Stellar objects
By the end of this topic you should be able to:
-describe the method of parallax, the method of
spectroscopic parallax and the Cepheid method for
determining distances in astronomy;
-define the parsec;
-state the definitions of absolute and apparent magnitude.
Parallax method
The parallax method takes advantage of
the fact that, when an object is viewed
from two different positions, it appears
displaced, relative to a fixed
background of stars, because of the fact
that in the intervening time the Earth
has moved in its orbit around the Sun.
Since the parallax angle is very small,
tan p ≈ p
Parallax method
The parallax angle is the angle at the
position of the star that subtends a
distance equal to the radius of the
Earth’s orbit around the Sun, a
distance known as one astronomical
unit, 1 AU = 1.5 x 1011 m.
Parallaxes are measured quite
accurately provided they are not
too small. For example, parallaxes
down to 1 arcsecond (1/3600 of a
degree) are easily measured.
If the star is too far away, the
parallax is too small and the
method fails.
Typically, measurements from
observatories on Earth allow
distances up to 300 ly to be
determined with the parallax
method. Using measurements from
satellites above the Earth’s
atmosphere, larger distances can
be determined.
Parsec
The parallax method can be used
to define a common unit of
distance in astronomy, the parsec.
One parsec is the distance to a star
whose parallax is 1 arcsecond.
1 pc = 3.09 x 1016 m = 3.26 ly
This means that if the parallax of a star is
known to be p arcsecond, the distance is 1/p
parsecs
Absolute and apparent magnitudes
The ancient astronomers devised a
relative system of classifying stars
according to how bright the appeared
to an observer on Earth. Each star was
given a number called the apparent
magnitude m-the higher the apparent
magnitude, the dimmer the star. In this
system six classes of brightness were
defined, and assigned numbers from 1
to 6. A magnitude 6 star was supposed
to be 100 times dimmer than a
magnitude 1 star.
The modern magnitude scale is
defined as follows:
Given a star of apparent brightness
b, we assign to that star an apparent
magnitude m defined by
b/b0 = 100-m/5 = 2.512-m
(The value b0 = 2.52 x 10-8 W/m2 is
taken as the reference value for
apparent brightness).
Taking logarithms (to base 10) gives
the equivalent form
m = -5/2 log(b/b0)
Absolute and apparent magnitudes
The human eye can detect a
star of apparent magnitude
not longer than about 6.
With simple binoculars the
limit is raised to stars of
magnitude 9. The largest
telescopes can record
images of objects of
apparent magnitude as faint
as 27.
Absolute and apparent magnitudes
Two stars that have the same
apparent magnitude are not
necessarily equally bright
intrinsically, since they may be at
differente distances. To establish a
system of absolute magnitudes that
will tell us if one star is intrinsically
brighter than another, we imagine
that all stars are positioned at the
same distance from Earth.
The apparent magnitude a star
would have if placed at a distance
of 10 pc from Earth is called the
absolute magnitude M of the star.
Thus, it can be proved that:
Absolute and apparent magnitudes
Common Name
Scientific Name
Distance (light years)
Apparent Magnitude
Absolute Magnitude
Sun
Sirius
Canopus
Alpha CMa
Alpha Car
8.6
74
-26.72
-1.46
-0.72
4.8
1.4
-2.5
Rigil Kentaurus
Alpha Cen
4.3
-0.27
4.4
Arcturus
Vega
Capella
Rigel
Procyon
Achernar
Betelgeuse
Hadar
Acrux
Altair
Aldebaran
Antares
Spica
Pollux
Fomalhaut
Becrux
Deneb
Regulus
Adhara
Castor
Gacrux
Shaula
Alpha Boo
Alpha Lyr
Alpha Aur
Beta Ori
Alpha CMi
Alpha Eri
Alpha Ori
Beta Cen
Alpha Cru
Alpha Aql
Alpha Tau
Alpha Sco
Alpha Vir
Beta Gem
Alpha PsA
Beta Cru
Alpha Cyg
Alpha Leo
Epsilon CMa
Alpha Gem
Gamma Cru
Lambda Sco
34
25
41
~1400
11.4
69
~1400
320
510
16
60
~520
220
40
22
460
1500
69
570
49
120
330
-0.04
0.03
0.08
0.12
0.38
0.46
0.50 (var.)
0.61 (var.)
0.76
0.77
0.85 (var.)
0.96 (var.)
0.98 (var.)
1.14
1.16
1.25 (var.)
1.25
1.35
1.50
1.57
1.63 (var.)
1.63 (var.)
0.2
0.6
0.4
-8.1
2.6
-1.3
-7.2
-4.4
-4.6
2.3
-0.3
-5.2
-3.2
0.7
2.0
-4.7
-7.2
-0.3
-4.8
0.5
-1.2
-3.5
Example questions
Calculate the absolute
magnitude of a star whose
distance is 25 ly and whose
apparent magnitude is 3.45
Calculate the distance to
Sirius using that m = -1.43
and M = 1.4
Spectroscopic parallax
The term spectroscopic parallax
refers to a method of finding the
distance to an star given the star’s
luminosity and apparent brightness.
The term is misleading in that no
use of parallax is being made.
The question is then how to
determine the luminosity of the star.
This is done by examining its
spectrum, from which the
temperature can be deduced.
Knowing the temperature and using
the HR diagram allow us to
determine the luminosity as well.
λ0T = 2.9 x 10-3 Km
Example question
A main sequence star emits most of its
energy at a wavelength of 2.4 x 10-7 m.
Its apparent brightness is measured to
be 4.3 x 10-9 Wm-2. How far is the star?
The Cepheids
Cepheid variable stars are stars
whose luminosity is not constant
in time but varies from a
minimum to a maximum
periodically, the periods being
typically from a 1 to 70 days. The
brightness of the star increases
sharply and then fades off more
gradually.
The reason for the periodic
behaviour of the brightness of
Cepheid stars has to do with the
interaction of radiation with
matter in the atmosphere of the
star. This interaction causes the
outer layesrs of the star to undergo
periodic expansions and
contractions. The star is at its
brightest when the surface of the
star expands outward at high
velocity. It is at its dimmest when
the surface moves inward.
The Cepheids
Henrietta Leavitt discovered a
remarkable relationship between the
peak luminosity of Cepheids and their
period. The longer the period, the
larger the luminosity.
Observing a Cepheid and finding its
period allows the determination of its
luminosity. This, in turn, allows the
determination of its distance.
Range of efectiveness for each method
The ordinary parallax method allows the determination
of distances up to about 100 pc. The spectroscopic
parallax method extends distance measurements to
about 10000 pc. The Cepheid variable star method
extends the scale further to 15 Mpc.
Questions
Questions