Astronomical Astronomical DistancesDistances

or or Measuring the UniverseMeasuring the Universe

The ProblemsThe Problems

by by Rastorguev Alexey,Rastorguev Alexey,professor of the Moscow State professor of the Moscow State

University and Sternberg University and Sternberg Astronomical Institute, RussiaAstronomical Institute, Russia

Sternberg AstronomicalSternberg Astronomical InstituteInstitute

Moscow UniversityMoscow University

I. Main Sequence scatterI. Main Sequence scatter• Estimate an intrinsic scatter of the Main Estimate an intrinsic scatter of the Main

Sequence for Pleiades cluster, due to its Sequence for Pleiades cluster, due to its finite size (neglect other sources of MS finite size (neglect other sources of MS scatter, such as metallicity and age scatter, such as metallicity and age differences and unresolved binaries)differences and unresolved binaries)

• Given are:Given are:• Apparent distance modulus (m-M)≈5.57m

• Color excess E(B-V) ≈ 0.04m

• Apparent angular size d ≈ 110 arc minutes

II. Mass and the distance of M4 II. Mass and the distance of M4 globular cluster (HST data & globular cluster (HST data &

VVRR))• Estimate the distance and the mass of M4 Estimate the distance and the mass of M4

globular cluster based on measured radial globular cluster based on measured radial velocities and proper motions of large sample of velocities and proper motions of large sample of cluster memberscluster members

• Given are:Given are:• Measured radial velocity dispersion σVR ≈ 3.75

km/s, with typical accuracy of one VR measurement as ±1.0 km/s

• Measured dispersion of full (2D) proper motions is σμ ≈ 0.6 mas/year, with typical accuracy of one PM measurement as ±0.1 mas/year

• Apparent effective radius rE ≈ 7.3 arc minutes

Hints to the Problem IIHints to the Problem II• (a) Correct VR 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 rE 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)

III. The distance to the Galactic III. The distance to the Galactic centercenter

• Estimate the distance to the Galactic Estimate the distance to the Galactic center based on proper motions center based on proper motions measurement of the bulge stars relative to measurement of the bulge stars relative to quasar and known space velocity of the quasar and known space velocity of the SunSun

• Full space velocity of the Sun (due to disk rotation ) with respect to the galactic center is V0 ≈ 220 km/s

• All other data can be taken from the diagram (see next slide)

• Bulge PM with respect to distant quasar

• μl component: along the galactic longitude (parallel to the galactic plane)

μl (mas/yr)

Hints to the Problem IIIHints 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)

IV. Incorrect Cepheid classificationIV. 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 <V>, mean apparent color <B-V>, pulsation period P

Hints to the Problem IV: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

V. Question:V. Question:

• Which methods of the distance measurement cannot be applied to individual objects and why?

VI. Question: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

VII. Question:VII. Question:

• Explain, why accreting White Dwarf model have been chosen to explain Supernova Ia explosion

VIII. Question:VIII. Question:

• What characteristics of globular cluster could be used as the distance indicators ?

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