Download determining stellar parameters from star`s

University of Ljubljana
FACULTY OF MATHEMATICS AND PHYSICS
DEPARTMENT OF PHYSICS
DETERMINING STELLAR PARAMETERS FROM
STAR'S SPECTRUM
Seminar
Sašo Palma
Mentor: dr. Tomaž Zwitter
April 21, 2006
Abstract
The most fundamental properties of a star are its mass, age and chemical composition. Unfortunately, age is not
directly observable and masses can only be determined directly (i.e. dynamically) in select binary systems.
Nonetheless, important parameters, in particular the effective temperature, surface gravity, metallicity, rotational
velocity and radial velocity can be obtained from the stellar spectral energy distribution (stellar spectra). The
GAIA Galactic survey satellite will obtain photometry of over 109 stars in our Galaxy across a very wide range
of stellar types. No other planned survey will provide so much photometric information on so many stars. Given
the size, multidimensionality and diversity of this dataset, this is a challenging task beyond any encountered so
far in large-scale stellar parameterization. The goal of this paper is to introduce the reader to basics of line
formation and line broadening thus help him better understand and (or) analyze stellar spectra. It briefly touches
the methods and exposes the problems with determining stellar parameters, but it gives no solutions to this
problem since this topic should be dealt with with much care, therefore exceeding the scope of this paper.
1. INTRODUCTION ............................................................................................................................................ 1
2. ABSORPTION .................................................................................................................................................. 2
2.1 TRANSITIONS................................................................................................................................................. 2
2.2 SCATTERING .................................................................................................................................................. 3
3. SPECTRAL ABSORPTION LINES ............................................................................................................... 4
3.1 THE BASIC LINE PROFILE ............................................................................................................................... 4
3.2 LINE BROADENING ........................................................................................................................................ 5
3.2.1 Natural broadening .............................................................................................................................. 5
3.2.2 Thermal broadening ............................................................................................................................. 6
3.2.3 Pressure broadening............................................................................................................................. 6
3.2.4 Rotational broadening.......................................................................................................................... 7
4. DETERMINATION OF STELLAR PARAMETERS................................................................................... 7
4.1 THE SYNTHETIC SPECTRA .............................................................................................................................. 7
4.2 THE GAIA RANGE (8480-8740 Å) ................................................................................................................ 8
4.3 EVOLUTION OF LINES WITH TEMPERATURE ................................................................................................... 8
4.4 EVOLUTION OF LINES WITH PRESSURE (GRAVITY) ....................................................................................... 10
4.5 EVOLUTION OF LINES WITH CHEMICAL COMPOSITION [FE/H]...................................................................... 11
4.6 EVOLUTION OF LINES WITH ROTATIONAL VELOCITY (VROT).......................................................................... 11
5. CONCLUSION ............................................................................................................................................... 12
LITERATURE .................................................................................................................................................... 15
i
1. Introduction
We can get a lot of information out of the stellar spectrum. Indelibly recorded on every photograph of stellar
spectrum is a detailed account of the atmospheric conditions at the surface of a star. Strictly speaking, the
spectrum tells us only which radiations the atoms are absorbing or emitting and how intensely. With present
knowledge of atomic structure, the astronomer may now predict just what influence the stellar climate exerts on
a particular atom, and thereby infer the stellar atmospheric conditions from the spectrum. [1]
The most important attribute of stars, and indeed the one that makes it possible for us to see them at all, is high
temperature. The stars are so hot that their material cannot possibly exist in solid or liquid form but must be
entirely gaseous. We shall talk about the effects of the temperature on a stellar spectrum in the following
chapters.
A body at any temperature above the absolute zero always radiates energy. That we call a black body radiation
spectrum which is given by Planck's law. If we integrate the spectrum over all possible wavelengths we get so
called Stefan-Boltzmann's law (the amount of energy emitted per second from each square meter of the star's
surface is ), thus we can calculate the star's luminosity L (assuming a sphere-like object):
L = 4π R 2σ T 4
(1.1)
With help of the eq. (1.1) we can calculate the temperature of the sun. But not everything is as straightforward as
it seems.
In figure 1.1 we compare the energy curve observed from the sun with a theoretical curve for a temperature of
5800K. Although the shapes of the two curves are somewhat similar, the deviations are real. Quite generally,
when proper allowance is made for distortions produced by the absorption lines, it is found that the stellar energy
curves differ from those of an ideal radiator, that is, from black-body curves calculated by theory according to
Planck's law. The stars do not radiate as black or even gray bodies. There are two reasons for these deviations.
