1. Astronomical Distances or Measuring the Universe The Problems by Rastorguev Alexey, professor of the Moscow State University and Sternberg Astronomical Institute, Russia Sternberg Astronomical Institute Institute Moscow University

2. I. Main Sequence scatter Estimate an intrinsic scatter of the Main Sequence for Pleiades cluster, due to its finite size (neglect other sources of MS scatter, such as metallicity and age differences and unresolved binaries)Estimate an intrinsic scatter of the Main Sequence for Pleiades cluster, due to its finite size (neglect other sources of MS scatter, such as metallicity and age differences and unresolved binaries) Given are:Given are: Apparent distance modulus (m-M)≈5.57 m Color excess E(B-V) ≈ 0.04 m Apparent angular size d ≈ 110 arc minutes

3. II. Mass and the distance of M4 globular cluster (HST data & V R ) Estimate the distance and the mass of M4 globular cluster based on measured radial velocities and proper motions of large sample of cluster membersEstimate the distance and the mass of M4 globular cluster based on measured radial velocities and proper motions of large sample of cluster members Given are:Given are: Measured radial velocity dispersion σ VR ≈ 3.75 km/s, with typical accuracy of one V R measurement as ±1.0 km/s Measured dispersion of full (2D) proper motions is σ μ ≈ 0.6 mas/year, with typical accuracy of one PM measurement as ±0.1 mas/year Apparent effective radius r E ≈ 7.3 arc minutes

4. Hints to the Problem II (a) Correct V R and PM dispersions for observational errors (b) Use simplest cluster representation as having isotropic space velocity distribution for cluster members (c) Use virial theorem to estimate cluster mass, with r E as the characteristic radius Based on data from:Based on data from: R.Peterson et al. (ApJ V.443, P.124, 1995) L.Bedin et al. (ASP Conf. Proc. V.296, P.360, 2003)

5. III. The distance to the Galactic center Estimate the distance to the Galactic center based on proper motions measurement of the bulge stars relative to quasar and known space velocity of the SunEstimate the distance to the Galactic center based on proper motions measurement of the bulge stars relative to quasar and known space velocity of the Sun Full space velocity of the Sun (due to disk rotation ) with respect to the galactic center is V 0 ≈ 220 km/s All other data can be taken from the diagram (see next slide)

6. Bulge PM with respect to distant quasar μ l component: along the galactic longitude (parallel to the galactic plane) μ l (mas/yr)

7. Hints to the Problem III Bulge proper motion with respect to very distant quasar can be considered as the reflected solar motion in the Galaxy Based on data from:Based on data from: L.Bedin et al. (MemSAIt V.74, P.436, 2003)

8. IV. Incorrect Cepheid classification Estimate the systematic error induced to the distance of the Cepheid (calculated by the P-L relation for fundamental-tone pulsations), given that it is incorrectly classified and really pulsate in the first overtone Given are:Given are: mean apparent magnitude, mean apparent color, pulsation period P

9. Hints to the Problem IV: Calculate also the difference in normal color (derived from Period – Color relation for fundamental-tone Cepheids) Take into account that the difference with true color will induce the error in the estimated interstellar extinction in V color band, and hence in the distance emodulus

10. V. Question: Which methods of the distance measurement cannot be applied to individual objects and why?

11. VI. Question: Construct Wesenheit index W(BV) for V and B color bands. Estimate the slope of W(BV) - lg P relation Hints:Hints: Use some of normal-color – period relations given in the Cepheids chapter

12. VII. Question: Explain, why accreting White Dwarf model have been chosen to explain Supernova Ia explosion

13. VIII. Question: What characteristics of globular cluster could be used as the distance indicators ?

14. E-mail: rastor@sai.msu.ru alex.rastorguev@gmail.com