Good Vibrations and Stellar Pulsations Brad Carroll Weber State University April 11, l = 2, m = 0 l = 5, m = 3

In August of 1596, David Fabricius (Lutheran pastor and amateur astronomer) observed  Ceti, a 2 nd 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  Cephei is variable:  = 5 days, 8 hours

 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 (  m/5) = 100 (0.9/5) = 2.3

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). Edward Charles Pickering’s Computers at Harvard Observatory

Henrietta Swan Leavitt Found 2400 Classical Cepheids In 1912, discovered the Period-Luminosity Relation (1868 – 1921)

Small Magellanic Cloud

Cepheids in the SMC From Shapley, Galaxies, Harvard University Press, Cambridge, MA, 1961.

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  = 6.6 days. d (pc) = 4.16/slope (in arcsec/yr) (4.16 AU/yr is the Sun’s motion) Calibration: The Distance to a Cepheid

Sandage and Tammann, The Astrophysical Journal, 151, 531, Period – Luminosity Relation d (pc) = 10 (m-M+5)/5 M = log 10  d – 1.43

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

antwrp.gsfc.nasa.gov/apod/ap html What is the Milky Way?

Thomas Wright, An Original Theory on New Hypothesis of the Universe, 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.

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. The Great Debate: Are the spiral Nebulae (such as M31 = Andromeda) comparable in size with the Milky Way, or are they much smaller and near? Harlow Shapley (left) vs Heber D. Curtis (right)

Classical Cepheids are the standard candles of the universe

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  T ~ 600 – 1100 K density period incr 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 TypeRange of Periods Population Type Type of Oscillation Long-Period Variables 100 – 700 days I, IIR Classical Cepheids 1 – 50 daysIR W Virginis stars 2 – 45 daysIIR RR Lyrae stars1.5 – 24 hoursIIR  Scuti stars 1 – 8 hoursIR, NR  Cephei stars 3 – 7 hoursIR, NR DAV stars100 – 1000 sINR R = radial oscillations NR = nonradial oscillations

RR Lyrae variables in the globular cluster M3 (one night’s observation) cfa-

Schwarzschild, Harvard College Observatory Circular, 431, 1938 Light and radial velocity curves for  Cephei approaching receding

 Cephei radius 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 gas- filled shell. Linearize the equation of motion by setting to get

For adiabatic motion, Also, Set and plug into

The result is or dynamical instability 

Compare this with the time for sound to cross a star’s diameter:  Estimate!   for the Sun,

The Period – Mean Density Relation density period incr incr

Organ Pipes and Pulsating Stars

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. Pulsating Stars are Heat Engines

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? 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 + ) T = (1 – 1.5) x 10 4 K helium II ionization zone (He + He ++ ) T = 4 x 10 4 K 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.   1stovertone1stovertone fundamentalfundamental nopulsatIonnopulsatIon

A Cepheid is brightest when its surface is moving outward most rapidly, and not at minimum radius – a phase lag. Phase lag problem: 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  f to equilibrium model scalar quantities f 0 go as (the real part of) l = 0 radial m > 0 retrograde m < 0 prograde m = 0 standing click here

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

Two Types of Frequencies The acoustic frequency: The (buoyancy) frequency:

or l = 2

p modes a surface gravity wave

g modes

Seismology and Helioseismology 5-minute p 15 mode with l = 20 and m = 16 Courtesy NOAO /research/seismology

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

sohowww.nascom.nasa.gov/publications/CDROM1/papers/rhodes/ 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 click

5-minute p modes have a very low amplitude, ~ 10 cm/s  L/L ~ incoherent superposition of 10 million modes f p1p1 p2p2 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 R sun, where T = 2.18 x 10 6 K mass fraction of helium at surface is Y = helioseismologically measured sound speed and calculated sound speed for standard solar model agree to within 0.1%

Liebbrecht, The Astrophysical Journal, 336, 1092, 1989 Rotational Frequency Splitting in Solar p-Mode Power Spectra l = 20

(a) angular velocity profile in the solar interior inferred from helioseismology (b) angular velocity plotted as a function of radius for several latitudes The Sun’s Internal Rotation Brandenberg, arXiv:astro-ph/ , 2007

– 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. ∂  /∂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/ , 2007

Delta Scuti Stars 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  2 Tauri

DAV White Dwarfs hydrogen atmospheres mass ~ 0.6 M sun  ~ 10 6 g cm -3 T e = 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 Winget et al, The Astrophysical Journal Letters, 245, L33, 1981  r/r log(1-r/R)

McGraw et al, The Astrophysical Journal, 250,349,1981 G G very complex!

Don Winget predicted that the helium partial ionization zone could drive oscillations in DB (helium atmosphere) white dwarfs with T e ~ 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:  g 1/T white dwarf cosmochronology! theoretical cooling rates + observed # white dwarfs of different luminosities history of star formation! white dwarfs are fossil stars

This suggests an age of 11 Gyr or less for the local disk

Detailed Asteroseismology of Other Stars The COROT (COnvection, ROtation, and planetary Transits) satellite was launched on December 27, 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!)