One is that temperature increases with depth in stars, so radiation from deeper layers corresponds to higher
temperatures. The other is that the material in the star's atmosphere is not gray but may even be strongly colored;
another way of saying this is that emissivity depends strongly on σ T 4 wavelength. [1] What then do we mean by
the temperature of the star? One way is by making the best fit of the energy curve to the theoretical black-body
curve, the other is to determine the color in which the star radiates the most energy. From many points of view
the most satisfactory definition is the effective temperature, which is the temperature of a perfectly black sphere
of the same size as the star that would have exactly the same total energy output L. So with the eq. (1.1) we get
the effective temperature Te which we than take as the temperature of the star. In the future we won't talk about
other temperatures so we will lose the index and talk only about temperature T.
The shape of the star's spectrum largely depends on the photosphere. The photosphere is the lowest layer of the
stellar atmosphere. Here, the stellar material becomes transparent to electromagnetic radiation. The radiation
reaching us from deeper layers is seen as coming from the photosphere; hence its name. The photospherical gas
is relatively cool; therefore, it absorbs part of the light coming from lower layers, giving rise to an absorption
and emission line spectrum. The dominant feature in most stars’ spectra is absorption line spectrum and that's
where we'll focus our attention on. Spectral lines give us crucial information about the star’s chemical
composition, rotation rate, magnetic phenomena, turbulence… In this seminar we will find out how can we
determine five parameters for the star: temperature (Teff), radial velocity (vr), gravity (log g), metallicity ([Fe/H])
and rotational velocity (vrot).
1
dj/dλ [1013Wm'2Å'1]
Wavelength (Å)
Fig 1.2: The energy distribution in the solar spectrum. At the longer wavelengths the observational data are included by
individual points. The dashed curve is the Planckian (perfect–radiator or black-body) curve for 5800 K in the same units. [1]
2. Absorption
Under the term absorption we mean both absorption as well as scattering of photons. The relevant processes can
be divided into four categories. We separate them according to the initial and final state of the electron. Thus at
the bound-free transition the electron is initially in bound state and afterwards in free state. Therefore we have
rather descriptive names for these categories [3] :
2.1 Transitions
¾
Bound-bound (b-b) transitions. These are atomic or molecular transitions between two bound states (i.e. two
electronic excitation levels, two molecular vibrational levels, etc.). As the energy levels for bound states are
discrete, the b-b contribution to absorption will be heavily dependent on the radiation frequency ν , and the
subsequent spectrum will show sharp, well defined, small-scale features that will be more or less intricate
depending on the structure of available levels (i.e., of the chemical composition of the medium). Only in
rare cases (i.e., transitions between different states of hydrogen atom) we can describe the process with a
simple equation, but other than that we don't have the equation that could describe every line in star's
spectra.
¾
Bound-free (b-f) transitions. This name spans such phenomena as photo ionization and photolysis, in which
a transition occurs between a bound and a free state. Since free states can have any energy, the b-f spectrum
will not depend so strongly on radiation frequency, so it will be relatively smooth, without the small-scale
features that b-b transitions show. There is a minimum energy threshold for b-f transitions: the binding
energy ε n of the state. No radiation with λ > hc ε n will be absorbed at all; once the threshold is reached,
absorption will sharply jump to a maximum and, after that, it will drop as the cross-section for absorption
diminishes with increasing energy. One of the most widely known jumps is Balmer jump at 3646 Å, which
corresponds to the first excited state (n=2) of the hydrogen. These jumps can be seen in Figure 2.1. The
most usual sources of b-f opacity in a photosphere are: neutral hydrogen H I; the negative hydrogen ion H-;
hydrogen molecules H2 and H -2 (for cool stars, where temperature is low enough to allow the existence of
molecules); and metals such as C, Si, Al, Mg and Fe (which give important contributions in the ultraviolet).
2
For very hot stars, helium can become an important source of b-f processes; for very cool stars, molecules
such as CN-, H2O- can also contribute.
¾
Free-free (f-f) transitions. These are processes that involve the absorption (or emission, called
bremmstrahlung or braking radiation) of radiation by a particle, which is accelerated. As f-f processes
involve particles with arbitrary energies, their contribution to opacity will also be smooth. The main
contributors to f-f processes are neutral hydrogen, H-, free electrons, and helium (at long wavelengths in
cool stars).
