Lecture 7
Stellar Distance and Velocity
I. Stellar Distance
A. The cosmic distance scale: in order to understand
the true physical nature of celestial objects we must be able to measure
an accurate distance to those objects; the distance scale of the universe
is set by a series of techniques, each one dependent on the previous technique
for an accurate calibration; also known as the cosmic distance ladder,
any recalibration of the lower "rungs" has a "domino effect" throughout
all the higher "rungs" possibly changing our fundamental ideas about the
universe; understandably, the cosmic distance scale is one of the most
fundamental issues in astrophysics
1. before distances could be measured to even the closest stars,
the absolute scale of the solar system had to be determined; sometimes
known as the "zeroth rung", the determination of the astronomical unit
(1AU
= average radius of the Earth's orbit) is the foundation on which the cosmic
distance ladder rests
a. during the 18th and 19th centuries the most accurate method of
determining the absolute scale of the solar system was to observe the ingress
and egress times of a transit of Venus across the face of the Sun from
two locations separated by as much of the Earth's diameter as possible;
unfortunately, Venus transits are relatively rare with a pattern of two
consecutive occurrences separated by more than a century followed by a
third transit less than a decade later - many scientific careers were won
or lost in this first attempt to climb the distance ladder (note:
there is an upcoming Venus transit in 2004, the first in over a century)
b. presently, the measure of an AU (and therefor the absolute scale
of the solar system) is determined to within approximately 100 meters by
RADAR techniques (time of flight measurements radio pulses bounced of solar
system bodies); 1 AU = 149,597,870.7 km
2. the first step in determining interstellar distances is stellar
parallax (see below); the following short non comprehensive overview
lists some of the distance measuring techniques used in determining the
cosmic distance scale: (note on units: one special astronomical
unit of length that many people are familiar with is the light year (ly)
= the distance light travels in 1 year, about 6 trillion miles; astronomers
tend to use the distance unit known as a parsec (pc), note 1 pc = 3.26
ly and 1 kpc = 1000 pc = 3260 ly)
parallax - nearby stars (with recent HIPPARCOS satellite data, out to ~
1kpc); spectroscopic parallax - stars with resolvable spectral type, mostly
within 10 kpc; cluster main sequence fitting - star clusters within 15
kpc; cepheid pulsating variable stars - previously within the local group
(galaxy cluster), now with Hubble Telescope out to Virgo Galaxy Cluster
~25,000 kpc; Type Ia supernova - galaxy clusters well beyond Virgo Cluster,
several 100,000 kpc to perhaps a few million kpc in a few cases; Hubble's
Law - theoretically anywhere within the observable universe if possible
to measure a redshift
B. Stellar parallax: 1st rung of the cosmic distance ladder,
a direct geometry based method employing the measurement of the apparent
angular shift of a star (stellar parallax angle) due to the Earth's orbital
motion around the Sun
1. using images taken 6 months apart, the apparent shift in angular
position of a nearby star relative to background stars in the field of
view may be measured, ½ of the angular shift is defined as the stellar
parallax angle (P); the background stars must be many, many times further
away than the nearby foreground star for the geometric assumptions to be
valid; if the diagram below were actually to scale, the nearby star would
be well over 2 miles to the right of the Sun for even the closest star
to the Sun; then P would be extremely small and the small angle approximation
holds allowing for the simple equation shown boxed in the diagram below
2. the simple equation shown boxed
in the above diagram must be used with the special units indicated in the
parenthesis: arcsec = "second of arc" = angular unit, 1 arcsec
= 1/3600 of a degree; pc = "parsec" = special distance unit defined as
the distance a star would be if its measured parallax angle was
exactly equal to 1 arcsec (hence, "parsec"); note: since both
light year and parsec sound somewhat like "time" units, many
people confuse them with units of time - remember they are both units of
length,
NOT time
II. Stellar Velocity
A. The doppler effect and radial velocity
1. doppler effect (see
diagram): the observed frequency and wavelength of light emitted
from an object moving toward an observer is shifted to higher frequency
and shorter wavelength ("blue-shifted") by the relative motion; for an
object moving away from an observer the light is observed to shift to lower
frequency and longer wavelength ("red-shifted"); in terms of wavelenght,
the non-relativistic equation which describes the doppler effect is given
by:
v/c = (l-lo)/lo
where v = relative velocity of the object
emitting the light to the observer; c = speed of light; l
= the measured wavelength of light from the moving object; lo
=
the rest wavelength (ie: the wavelength the light would be if the
object were at rest relative to the observer) {typically, the radial
velocity (vR) is given in units of kilometers
per second (km/s), to convert the above fraction of the speed of light
(v/c) into km/s multiply by the speed of light in km/s: c = 300,000
km/s}
2. note the convention that if
the light is "blue-shifted", (l-lo)
has a negative value since the observed wavelength (l)
is smaller than the rest wavelength (lo)
and the calculated line-of-site velocity or radial velocity of the
object is also negative, if the light is "red-shifted" the calculated radial
velocity is positive; ("blue-shift" = neg. radial vel. = motion toward
observer, "red-shift" = pos. radial vel. = motion away from observer)
B. The proper motion of a star
1. a star's intrinsic motion through space causes it to change position
with respect to other stars on the sky. The component of a star's
velocity perpendicular to our line of sight to the star causes an angular
rate of change in position as seen on the sky. This (extremely slow!)
angular motion across the sky is called proper motion(m);
the units of proper motion are arcsec/year and as noted above the
motion is very small, the larges know proper motion is that of Barnard's
Star which has a motion of 10.3"/year
2. relationship between a star's transverse velocity (velocity perpendicular
to our line of sight) and its proper motion:
vT
= (1.49598 x 108)mr
where vT
= transverse velocity in km/s, m = proper motion
in arcsec/yr, r = distance to star in parsecs, (note: the constant
in parenthesis is a conversion factor so that the transverse velocity is
in km/s when proper motion is in arcsec/yr and distance is expressed in
pc)
C. The space velocity of a star relative to the
Sun
1. the space velocity (v*)
of a star is the speed and direction of motion of a star through space
relative to the Sun; the space velocity can be determined by measuring
the distance, proper motion, and radial velocity of a star; the distance
and proper motion determines the tranverse velocity (speed and direction
in the plane of the sky) and the radial velocity gives the speed in the
direction perpendicular to the plane of the sky (line of sight direction)
2. the space velocity magnitude (speed) is then:
v*
= [(vR)2
+ (vT)2]1/2
where all speeds are in km/s; the direction
is determined by the components of the transverse velocity in the plane
of the sky and the radial velocity componet