Download Good Vibrations and Stellar Pulsations - Physics

l = 2, m = 0
Good Vibrations
and
Stellar Pulsations
Brad Carroll
Weber State University
April 11, 2007
www.univie.ac.at/tops/intro.html
l = 5, m = 3
In August of 1596, David Fabricius (Lutheran pastor
and amateur astronomer) observed o Ceti,
a 2nd magnitude star in the constellation Cetus. As it
declined in brightness, the star vanished by October.
Later it reappeared, and was renamed Mira
(“the Wonderful”)
By 1660 its 11-month period had been established.
The light variations were believed
to be caused by “blotches” on the
surface of a rotating star.
web.njit.edu/~dgary/728/Lecture12.html
In 1784, John Goodricke of York discovered that
d Cephei is variable: P = 5 days, 8 hours
d Cephei is the prototype of the Classical Cepheids
From Wycombe Astronomical Society, wycombeastro.org.uk/news.shtm
magnitude varies from 3.4 to 4.3,
so luminosity changes by factor of
100(Dm/5) = 100(0.9/5) = 2.3
Edward Charles Pickering’s Computers
at Harvard Observatory
From left to right: Ida Woods, Evelyn Leland, Florence Cushman, Grace Brooks,
Mary Van, Henrietta Leavitt, Mollie O'Reilly, Mabel Gill, Alta Carpenter,
Annie Jump Cannon, Dorothy Black, Arville Walker, Frank Hinkely, and
Professor Edward King (1918).
www.astrogea.org/surveys/dones_harvard.htm
Henrietta Swan Leavitt
(1868 – 1921)
Found 2400 Classical Cepheids
In 1912, discovered the
Period-Luminosity Relation
Small Magellanic Cloud
Cepheids in the SMC
From Shapley, Galaxies, Harvard University Press, Cambridge, MA, 1961.
Calibration: The Distance to a Cepheid
The nearest Cepheid is
Polaris (over 90 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 P = 6.6 days.
d (pc) = 4.16/slope (in arcsec/yr)
(4.16 AU/yr is the Sun’s motion)
www.cnrt.scsu.edu/~dms/cosmology/DistanceABCs/distance.htm
Period – Luminosity Relation
M<V> = -2.81 log10 Pd – 1.43
d (pc) = 10(m-M+5)/5
Sandage and Tammann, The Astrophysical Journal, 151, 531, 1968.
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
What is the Milky Way?
antwrp.gsfc.nasa.gov/apod/ap060801.html
Thomas Wright, An Original Theory on New Hypothesis of the Universe, 1750.
Kapteyn, The Astrophysical Journal, 55, 302, 1922.
In 1913, Hertzsprung calculated that the distance
to the Small Magellanic Cloud was 33,000 light years.
This was the greatest distance ever determined for an
astronomical object.
In 1917, Harlow Shapley used Hertzsprung’s calibration
of the period-luminosity relation to determine the
distance to the globular clusters (some of which contain
Cepheids).
The Milky Way has about 120 globular clusters,
each containing perhaps 500,000 stars.
One-third of all known globular clusters
covers only 2% of the sky, in the constellation
Sagittarius.
Shapley found
the globular
clusters had a
spherical
distribution.
homepage.mac.com/kvmagruder/bcp/aster/constellations/Sgr.htm
The Sun was removed from the center of the
universe, and placed at an inconspicuous spot
near the edge.
The Great Debate:
Harlow Shapley (left)
vs
Heber D. Curtis (right)
Are the spiral Nebulae (such as M31 = Andromeda)
comparable in size with the Milky Way, or are they
much smaller and near?
In 1925, Edwin Hubble discovered a
Classical Cepheid in M31.
Hubble used Hertzsprung’s calibration
of the period-luminosity relation to
calculate that M31 was over
300,000 pc distant. At this distance,
M31 would be 10 kpc in diameter.
The spiral nebulae are galaxies like
our own.
Classical Cepheids are the
standard candles of the universe
www.pas.rochester.edu/~afrank/A105/LectureXV/LectureXV.html
Embarrassments!
• Our galaxy seemed to be the largest.
• The globular clusters in M31 were underluminous
by a factor of 4.
In 1952, Walter Baade discovered that there are
two types of Cepheids and two period – luminosity
relations.
Population I Cepheids (Classical Cepheids) are relatively rich in heavy elements.
Population II Cepheids (W Virginis stars) are relatively poor in heavy elements.
Pop I Cepheids are four times more luminous than Pop II Cepheids.
outreach.atnf.csiro.au/education/senior/astrophysics/variable_cepheids.html
So …..
