2.4 Pulsation as Cause of Variability
2.4.1 Stellar Structure and Evolution
Nowadays, the theory of stellar structure and evolution is based on equations describing the hy- drostatics (or hydrodynamics) of the stellar interior. Major contributions were made by Atkinson (1931), Bethe (1939), Bethe and Marshak (1939) and Gamov (1939).
The description of stellar structure and evolution below is based mostly on Catelan and Smith (2015).
In the following, a spherically symmetric, self-gravitating star is assumed. The properties of the stellar matter at any point of the stellar interior are described with density ρ, temperature T , pressure P , entropy per unit mass S, coefficient of thermal conductivity per unit volume λ, and the chemical composition, based on the abundance of elements Xi. The mass of the star is assumed
to be M , its radius R.
The basic set of equations of a star in hydrostatic and thermal equilibrium then consists of the following four time-independent differential equations (Kippenhahn and Weigert 1990). Of them, the first two describe the mechanical structure of the star, and the last two its energetic and thermal structure:
∂r ∂Mr = 1 4πr2ρ (2.2) ∂P ∂Mr = −GMr 4πr4 (2.3) ∂L ∂Mr = − ν − g (2.4) ∂T ∂Mr = −GMrT 4πr4P∇ (2.5) with
Mr: stellar mass enclosed in radius r of a star with total mass M (2.6)
L: luminosity in units of J s−1 (energy per unit time) (2.7) : energy generation rate in the form of thermonuclear reactions (2.8) ν: energy loss rate in the form of neutrinos, important in late (2.9)
stages of evolution
g: work performed on the gas during any expansion or contraction (2.10)
of the star,
i.e., the total heat flux through a spherical shell with radius r, is given as luminosity L = 4πr2F =
−4πr2λ∂T /∂r.
The temperature gradient ∇ ≡ ∂ ln T
∂ ln P in Equ. (2.5) depends on the modus of energy transport,
being primarily radiative (∇rad), conductive (∇c) or convective (∇conv).
In the case of radiative transport, ∇ takes the form
∇ = ∇rad=
3 16πacG
κRLP
M T4 (2.11)
where a = 4σc = 7.5657× 10−16Jm−3K−4 is the radiation constant, with the Stefan-Boltzmann
constant σ. The Rosseland mean opacity κR (per unit mass) is defined as
κR= ∞ R 0 1 κν ∂Bν(T ) ∂T dν ∞ R 0 ∂Bν(T ) ∂T dν (2.12)
with the coefficient of monochromatic radiative opacity (per unit mass) κν = 4acT
3
3ρ c/ν1 , and the
Convective regions in the interior of a star are identified by the Ledoux criteria (Ledoux 1947), vrad > vad−
χµ
χT∇µ
(2.13)
with the molecular weight µ and
χµ ≡ d ln P d ln µ ρ,T (2.14) χT ≡ d ln P d ln T ρ,µ (2.15) ∇µ ≡ d ln µ d ln P . (2.16)
In the absence of a chemical composition gradient∇µ, this reduces to the Schwarzschild criterion
(Schwarzschild 1906) for the onset of convection ∇rad >∇ad.
In the above, partial derivatives have been used to emphasize the non-stationary nature of the physical solutions. They evolve over time as a consequence of the nuclear processes in the interior of the star.
The fact that stars are radiating away energy, because they are luminous, implies they are not stationary. Also, due to the nuclear processes providing the energy, their chemical composition has to change over time. Therefore, the equations above have to be supplemented by a set of equations to describe the evolution of the abundance ∂Xi/∂t and thus introducing a nuclear time
scale:
As the energy released by fusing a mass ∆M of H into He is ∼0.007 ∆Mc2, the time until the H is exhausted, given the star’s current luminosity, will be:
tnuc=
0.007 ∆M c2
L , (2.17)
which is≈ 1010− 1011yr for our Sun.
However, the actual lifetime of a star is only one tenth of tnuc because it changes its luminosity
to become brighter during its evolution.
In the following, the Equations (2.2) to (2.5) are modified to describe the time evolution of a spherical symmetric star having a given distribution of chemical abundances Xi(Mr).