Absorption for bound-free and free-free transitions is shown in Fig. 2.1 for total absorption as well as for
absorption of some elements mentioned above, which contribute the most to total absorption.
Figure 2.1: (a) Absorption coefficients per unit electron pressure are compared at unit optical depth in a solar model
photosphere. The total absorption is dominated by the H- ion. (b) The absorption coefficients are shown at one optical depth
for a hotter model. (c) The total absorption coefficient of this hot model is dominated by neutral hydrogen. Subscript denotes
bound-free and free-free transitions on all three pictures. [5]
2.2 Scattering
In these processes, a photon is scattered by a particle without being absorbed; the energy of the photon does not
appreciably change. As said before, although the photon is not destroyed, it is driven out of its path; if the
density of scattering particles is high enough, the number of these random deviations will be high enough to
enormously slow down the movement of the photon in a given direction. Scattering is also a source of
continuous opacity, for the same reasons as b-f and f-f processes.
1. If the scatterer is a free particle with a nonzero electric dipolar momentum, it will absorb the incoming photon
and undergo forced oscillations, thereby re-radiating it back in a random direction. This process is called
Thompson scattering when the energy of the radiation is low, and Compton scattering when it is high.
3
2. When the scatterer is a bound particle with a binding energy ε 0 = hc λ0 , the incoming radiation will force it
to oscillate (hence re-radiate) at frequency λ, which in general will be different from λ0; it is said that Rayleigh
scattering takes place:
8π
σR =
3
2
⎛
⎞ λ04
λ04
e2
=
σ
,
⎜
⎟
T
2
4
λ4
⎝ 4πε 0 me c ⎠ λ
(2.1)
where σT is the Thompson scattering cross-section. It can be seen that Rayleigh scattering is more important for
radiation of shorter wavelength. In fact, this higher sensitivity to blue light in respect to red light is responsible
for the blue color of the sky on the Earth: blue light is scattered more efficiently than red one, thus giving the sky
a blue color away from the Sun.
3. Spectral absorption lines
3.1 The basic line profile
Each element has its own spectrum, unique as a human fingerprint. Spectral lines correspond to transitions in
atom, ion or molecule, when an electron changes energy levels. Transitions from higher states to the ground one
(first one, n =1) is called the Lyman series; those to the second energy level, the Balmer series; to the third
Paschen series, to the fourth Brackett series and to the fifth energy level, the Pfund series.
A simplified profile of a spectral absorption line centered on wavelength λ0 is shown in Fig. (3.1). It represents
the detected radiation flux Fλ, divided by the estimated continuum flux Fλc ; this representation is known as a
normalized spectrum.
The deep region near λ0, where absorption is strongest, is the core of the line; the two shallow regions beyond
the core are the wings. The depth D of the line at any λ can be evaluated by
D=
Fc − F
,
Fc
(3.1)
where the subscript λ has been dropped from eq. (3.1) for clarity.
Spectral lines exhibit a wide variety of depths, widths and shapes. All this information must be reduced to a
common magnitude for strengths of different lines (or the same line in different stars) to be compared. This
magnitude is the equivalent width W of the line. It is simply the depth D of the line at each wavelength
integrated over all wavelengths:
∞
W =∫
0
Fc − F
d λ.
Fc
(3.2)
W can be interpreted as the width (in Å) of a box with unit depth; it is marked in grey in Fig. (3.1). The higher
the equivalent width is, the stronger the line will be. So when we talk about the strength of the line, that actually
means higher equivalent width. Typical values of W are of the order of 0.1 Å. Another, less useful measure of
line width is the full width at half-maximum (FWHM), defined as the width for which the depth of the line is 1/2
of its maximum value. [3]
4
3.2 Line broadening
Contrary to what one might expect from the discussion on b-b processes in sec. (2.1), fig. (3.1) shows that a
spectral line is not infinitely thin. It has a finite width, which is caused by several physical phenomena. Each of
these phenomena will “deform” the original line profile to give the one that is observed, but the equivalent width
W will stay the same. If a line is especially sensitive to one or more of these influences, the deformed shape can
yield valuable information about the deforming phenomenon.
