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ASTR 551, Majewski [SPRING 2003]. Lecture Notes

## ASTR 551 (Majewski) Lecture Notes

### DYNAMICAL METHODS FOR DISTANCE DETERMINATION

1. Trigonometric Parallax

• Apparent shift in position of a nearby star because of the orbital motion of the Earth about the Sun.
• π = difference between the geocentric and heliocentric positions of the star
• Effect is small: π < 1" for all stars
• tanπ ~ π (radians) = r/d

To convert to arcsec: π (arcsec) = 206265π (radians)

then:

If r = 1 Astronomical Unit (A.U.) = 150,000,000 km:

• We define 1 parsec (pc) = 206265 A.U. = 3x1013 km = 3.26 light years (l.y.) Then:

• First parallax measured by Bessell in 1838

• NOTE: problem is more difficult than it appears since background reference stars also have parallax. Thus, we typically measure relative parallax and correct to absolute parallax by statistical models of expected distance distribution of background stars.

• If have QSO or compact galaxy in field, have absolute reference

• Not typical situation, but can often bootstrap to absolute frame - major enterprise

• Fundamental stars set up and measured angles referred to them (Note: we are setting up such a system for SIM)

• Before 1999: often meridian circle work FK5 (Fundamental Katalog)

• Now: International Celestial Reference System to extragalactic objects -- use radio interferometry to get position

2. Reduced Proper Motion Diagram

• Luyten (1922)

• Note:

m - M = 5logd + 5

M = m - 5logd + 5

• What if all stars we cared about had same V (or a well defined average V)?

• Then:

where:

VS = space velocity (total velocity of a star)

VT = transverse velocity (velocity perpendicular to line of sight)

VR = radial velocity (velocity parallel to line of sight = Doppler velocity)

• Note:

• VR can be measured directly by the Doppler effect.

VR is seen as a blueshift (if approaching us) or redshift (if receding).
• VT cannot be measured directly.

Only the angular change, the proper motion, can be observed.

• To convert from the proper motion to the transverse velocity, one needs to know the distance, d, to the star.

When one works out the math one finds that:

and:
5logd = 5log [ VT / 4.74 μ]

M = m - 5logVT + 5logμ + 5log(4.74) + 5

collect all observed data together and define:

Hm = m + 5logμ + 5 = M + 5logVT - 5log(4.74)

e.g.

HV = V + 5logμ + 5 (observed)

HV = MV + 5logVT - 3.378

(presumably a constant for each spectral type)

• For different spectral types, there is an MV - color relation, so one can define ridge lines that depend only on assumed VT

• So, e.g.:

MV(B-V) --> HV(B-V, VT)

• For constant VT:

• Thus, one can sort stars into a version of CMD only based on observables of apparent magnitude, color, and μ - easier to measure than parallax; assuming a constant mean VT and MV(B-V) relation

• Normally have different MV(B-V) and VT for each population (e.g. Pop I - disk; Pop II - halo)

Disk, say VT ~ 10's km/s for disk ~ 100's km/s for halo

Precision not important, since taking log

3. Moving Cluster Method - useful for large, nearby clusters of stars

• In this case, we see the motions of individual stars as they approach or recede from us as a group

• Simple case - cluster moving along line of sight:

cluster diameter = d

presently at distance r

cluster's radial velocity is = vr

cluster's angular size is θ = d/r

Can show that:

where vr is recessional velocity, theta-dot is the rate at which the cluster is shrinking

• In general, the cluster will not be moving along the line of sight. Then we see the cluster apparently moving toward or away from a convergence point (like a radiant of a meteor shower).

the point at infinity at which all parallel paths of motion intersect

• Many clusters have been found this way, by searching proper motion catalogs for stars whose μ vectors intersect at a point.

• Convergence point is defined by proper motion data, but the angular distance it lies from a cluster star contains key information on the space velocity because it tells us the direction of the complete space velocity vector - knowing this direction allows us to get VR immediately from VT and then r follows from μ, VT:

only need VR, μ, and ψ for each star

• Assumptions:

• cluster is not expanding/contracting in linear size

• cluster is not rotating

• proper motions are accurate

• cluster is large in sky (close) so that μ accurate; convergence point clear

• ψ not close to 0° or 90°

• Most important cluster this has been done for is the Hyades:

• about 200 stars at ~ 45 pc

• standard cluster for which trigonometric parallax were not possible for all stellar types until recently

• distance set the entire distance scale

• now π and μ so well known that we can see depth effects in cluster

• Also:

 Ursa Major Group 60 stars ~24 pc Pleiades 600 stars 115 pc (no good trig. π) Sco-Cen Group 100 stars 170 pc (no good trig. π)

4. Dynamical Parallaxes of Clusters - useful for getting MV (RR Lyrae) variables

• Easy case: for self bound cluster, assume isotropic velocity dispersion in cluster; this can be measured in core with radial velocities: σVR

• Then the velocity dispersion in plane of sky should be similar in either or δ direction:

σVT = σVT δ = σVR

• But for transverse velocity, only measure σμ. Then:

• First done by Cudworth (1979) for M3. Can be done to about 10-15 kpc. But:

• real clusters often do not have isotropic velocities; radial anisotropy

• rotation - inflates apparent σV

• σV is function of radius, so have to pick μ and VR sample with same ~ radius

• measurement errors inflate σ's:
true)2 = (σobs)2 - (εRMS)2

• Need to model cluster dynamically; use large samples; about 8 clusters done this way.