In the case of a pulsating star, the system is outside hydrostatic equilibrium, as there is no longer perfect balance between pressure gradient and gravity. In this case, Equ. (2.2) to (2.5) changes as the mass element will undergo acceleration, and becomes:
∂r ∂Mr = 1 4πr2ρ (2.18) ∂P ∂Mr = −GMr 4πr4 − 1 4πr2 ∂2r ∂t2 (2.19) ∂Lr ∂Mr = − T∂S ∂t = − ν− g (2.20) ∂T ∂Mr = − 3κLr 64π2acT3r4. (2.21)
Here, three kinds of derivatives appear: • ∂2
r/∂t2 in Equ. (2.19) describes hydrodynamical changes to the stellar structure. These
changes occur on the dynamical time scale τdyn =
R3
GM
1/2
' (G¯ρ)−1/2.
• T ∂S/∂t in Equ. (2.20) which is often written as an additional energy generation term g:
g =−T
∂s
∂t. (2.22)
This term describes changes to the stars thermal structure, resulting from contraction (g > 0)
or expansion (g < 0). Such changes occur on the Kelvin-Helmholtz timescale
tKH =
GM2/R
L . (2.23)
The Kelvin-Helmholtz time scale indicates the time required to radiate the current gravitational binding energy of the Sun at its current luminosity; this is the timescale on which the Sun would contract if its nuclear energy sources were turned off. For our Sun, tKH ∼ 3 × 107yr.
As τnuc τKH τdyn, changes in the stellar chemical abundance occur on much larger timescales
than the dynamical timescale.
If the time derivative in Equ. (2.19) vanishes, the star is in hydrostatic equilibrium. If the time derivative in Equ. (2.20) vanishes, the star is in thermal equilibrium.
It is always assumed – no matter if the star is in hydrostatic or thermal equilibrium – that the conditions of a local thermodynamic equilibrium are satisfied.
These equations have to be supplemented with boundary conditions (Catelan and Smith 2015). The boundary conditions for the differential equations of stellar evolution constitute an important part of the overall problem. At the stellar center, two adjustable parameters exist: The central density ρc, and the central temperature Tc. At the stellar surface, there are also two adjustable
Central boundary conditions
At the center (r = 0) of the star, the enclosed mass Mr, the radius r and the luminosity Lr have
to vanish and the energy generation rate must remain finite. Therefore, both Mr and L must
vanish at the center: Mr= 0, Lr= 0 at r = 0.
As nothing is known a priori about the central values of P and T , the remaining two boundary conditions must be specified at the surface rather than at the center.
Surface boundary conditions
At the surface (Mr = M ), the boundary conditions are generally much more complicated than
at the center. The simplest option is to take the zero boundary conditions T = 0 and P = 0 at the surface. However, in reality, T and P never become zero because the star is surrounded by an interstellar medium with low, but finite density and temperature. Another option is to set the outer boundary conditions to ρ = 0, T = 8πRL2σ
1/4
, where σ is the Stefan-Boltzmann constant.
A more realistic option is to identify the surface with the star’s photosphere, which is where the bulk of the radiation escapes and which corresponds to the visible surface of the star. The photospheric boundary conditions approximate the photosphere with a single surface at optical depth τ = 2/3 (Catelan and Smith 2015). One can write
τph= ∞ Z R κρ dr≈ κph ∞ Z R ρ dr, (2.24)
where κph is an average value of the opacity over all layers above the photosphere. Assuming
the atmosphere is geometrically thin, thus its extent is very small compared to R, it follows
dP
dr =−GMr2 ρ ⇒ P (R) ≈ GMR2
∞
R
R
ρ dr. Since τph = 2/3 and T (R) ≈ Teff, the boundary conditions
can be written as Mr= M : P = 2 3 GM κphRL , L = 4πR2σT4. (2.25)
The Hertsprung-Russell Diagram
In 1912, two astronomers – Ejnar Hertzsprung and Henry Norris Russell – found independently that when stars are plotted accordingly their temperature and luminosity, the majority of them fall on a smooth curve (Hertzsprung 1911). The properties plotted originally were properties which can be determined observationally, e.g. the absolute visual magnitude MV vs. the B − V color
index. The resulting diagram turned out to be a significant tool to understand stellar evolution, and was thus named after Hertzsprung and Russell.
