ASTR 400/700:
Stellar Astrophysics
Stephen Kane
Stellar Pulsation
Chapter 14.1, 14.2
Stellar Models:
The complete set of differential equations describing the interiors
of stars is therefore:
Equation of Continuity:
Hydrostatic Equilibrium:
Energy Generation:
Temperature Gradient:
dM ( r )
= 4π r 2 ρ
dr
dP − G M ( r ) ρ
=
dr
r2
dL
= 4π r 2 ρ ε
dr
− 3 κ ρ Lr
 dT 

 =
3
2
dr
4
ac
T
4
π
r

 rad
− 1 GM ( r )
 dT 

 =
2
dr
C
r

 ad
P
If a star is neither expanding nor contracting,
we may assume that throughout the star there is
a balance between pressure and
1.
2.
3.
4.
temperature
density
luminosity
gravity
RR Lyrae variables in the globular cluster M3
(one night’s observation)
http://www.astro.princeton.edu/~jhartman/M3_movies.html
Observations of Pulsating Variables
•
•
First noticed in 1595 by
David Fabricius. Mira
would be 2nd magnitude at
its brightest…then would
“vanish”…
Believed to be due to dark
splotches on rotating
star…
Delta-Cephei
John Goodricke discovered in 1784
that the brightness of Delta-Cephei
was variable with a period of about
5 days!!!!
Prototype of the classical cepheid
variable star
magnitude varies from 3.4 to 4.3,
 luminosity changes by factor of
100(∆m/5) = 100(0.9/5) = 2.3
Period-Luminosity Relation
Stars of the “Classical Cepheid Variable” type in the
Small Magellanic Cloud were observed…and found to
have a strong correlation between Period and apparent
magnitude…
• Henrietta Swan Leavitt discovered and
classified ~2400 classical cepheid
variable stars
• Periods 1-50 days
• She plotted luminosity vs. period for a
set of cepheids from the Small
Magellanic Cloud and found the
Period-Luminosity relation
Henrietta Swan Leavitt
(1868-1921)
• Ability to measure Distances !!!!!
Period-Luminosity Relation
Notice that there is scatter…
Calibration of Cepheids
The nearest Cepheid is
Polaris (~200 pc), too far
for trigonometric parallax.
d (pc) = 1/p (in arcsec)
In 1913, Ejnar Hertzsprung of
Denmark used least squares
mean parallax to determine the
average magnitude M = -2.3 for
a Cepheid with Π = 6.6 days.
d (pc) = 4.16/slope (in arcsec/yr)
(4.16 AU/yr is the Sun’s motion)
Calibration of Cepheids
• Relation between
average V band absolute
magnitude and Period
Cepheid Calibration-Infrared
• Improved calibration
at infrared
wavelengths that suffer
less from extinction
• Adding a color term gives
further improvement
How to Find the Distance to a Pulsating Star
•
•
•
•
Find the star’s apparent magnitude m (just by looking)
Measure the star’s period (bright-dim-bright)
Use the Period-Luminosity relation to find the stars
absolute magnitude M
Calculate the star’s distance (in parsecs) using
d (pc) = 10(m-M+5)/5
Pulsation Hypothesis for Brightness Variation
• Shapley proposed that the observed
variation in brightness and temperature
caused by radial pulsations of single
stars.
• Rhythmically “breathing” in and out!
• R varies-->causes Luminosity and
temperature to vary
• Sir Arthur Eddington provided
theoretical framework that could
explain the variations in Brightness,
temperature, radius and surface velocity
• Delta-Cephei: supergiant star. Radius
varies by 5%-10% (~1 solar radius).
F5(hottest)-G2(coolest)
• Star is brightest when its surface is
expanding outward most rapidly, after it
has passed through its minimum
radius…phase lag
Delta-Cephei
The Instability
Strip
• Majority of pulsating
stars lie in the instability
strip on the H-R diagram
• As stars evolve along
these tracks they begin
to pulsate as they enter
the instability strip and
cease oscillations once
they leave it.
∆T ~ 600 – 1100 K
Classes of Pulsating Stars
Type
Range of
Periods
Population Type
Type of
Oscillation
Long-Period Variables
100 – 700 days
I, II
R
Classical Cepheids
1 – 50 days
I
R
W Virginis stars
2 – 45 days
II
R
RR Lyrae stars
1.5 – 24 hours
II
R
d Scuti stars
1 – 8 hours
I
R, NR
b Cephei stars
3 – 7 hours
I
R, NR
DAV stars
100 – 1000 s
I
NR
The Physics of Stellar Pulsation
• The radial oscillations
of a pulsating star are
the result of sound
waves resonating in the
stars interior
• Pulsation period can be
roughly estimated from
how long it takes a
sound wave to cross
the star’s diameter
Standing sound waves in an organ pipe.
Radial modes for a pulsating star