Figure 3.1: Schematic absorption line profile. Normalized flux (F / Fc) is plotted against wavelength. Equivalent
width W is indicated by the shaded rectangle. The frontier between the line core and the wings is delimited by
the two dotted lines around λ0 (the position of this frontier is arbitrary). [3]
3.2.1 Natural broadening
According to the energy-time uncertainty relation of quantum mechanics,
ΔE Δt ≈ ,
the energy of the excited state must be undetermined by an amount ΔE ∼
(3.3)
Δt . If we now derivate the equation
for the energy of the photon E = hν = hc λ , we get
ΔE ≈
hc
λ2
Δλ .
(3.4)
Now we can combine eq. (3.3) and (3.4) to get the estimation for the natural broadening of the line
Δλ
λ
≈
λ ⎡ 1
1 ⎤
+
⎢
⎥,
2π c ⎢⎣ Δti Δt j ⎥⎦
(3.5)
which corresponds to transition from i-th into j-th state. Average lifetime of the first two excited levels of H,
which give rise to the well-known Balmer α line at 6562.797 Å, is about 10-8 s, thus eq. (3.5) gives an estimate
of
Δλ
λ
≈ 6.96 × 10−8.
5
(3.6)
3.2.2 Thermal broadening
The particles in the stellar medium are moving with a velocity distribution given by the Maxwell-Boltzmann
distribution
3
2
mv
−
dn
⎛ m ⎞ 2
2
2 kT
v
= n0 ⎜
4
π
.
e
⎟
dv
⎝ 2π T ⎠
(3.7)
The most likely velocity for the above distribution function is
v=
2kT
.
m
(3.8)
Recalling the expression for a Doppler wavelength shift Δλ λ = vR c and combining it with eq. (3.8) it is easily
shown that the thermal broadening is
Δλ
λ
=
1 2kT
≈ 1, 6 × 10−5 ,
c m
(3.9)
which is about 250 times higher then the natural broadening given by the eq. (3.6).
3.2.3 Pressure broadening
Pressure broadening is due to the collisional interaction of the particles that form the lines and those that
surround them. The collisions alter the energy levels of all the particles involved (specially the less bound ones).
Average time between two collisions approximately equals to the quotient of the mean free path of the particle
l=
1
nσ
and its most likely velocity v = 2kT m . When we combine these two values we get
Δt ≈
l
1
=
v nσ 2kT m
(3.10)
and
Δλ
λ
=
λ 1
λ nσ
≈
π c Δt 0 c π
2kT
m
.
(3.11)
For the Hα line mentioned above, at the solar effective temperature and with a hydrogen number density of 1.5
1017 cm3, eq. (3.11) yields
Δλ
λ
≈ 3,59 × 10−8 ,
(3.12)
which is comparable to the result for natural broadening. However, this modest value is due to the low density of
the medium; as density grows, pressure broadening can become a dominant feature in a spectral line. In addition
we have to mention broadening caused by the Stark effect (the splitting of spectral lines in an electric field)
which is most pronounced in the lines of hydrogen and helium. Because each atom in star's atmosphere is
subjected to changing electric field, produced by the electrons and ions that happen to be dashing nearby, the
superposition of the radiations from the different atoms of the same element will not coincide, but will overlap to
produce a broad, fuzzy spectral line. This effect depends on the gravity at the surface of a star. In the relatively
dense atmospheres of a dwarf (high pressure), radiating atoms and their disturbing charges are close together;
consequently the momentary electric fields are larger and the lines become broadened. In the rarefied supergiant
atmospheres (low pressure), the density is usually so low as to render Stark broadening of minor importance.
Hence the lines of hydrogen and helium, although broad and fuzzy in hot dwarf stars, are relatively sharp and
narrow in the supergiants.
6
3.2.4 Rotational broadening
As a star rotates, part of the material is moving toward the observer, and another part is moving away from him.
The radiation due to the approaching part will be Doppler-shifted to shorter wavelengths, and that coming from
the receding part will be shifted to longer ones. The net effect is that any spectral feature will be “stretched”, so
to speak, over a wider wavelength range. A thin spectral line will thus be broadened.
The rotational shift of a spectral line of wavelength λ is given by
Δλ
λ
=
v0 sin i
c
(3.13)
where v0 is the rotation speed at the star’s equator and i is the angle of the rotation axis to the observer’s line of
sight. The combination v0·sin i is often referred to as projected equatorial velocity. The rotational speed of the
Sun is about 1.88 Km/s, thus, assuming a maximum value of i =π/2,
Δλ
(3.14)
≈ 6,3 × 10−6 .