Fig. 2.9 shows a Hertzsprung-Russell diagram (HR diagram) in log L/L vs. log Teff with common
types of stars. Also, different types of pulsating stars within the region of instability, the instability strip, are shown.
The basic HR diagram is a luminosity vs. temperature graph. The temperature may be replaced or supplemented with spectral class or color index. The main spectral classes in order from hottest to coolest are O, B, A, F, G, K and M. Within the HR diagram, the stars which lie along the nearly straight diagonal line are known as main sequence stars. Those stars are still burning hydrogen in their cores. The main sequence accounts for about 90 percent of the stellar population above 0.5 M (Arnett 1996).
There is a correlation between a main sequence star’s mass and luminosity. Stars that are high up on the main sequence are more massive. The relation, the mass-luminosity relation (Catelan and Smith 2015), says that L∝ M3.5.
Stars younger and older than main sequence stars are called pre- and post-main sequence stars. Stars that have evolved well beyond the main sequence are often on the red giant branch of the HR diagram, or might be asymptotic giant branch stars. RR Lyrae are found on the horizontal branch, or β Cephei stars on the upper main sequence.
The cooler, dimmer stars are found towards the lower right of the HR diagram, and the hotter, more luminous stars in the upper left. Old red giants are found at the red giant branch (RGB). Our own star, the Sun, is located nearly in the middle of both the temperature and luminosity scales relative to other stars.
Because of the relation L = 4πR2σT4, stars above the main sequence (having higher luminosity, with the same temperature as cooler main sequence stars) have larger radii. Also, stars having the same luminosity as dimmer main sequence stars, but are to the left of them (being hotter), have smaller radii.
Stellar Evolution and the Instability Strip
One of the key concepts of modern astronomy is that stars change over time – they are born from clouds of interstellar gas and dust, they shine over billions of years by light created through nuclear fusion of hydrogen (H) in their cores, and eventually run out of their nuclear fuel and die. During their last stages, they return some of their mass back to interstellar space, that will be taken up into new generations of stars. The process of change a star undergoes during its lifetime is called stellar evolution.
As such processes take millions to billions of years for a star, we can’t observe them directly. However, there are many pieces of evidence that formed the current understanding of stellar evolution. One was the understanding of the nuclear physics responsible for why stars shine over such a long time, and the subsequent realization that their fuel is nuclear, and thus large but finite.
Another piece of evidence was the observational study of star clusters – groups of stars all born at the same time and place, and thus from the same composition under the same conditions –
log T
efflog (L/L
)
5.0 4.5 4.0 3.5 6 4 2 0 -2 -4 O B A F G K MRNS LT Solar-like ZAMS CO-WD He-WD ELM-HeV hot-DAV V777 Her = DBV DQV ZZ Cet = DAV EHM-DAV Hya x GW VirGW Vir sdOV DW Lyn
ZA HB RR Lyrae roAp Sct d SX Phe Dor g SR Mira RV Tau Cep d W Vir BL Her BL Boo R CrB LBV WR Cyg a PV Tel Cep b 53 Per = SPB
WHITE DWARFS
SUPERGIANTS
GIANTS
MAIN SEQUENCE
INST
ABILITY
STRIP
Figure 2.9 Schematic distribution of different types of pulsating stars across the Hertzsprung-Russell diagram. The instability strip, including RR Lyrae and Cepheids, is indicated. Own figure; main features are taken from Catelan and Smith (2015).
and the eventual realization that the properties of star clusters differ depending upon how old they are.
During their evolution, almost all stars have phases of instability, leading to variability in their light curves. Above, theoretical models of stellar structure and evolution and statements for instability were given. Here, the problem is described and discussed from a more observational view.
Stars being in a state of instability can be found in an area of the HR diagram called the instability strip (Gautschy and Saio 1996). This is an area around 1000 K wide just above the main sequence, as shown in Fig. 2.9. It includes Cepheid variables where it intersects the supergiants, RR Lyrae variables where it crosses the horizontal branch, as well as δ Scuti stars, rapidly oscillating Ap (roAp) stars and others near the main sequence.
Stellar Mass and Stellar Evolution
The HR diagram indicates that a star of a given brightness could only lie within a certain range of colors, and a star with a given color could only be found within a certain range of brightnesses. More observational and theoretical research showed that the HR diagram represents a snapshot of the evolutionary states of the stars within the diagram. As a star evolves, it changes in brightness and color in a very predictable way, and stars of different masses change in very different ways. The progress of a star’s life is predetermined by its mass, because mass is what determines the amount of energy being produced and how fast its evolution will be.