•
Hertzspring’s Classical Cepheids (Pop I) were obscured by
dust in the plane of the Galaxy, so luminosities of Classical
Cepheids were calibrated too low by a factor of 4.
• Shapley mistook the Pop II Cepheids in globular clusters for
Pop I Cepheids, so his Pop II Cepheids in the globular
clusters were properly calibrated (luck!).
• Shapley’s distances to the globular clusters were correct.
• Hubble’s Pop I Cepheids in M31 were underluminous by a
factor of 4, so M31(and all other galaxies measured using
Classical Cepheids) was twice as far away as previously
believed, and twice as large.
• The globular clusters around M31 are as bright as those
surrounding our own galaxy.
The Instability Strip on the HR Diagram
DT ~ 600 – 1100 K
density
incr
< hotter
cooler >
period
incr
Luminous Blue Variables
Wolf-Rayet stars
 Cephei stars
Planetary Nebula Nuclei Variables
Miras, Semi-Regular variables
Slowly Pulsating B stars
DO-type Variable white dwarfs
DB-type Variable white dwarfs
DA-type Variable white dwarfs
Some Pulsating Variables
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
II
R
RR Lyrae stars 1.5 – 24 hours
d Scuti stars
1 – 8 hours
I
R, NR
 Cephei stars
3 – 7 hours
I
R, NR
DAV stars
100 – 1000 s
I
NR
R = radial oscillations
NR = nonradial oscillations
RR Lyrae variables in the globular cluster M3
(one night’s observation)
cfa-www.harvard.edu/~jhartman/M3_movies.html
Light and
radial velocity curves
for d Cephei
receding
approaching
Schwarzschild, Harvard College Observatory Circular, 431, 1938
d Cephei radius
Schwarzschild, Harvard College Observatory Circular, 431, 1938
Schwarzschild, Harvard College Observatory Circular, 431, 1938
The star is brightest when
its surface is moving
outward most rapidly, and
not at minimum radius –
a phase lag.
Schwarzschild, Harvard College Observatory Circular, 431, 1938
Consider the adiabatic, radial pulsation of a gasfilled shell.
Linearize the equation of motion
𝑑2 𝑅
𝐺𝑀𝑚
2
𝑚
= 4𝜋𝑅 𝑃 −
2
𝑑𝑡
𝑅2
by setting
𝑅 = 𝑅0 + 𝛿𝑅 and 𝑃 = 𝑃0 + 𝛿𝑃
to get
𝑑2
2𝐺𝑀𝑚
2
𝑚 2 𝛿𝑅 = 8𝜋𝑅0 𝑃0 𝛿𝑅 + 4𝜋𝑅0 𝛿𝑃 +
𝛿𝑅
2
𝑑𝑡
𝑅0
For adiabatic motion,
𝑃 𝑅3
𝛾
= constant
𝛿𝑃
𝛿𝑅
= −3𝛾
𝑃0
𝑅0
Also,
4𝜋𝑅02 𝑃
𝐺𝑀𝑚
=
𝑅02
Set
𝛿𝑅 ∝ 𝑒 𝑖𝜎𝑡
and plug into
𝑑2
2𝐺𝑀𝑚
2
𝑚 2 𝛿𝑅 = 8𝜋𝑅0 𝑃0 𝛿𝑅 + 4𝜋𝑅0 𝛿𝑃 +
𝛿𝑅
2
𝑑𝑡
𝑅0
The result is
−𝑚𝜎 2 𝛿𝑅
𝐺𝑀𝑚
= −3𝛾 + 4
3 𝛿𝑅
𝑅0
or
𝜎2
𝐺𝑀
= (3𝛾 − 4) 3
𝑅0
4
If 𝛾 < , 𝜎 is imaginary
dynamical instability
3
4
If 𝛾 > , the oscillation period is
3
2𝜋
=
P =
𝜎
2𝜋
3𝛾 − 4
4
𝜋𝐺𝜌0
3
5
For 𝛾 = and 𝜌 = 1.41 g cm-3 for the Sun,
3
2𝜋
= 2.78 hours
P=
4
𝜋𝐺𝜌0
3
Compare this with the time for sound to cross a star’s diameter:
2𝑅
=
P=
𝑣𝑠
2𝑅
𝛾𝑃/𝜌
Estimate!