5. Dynamical π of Binary stars

• Method used to obtain stellar masses for binaries of known distance, or distance with reasonable guesses of binary mass.

• Fundamental determinations of stellar masses based on Kepler's 3rd law:

G (1 + 2) P 2 = 4 π2 a 3

where a is the semi-major axis

• For planets around the Sun, derive units that are convenient:

Then, if units are years, A.U.'s, and , then:

(1 + 2) P 2 = a 3

Note:

Then:

where (") is the angular size (measurable) of the semi-major axis.

• Thus, if we can observe period of binary and ("), and make a reasonable guess for 1 and 2, we can get d.

• Since (1 + 2) is taken to 1/3 root, even crude guesses for mass give good d.

• d --> MV

• MV --> mass from -L relation

• reiterate

• Converges quickly.

• Note: SIM planet finding - measure the " of displaced stars (can't see planets).

• At least 50% of "stars" within 5 pc of Sun are multiple, bound systems.

• Need visual binaries t get ", but these tend to be long P.

• Have to assume stars are "normal" so that we can guess 1 and 2

• Can provide check on other methods:

e.g. in 1960's the Hyades binaries gave discordant values compared to moving cluster -- led to revision of d

• Note: If we can measure the velocities of the stars (spectroscopic binary) then we can get the linear dimension of the semi-major axis.

• Then d = a / as before.

• But have to get sin i (not normal)

• Only in case of eclipsing binary do we know i ~= 90 °, sin i ~= 1 (unless we have very close binaries, in which case eclipsing can occur for i not equal to 90 °).

• Thus, eclipsing binaries are very important, finding more of them in microlensing surveys; also can get radius of stars - useful for distances

• Note: if desire masses:

• Note, because sin i <= 1, can only get lower limit on 1 and 2.

• Can make a statistical correction by noting < sin3i > = 0.59 for randomly oriented orbital planes.

• But 0.59 correct? Probably more likely to see large sin i 's.

##### From Binney & Merrifield.

• L = 4πR 2σ Teff 4

• If we know R and Teff, we get:

L --> MV, Mbol --> gives photometric π

Mbol = -2.5 log(L) + C = -10 log Teff - 5 log(R) + C

• But getting R is hard -- eclipsing binaries very useful for this (see section 3.2.5).

• But pulsational variables help:

• Lines in spectrum of a variable star show Doppler shifts that vary cyclically with pulsations.

• So do L and Teff.

• Change in radius between times to and t1 -- integrate over l.o.s. velocity:

• Thus, if star starts at ro at to, it is ro + δr1 at t1.

• Then:

mbol,1 - mbol,0 = Mbol,1 - Mbol,0 = -5 [log(ro + δr1) - log ro] - 10 [log Teff,1 - log Teff,0]

• If we know Teff and mbol at both to and t1, we can get ro

• Pick a time, t1, when Teff is the same as at to:

• Then:

• SO:

1. Make measures of vR throughout oscillation; integrate vR dt to get δro

2. Measure apparent mags to estimate mbol (correcting for extinction), and pick out pairs of observations where color is same --> Teff same.

From ro equation above, get δr --> gives us values of R for star.

3. Get Mbol = -10 log Teff - 5 log R + C.

4. mbol - Mbol --> gives d

• Shortcomings:

1. How to weight average r values to get mean R.

2. Star atmospheres are not at a single surface.

3. Colors / wavelengths observed come from different layers, and these may differ from vR layer.

5. Color --> Teff, mx --> mbol can be tricky.

• Still, very commonly used method for RR Lyrae, Cepheids, even Supernovae.

MK (P)

• Theory predicts MK = -2.22 logP

• Observations: MK = -2.3 logP - 0.88

• Thus, conclude BW method is working

-->
GETTING MV (RR)

As a demonstration of how the various distance-gauging methods work, compare the following derived values for the absolute magnitude of RR Lyrae stars (at one fiducial metallicity).

Most common methods:

1. Statistical parallax

2. MS fitting/WD fitting of globulars

3. Distance to Mag. clouds

• MS fitting

• Cepheids

Can also compare the distances to the LMC and M31 from two types of pulsational variables.

SUMMARY

Main techniques used for distance-gauging different stellar types:

 Supergiants Open cluster fitting O-A stars Open clusters, secular & stat π F-M dwarfs Trig π, moving cluster F-M giants Moving cluster, cluster fitting, secular & stat π White dwarfs Trig π, binaries, cluster fitting Cepheids, RR Lyrae Stat π, Baade-Wesselink, MS fitting to clusters

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All material copyright © 2003 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 at the University of Virginia.