In the HR diagram, four large regions can be identified (Catelan and Smith 2015): • red dwarfs: M < 0.7 M , their main sequence lifetime exceeds age of the Universe
• low-mass stars: 0.7 M < M < 2 M , they end lives as white dwarfs and possibly planetary
nebulae
• intermediate-mass stars: 2M < M < 8− 10 M , similar to low-mass stars, but higher L,
they end as higher mass white dwarfs and planetary nebulae
• high-mass stars: M > 8 − 10 M , distinctly different evolution paths, end as supernovae,
leaving neutron stars or black holes.
The age of a star tells which evolutionary stages it has passed. Both of this quantities are hard to measure directly, but are related to temperature and luminosity. Within the HR diagram, evolutionary tracks can be identified. Fig. 2.10 to 2.12 show them depending on the mass ranges specified above. The following description of the evolutionary tracks of stars as depending on their mass is oriented on Catelan and Smith (2015), with the evolutionary tracks therein based on simulations by A. V. Sweigart using a 1D hydrostatic code.
Low-mass stars (Fig. 2.10) start from a molecular cloud that becomes unstable according to the Jeans criterion for instability (Jeans 1902) and thus collapses and fragments (Bodenheimer 2003). The temperature at this early phase (isothermal phase) is of order 10 K and the cloud is optically thin.
Eventually the opacity increases and the temperature rises, forming a proto-star1l. A hydrostatic core forms after 1.5× 105 years (Wuchterl and Klessen 2001) 2l, luminosity and temperature
increase steadily. When accretion stops, the photosphere of the proto-star becomes visible 3l. The star becomes fully convective, and luminosity decreases at about constant temperature, leading to the vertical segment of the evolutionary track, the Hayashi track. After this point, the proto-star becomes hotter and increases in brightness, while shrinking in size. Increasing core temperature leads to incomplete CNO processing, with 12C being consumed into the core. Until 12C is exhausted, additional 5×107 years will have passed. The star has now reached the zero-age
main sequence (ZAMS)4l. The star evolves along the much longer nuclear time scale, which is of order 1010years for low-mass stars. Temperature and luminosity slightly increase with time as H is steadily transformed into He. When the star reached the turn-off point5l, H burning stops to be
a central process and instead becomes a shell-burning process. Not all the energy released by the H-burning shell reaches the surface: part of it expands the star’s envelope. The star begins to cool down and enters the subgiant phase. When the convective envelope reaches its maximum inwards extension after additionally 109years6l, the dredge-up of material occurs. This material has been
partially processed nuclearly and includes a small amount of He. The H-burning shell continues to advance outward in mass 7l, thus leading to a continued increase of mass in the He core. The H-burning shell actually encounters the chemical composition discontinuity that was left behind from the dredge-up phase. The remaining time on the RGB 8lis characterized by the increasing size of the He core, which keeps on contracting and heating up. At this stage, large amounts of energy are lost in the form of neutrinos. When temperature finally becomes high enough for the He-burning reactions, this happens in a shell inside the He-rich core, leading to the helium flash. The zero-age horizontal branch (ZAHB) 9lmarks the onset of the He-burning phase. In the case of low-mass stars, it is referenced to as the horizontal branch (HB) phase, because HB stars have very nearly the same luminosity irrespective of mass. When He has been exhausted at the center after additionally 108 years10l, the star continues as a low-mass asymptotic giant branch (AGB)
star. An AGB star has an inert core comprised mostly of C and O, that burns He in a shell and H in another shell further out. The onset of He-shell burning causes a temporary reversal in the star’s evolutionary direction. This leads to the AGB clump in the observed color-magnitude (CM) diagrams of well-populated globular clusters and old Local Group galaxies. The details of the AGB phase, which lasts of order 107 years, depend strongly on the poorly known mass loss rates. Once the AGB star’s envelope mass has become very low, a final mass ejection phase may take place, the so-called superwind phase, leading to the formation of a post-AGB star. After a quick evolution to the blue, the star finally settles on the white dwarf (WD) cooling sequence.