force
𝐺𝑀𝑀/𝑅2
mass
𝑀
𝑃 =
≈
and 𝜌 =
= 3
2
area
𝑅
volume
𝑅
P ≈
2
𝛾𝐺𝜌
The Period – Mean Density Relation
1
period ∝
mean density
density
incr
period
incr
Organ Pipes
and
Pulsating Stars
𝛿𝑟
= 1 at the surface
𝑅
Pulsating Stars are Heat Engines
The Otto cycle. 1. In the
exhaust stroke, the piston
expels the burned air-gas
mixture left over from the
preceding cycle. 2. In the
intake stroke, the piston
sucks in fresh air-gas
mixture. 3. In the
compression stroke, the
piston compresses the
mixture, and heats it. 4. At
the beginning of the power
stroke, the spark plug fires,
causing the air-gas mixture
to burn explosively and heat
up much more. The heated
mixture expands, and does
a large amount of positive
mechanical work on the
piston.
www.lightandmatter.com/html_books/0sn/ch05/ch05.html
• In
1918, Arthur Stanley Eddington proposed that pulsating stars
are heat engines, transforming thermal energy into mechanical
energy. He proposed two mechanisms:
• Energy Mechanism Eddington suggested that when the star is
compressed, more energy is generated by sources in the stellar
core. Ineffective. The core pulsation amplitude is very small.
• Valve Mechanism “Suppose that the cylinder of the engine leaks
heat and that the leakage is made good by a steady supply of
heat. The ordinary method of setting the engine going is to vary
the supply of heat, increasing it during compression and
diminishing it during expansion. That is the first alternative we
considered. But it would come to the same thing if we varied the
leak, stopping the leak during compression and increasing it
during expansion. To apply this method we must make the star
more heat-tight when compressed than when expanded; in other
words, the opacity must increase upon compression.”
But this does not work for most stellar material! Why?
𝜌
opacity ∝ 3.5
𝑇
The opacity is more sensitive to the temperature than to the
density, so the opacity usually decreases with compression
(heat leaks out).
But in a partial ionization zone, the energy of compression
ionizes the stellar material rather than raising its temperature!
In a partial ionization zone, the opacity usually increases with
compression!
Partial ionization zones are the direct cause of stellar
pulsation.
• hydrogen ionization zone (H
H+ and He
He+)
C
T = (1 – 1.5) x 104 K
• helium II ionization zone (He+
C
He++)
T = 4 x 104 K
f
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If the star is too hot, the
ionization zones will be too
near the surface to drive
the oscillations. This
accounts for the
“blue edge” of the instability
strip. The “red edge” is
probably due to the onset
of convection.
www.univie.ac.at/tops/dsn/texts/nonradialpuls.html
Phase lag problem:
A Cepheid is brightest when its surface is moving outward
most rapidly, and not at minimum radius – a phase lag.
• the emergent luminosity varies inversely with the
mass lying above the hydrogen ionization zone
• the luminosity on the bottom of the hydrogen
ionization zone is largest at minimum radius
• the hydrogen ionization zone is moving outwards
(through mass) fastest at minimum radius
• the hydrogen ionization is farthest out ~ ¼ cycle later
• the luminosity peaks ~ ¼ cycle after minimum radius
Nonradial Oscillations
Pulsational corrections df to equilibrium model scalar
quantities f0 go as (the real part of)
𝛿𝑓(𝑟, 𝜃, 𝜑, 𝑡) = 𝛿𝑓 𝑟 𝑌𝑙𝑚 𝜃, 𝜑 𝑒 𝑖𝜎𝑡
l = 0 radial
m > 0 retrograde
m < 0 prograde
m = 0 standing
click here
http://gong.nso.edu/gallery/images/harmonics
Smith, The Astrophysical
Journal, 240, 149, 1980
to Earth
In a rotating star,
frequencies are
rotationally split
(~ Zeeman).
Si III
l = 2, m = 0, -1,
-2
Two Types of Nonradial Modes
www.astro.uwo.ca/~jlandstr/planets/webfigs/earth/slide1.html
Two Types of Frequencies
The acoustic frequency:
2
𝑆𝑙2
𝛾𝑃 𝑙(𝑙 + 1)
=
≈
2
𝜌
𝑟
2𝜋
time for sound to travel
one horizontal wavelength
The Brunt-V@is@l@ (buoyancy) frequency:
𝑁2
1 𝑑𝜌
1 𝑑𝑃
= −𝑔
−
𝜌 𝑑𝑟 𝛾𝑃 𝑑𝑟
2
2𝜋
≈
time for one vertical osc ' n
of displaced mass element
A wave of frequency 𝜎 can propagate only if
𝜎 2 > 𝑆𝑙2 and 𝑁 2 (p modes)
or
𝜎 2 < 𝑆𝑙2 and 𝑁 2 (g modes)
l=2
p modes
a surface
gravity
wave
g modes
Seismology and Helioseismology
5-minute p15 mode with l = 20 and m = 16
www.geophysik.uni-muenchen.de
/research/seismology
Courtesy NOAO
GONG
(Global Oscillation Network Group)
a six-station network of extremely sensitive
and stable velocity imagers
located around the Earth to obtain
nearly continuous observations
of the Sun's "five-minute" oscillations
SOHO
(Solar and Heliospheric
Observatory)
Michelson Doppler Interferometer
(MDI)
- measures vertical motion of
photosphere at one million points
-can measure vertical velocity
as small as 1 mm/s
click
5 hours of MDI Medium-l data
96/09/01
Measurements of Frequencies of
Solar Oscillations from the MDI
medium-l Program by E.J.
