Photography & Science

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As a first approximation, we can estimate the distance to the h and χ Persei using the distance modulus formula, Equation 4.12 from Reference 5;

m-M = 5log(r/10 pc),               (1)

where m is the apparent magnitude, M is the absolute magnitude, m-M is the distance modulus, 10 parsecs is the reference distance, and r is the distance to the object of interest in parsecs. Extinction by the interstellar medium due to reddening, scattering, absorption, and the Earth's atmosphere is assumed to be negligible in this equation. Note that 1 parsec (pc) = 3.26 light-years (lys).

We start with distance estimates to h Persei (NGC 869) and χ Persei (NGC 884) by R.J. Trumpler (Ref. 6). His distance moduli using Eq. 4.12 were 11.4 and 11.9, respectively. Inverting Eq. 1 and solving for the distance r in parsecs gives

r = 10{10^[(m-M)/5]}.               (2)

Thus, his distances to h and χ Persei were approximately 1900 and 2400, respectively. Trumpler simply takes the average of these two distances to conclude that both clusters are about 2200 parsecs away (~ 7200 lys).

Trumpler also attempts to correct for interstellar absorption using an absorption constant k as follows

m-M = 5log(r/10 pc) +kr,        (3)

where his value for k is 0.79 magnitudes per 1000 parsecs. This leads to corrected distance estimates of 1350 parsecs (~4400 lys) for both clusters. These distances do not compare well with predicted distances by follow-on research teams whose dereddened distance moduli estimates became more sophisticated. Here are some examples from the academic literature. Separate distances for the two clusters are noted, otherwise the authors concluded that they are at the same distance. I have converted published dereddened distance moduli to distances in parsecs and lightyears using Eq. 1 above.

• 1920 parsecs (6260 lys), Ref. 7
• 2150 parsecs for h Persei (7000 lys), 2500 for χ Persei (8150 lys), Ref. 8
• 2090 parsecs (6800 lys), Ref. 9
• 2240 parsecs (7300 lys), Ref. 10
• 2190 parsecs (7140 lys), Ref. 11
• 2330 parsecs (7600 lys), Ref. 12
• 2340 parsecs (7640 lys), Ref. 13
• 2340 parsecs for h Persei (7640 lys), 2310 for χ Persei (7530 lys), Ref. 14
• 2290 parsecs for h Persei (7470 lys), 2340 for χ Persei (7640 lys), Ref. 15

References

5. Fundamental Astronomy, 6th Edition, Hannu Karttunen, Pekka Kröger, Heikki Oja, Markku Poutanen, and Karl Johan Donner, Editors, Springer-Verlag, Berlin, 2017, p. 96.

6. "Preliminary results on the distances, dimensions, and space distributions of open star clusters," Robert J. Trumpler, Lick Observatory Bulletin ; No. 420, University of California Press, 1930, pp. 154-188.

7. " A spectroscopic study of the region of the double cluster in Perseus," William P. Bidelman, Astrophysical Journal, 98, 61-81 (1943).

8. "Ages and structures of strain the h and χ Persei association," Rudolph Schild, The Astrophysical Journal 148, 449-458 (1967).

9. "Photometric study of the double cluster h & χ Persei," A. Marco and G. Bernabeu, Astronomy & Astrophysics 372, 477-494 (2001).

10. "UBVI and Hα photometry of the h and χ Persei cluster," Stefan C. Keller, Eva K. Grebel, Grant J. Miller, and Kenneth M. Ross, The Astronomical Journal 122, 248-256 (2001).

11. "CCD uvbyβ photometry of young open clusters: I. The double cluster h and χ Persei," G. Capilla and J. Fabregat, Astronomy & Astrophysics 394, 479-488 (2002).

12. "Membership in the region of the double cluster h and χ Persei working from proper motions: distance moduli and extinction in that galactic direction," Antonio Uribe, José-Alejandro García-Varela, Beatriz-Eugenia Sabogal-Martínez, Mario A. Higuera G., and Eduardo Brieva, Publications of the Astronomical Society of the Pacific 114, 233-248 (2002).

13. "The star formation history and mass function of the double cluster h and x Persei," Catherine L. Slesnick, Lynne A. Hillenbrand, and Philip Massey, The Astrophysical Journal 576, 880-893 (2002).

14. "Fitting the young main-sequence: distances, ages and age spreads," N.J. Mayne and Tim Naylor, Monthly Notices of the Royal Astronomical Society 386(1), 261-277 (2008).

15. "The stellar population of h and χ Persei: Cluster properties, membership, and the intrinsic colors and temperatures of stars," Thane Currie, Jesus Hernandez, Jonathan Irwin, Scott J. Kenyon, Susan Tokarz, Zoltan Balog, Ann Bragg, Perry Berlin, and Mike Calkins, The Astrophysical Journal Supplement Series 186, 191-221 (2010).