio.cfg
Output file: parameters.ans

11. Run mkobsfile to create a single file with instrument magnitudes combined from all filters for target stars

a. tolerance = 4
Input files: B-phot.txt, Vphot.txt, imageset.txt, shifts.txt, apercors.txt
Output file: observations.txt

12. Run invertfit to create calibrated magnitudes from instrumental magnitudes

Input file: observations.txt, parameters.ans, ctio.cfg
Output file: calibrated.txt

13. Plate solve one of the images and use to convert X, Y pixel locations to RA and DEC

a. Ra/Dec data from:

14. Map X, Y pixel to star number

15. Filter by membership probability

a. Probability data from: http://www.univie.ac.at/webda/cgi-bin/frame_data_list.cgi?ngc7092+prob+prob.mu

16. Use invertfit output file to map star number to standard magnitude


Cluster membership can be accomplished using one or more criteria: photometric, kinematic, statistical and spectroscopic. If the probable error of an individual proper motion is of the order of ±0.0001 arcsec, proper motion provides a reliable segregation of cluster members from field stars (Vasilevskis 1962). Using plates taken 70 years apart, Platais found common proper motions for stars in the M39 field on the order of 0.15 arcsec.

Photometry was performed on the 62 stars with M39 cluster membership probability > 50%. Of the 62 stars, 12 stars were discarded due to saturation, 6 stars were out of the field of view and 2 stars were not resolved from nearby neighbors. This left 42 stars and represents a sample for the cluster from about V=9.0 to about V=13.75 mag. All 42 remaining stars fall on or near the main sequence. The derived standard magnitudes of the 42 stars are shown in Table 2 along with the standard magnitudes and cluster membership probabilities given by Platais.

Table 2. Photometry

M39 Color Magnitude Diagram

A Color Magnitude diagram was constructed from standard magnitudes obtained from photometry. The Color Magnitude diagram [mV, (B-V)0] for the 42 members of M39 is plotted in Fig. 2. From the diagram it can be seen that no cluster member has yet reached the red giant phase of the stellar evolution which implies this is a young star cluster.

M39 Color Magnitude Diagram
Figure 2. M39 Color Magnitude Diagram

Photometry Error

Compared to values from Platais, this study has a mean difference of mV=0.03 mag and a B-V mean difference of 0.05 mag.

For the transformation equations, the RMS Fitting Error in V was 0.0106 and in B the RMS error was 0.0237.

M39 Reddening

Additional shifts in magnitude and color index result from interstellar absorption. E(B-V) = 0.025 (B. J. Anthony-Twarog 1984). Taking the value of R = 3.25 (Moffat & Schmidt-Kaler 1976) the estimated Av = R * E(B-V) is 0.08 mag.

M39 Distance

The normal points for the Zero Age Main Sequence (ZAMS) were given in Schmidt-Kaler, 1982. A third order polynomial fit was used to interpolate apparent magnitudes for the specific B-V points given by Schmidt-Kaler from the Color Magnitude diagram obtained from M39 photometry.

M39 distance modulus
Table 3. Results

The average distance modulus was 7.14. Using this distance modulus the cluster distance was determined to be 268 pc (872 ly).

Comparison to Literature

For this cluster Johnson (1953) gives a distance of 275±30pc, McNamara & Sanders (1977) gives 265 pc and Anthony-Twarog (1984) gives a distance modulus of 7.23 (279 pc).


1. Platais I. 1984, Pis’ma Astron. Zh. 10, 203-209
2. SIMBAD, 2000, A&AS,143,9 , The SIMBAD astronomical database, Wenger et al.
3. Vasilevskis, S., 1962, Astr. J., 67, 699
4. Schmidt-Kaler K., 1982, Landolt-Börnstein, Vol. 2b, p. 19, p. 453
5. Anthony-Twarog, 1984, Astron. J., Vol. 89, No. 5, p. 655 - 662
6. Moffat & Schmidt-Kaler, Th. 1976, Astr. Astrophys., 48, 115.