Rhodes, Jr., A.G. Kosovichev, P.H.
Scherrer, J. Schou & J. Reiter
sohowww.nascom.nasa.gov/publications/CDROM1/papers/rhodes/
• 5-minute p modes
have a very low
amplitude, ~ 10 cm/s
• dL/L ~ 10-6
• incoherent
superposition of
10 million modes
p2
p1
f
sohowww.nascom.nasa.gov
/publications/CDROM1/papers
/rhodes/
Theory (curves) vs. Data (circles)
Libbrecht, Space Science Reviews, 47, 275, 1988
Some Results for the Sun
• base of convection zone at 0.714 Rsun, where T = 2.18 x 106 K
• mass fraction of helium at surface is Y = 0.2437
• helioseismologically measured sound speed and calculated
sound speed for standard solar model agree to within 0.1%
www.sns.ias.edu/~jnb/Papers/Preprints/solarmodels.html
Rotational Frequency Splitting
in Solar p-Mode Power Spectra
l = 20
Liebbrecht, The Astrophysical Journal, 336, 1092, 1989
The Sun’s Internal Rotation
(a) angular velocity profile
in the solar interior inferred
from helioseismology
Brandenberg, arXiv:astro-ph/0703711, 2007
(b) angular velocity plotted
as a function of radius for
several latitudes
– The contours of constant angular velocity do not show a
tendency of alignment with the axis of rotation, as one would
have expected, and as many theoretical models still show.
– The angular velocity in the radiative interior is nearly
constant, so there is no rapidly rotating core, as has
sometimes been speculated.
– There is a narrow transition layer at the bottom of the
convection zone, where the latitudinal differential rotation goes
over into rigid rotation (i.e. the tachocline). Below 30◦ latitude
the radial angular velocity gradient is here positive,
i.e. ∂W/∂r > 0, in contrast that what is demanded by
conventional dynamo theories.
– Near the top layers (outer 5%) the angular velocity gradient
is negative and quite sharp.
Brandenberg, arXiv:astro-ph/0703711, 2007
Delta Scuti Stars
q2 Tauri
• A to early F stars
• Periods 30 min to 8 hrs
• radial, nonradial p (sometimes g)
Poretti et al, The Astrophysical Journal, 557,1021, 2002
DAV White Dwarfs
• hydrogen atmospheres
dr/r
• mass ~ 0.6 Msun
• r ~ 106 g cm-3
• Te = 10,000 K – 12,000 K
• periods = 100 – 1000 s
nonradial g-modes
trapped in hydrogen surface
layer
• hydrogen partial ionization
zone drives the DAV oscillations
< surface
center >
log(1-r/R)
Winget et al, The Astrophysical Journal Letters, 245, L33, 1981
G191-16
G185-32
G191-16
very
complex!
McGraw et al, The Astrophysical Journal, 250,349,1981
G185-32
Don Winget predicted that the helium partial ionization zone
could drive oscillations in DB (helium atmosphere) white
dwarfs with Te ~ 19,000 K
Te ~ 26,000 K
rotationally split frequencies
Winget et al, The Astrophysical Journal Letters, 262, L11, 1982.
White Dwarf Seismology
• Verify theories of white dwarf structure
• Determine white dwarf rotation rates
• Calibrate cooling rates: Pg ∝ 1/T
white dwarf cosmochronology!
white dwarfs are fossil stars
theoretical cooling rates
+
observed # white dwarfs
of different luminosities
history of star formation!
This suggests an age of 11 Gyr or less for the local disk
www.casca.ca/ecass/issues/2000-JS/fontaine.html
Detailed Asteroseismology of Other Stars
The COROT (COnvection,
ROtation, and planetary Transits)
satellite was launched on
December 27, 2006.
Equipped with a 27-cm diameter
telescope and a 4-CCD camera
sensitive to tiny variations of the
light intensity from stars.
smsc.cnes.fr/COROT/
The Basic Equations
(if you really want To know!)