λ
4. Determination of stellar parameters
Now that we have seen the basics behind the line formation and the line broadening it is time to turn our
attention to the main goal of this seminar, which is the determination of the five stellar parameters mentioned in
introduction: temperature (Teff), radial velocity (vr), gravity (log g), metallicity ([Fe/H]) and rotational velocity
(vrot). Out of this five parameters, determination of radial velocity is definitely the one which is the most trivial.
The effect of this parameter to the star's spectrum is merely in the shift of a whole spectrum to higher or lower
wavelengths due to the Doppler effect. In spite the simplicity of the previous sentence, there still exists many
problems which don’t allow us to determine radial velocity within error of about 2 km/s (see section 5). The
determination of other four parameters on the other hand is much more complicated. Since we already shown
that we know the physics behind the line formation and broadening it is possible to calculate how a spectrum
with specific astrometric parameters look like should. These calculated spectra are called synthetic spectra. To
determine the parameters for an observed star we then have to compare a whole set of synthetic spectra with our
observed spectrum to find the best match and thus we can determine the parameters we want. But of course not
everything is as straightforward as it may seem. Let's take it one step at a time…
4.1 The synthetic spectra
A grid of Kurucz spectra (normalized flux F / Fc vs. λ) has been computed in order to study the behavior of
photo spherical lines for different values of effective temperature Teff, surface gravity log g, metallicity [Fe/H]
and rotational velocity vrot. This spectra are based on the Kurucz models of stellar atmospheres. The use and
physical background of his program for calculating synthetic spectra are well discussed in [3]. This models
include some important simplifications:
¾
¾
¾
The thickness of the photosphere is much smaller than the whole stellar radius; therefore, the
photosphere is considered plane, and the photosphere layers, which contribute to stellar spectrum, are
considered to be parallel.
There is no formation or absorption of energy within layers.
We can say that there exists a local thermodynamic equilibrium; temperature in the layer is fixed by
local conditions and the photons from surrounding layers have no effect on it.
The wavelength range chosen is λ=8480 to 8740 Å, which can we call the GAIA range.
7
4.2 The GAIA range (8480-8740 Å)
As its name suggests, this is the wavelength range the GAIA mission will explore. This mission is expected to
provide a stellar catalog that will hopefully be the main reference for at least a few decades; therefore, its
spectral window will become of great importance in the near future. The reasons why this range was chosen for
GAIA is because this is a region with very low lines caused by the Earth’s atmosphere (“telluric” lines), thus
allowing precise ground-based observations for preliminary studies as well as for conformational observations
after the GAIA mission is completed. Besides, there are several strong and sharp, well-defined lines, which are
shown in Fig. 4.1. The most important of these lines are the Ca II triplet. Paschen P13, P14 and P15 lines and
numerous other metal lines (N, Si, Ti, Cr and Fe) are also present; they allow accurate comparison of observed
and model spectra, thus making possible to reliably classify observed stars [3]. Let us now take a closer look on
how each of the parameters (temperature, gravity, metallicity, rotational velocity and radial velocity) changes the
stellar spectra.
Figure 4.1: Some of the most important absorption lines in the GAIA range. The P13 – P15 are 13th to 15th
Paschen lines of hydrogen [3]
4.3 Evolution of lines with temperature
Temperature is the variable that most strongly affects spectral lines, due to the exponential dependence on T of
the ionization and excitation processes. The temperature (kinetic energy of atoms) must be high enough to excite
electrons to higher levels, but low enough not to ionize to many atoms. Excitation process is described with
Boltzmann equation and ionization process is described by the Saha equation.
¾ Boltzmann equation
Ratio between the number of atoms with energy Eb and the number of atoms with energy Ea is:
N b g b − ( Eb − Ea ) kT
(4.1)
=
.
e
Na ga
gb and ga are the number of degenerate states at energies Eb and Ea. Temperature dependence of the ratio
Nb N a for first and ground state for hydrogen is shown in Fig. 4.2. It can be seen that for high ratio of first
excited state one needs very high temperature (above 10,000 K). The cause of detection of hydrogen lines at
much lower temperatures must therefore be somewhere else.
8
¾ Saha equation
The degree to which the gas is ionized is also very important. The ratio between (i+1)-times and i- times ionized
atoms is described with Saha equation:
3
N i +1 2 Z i +1 ⎛ 2π me kT ⎞ 2 − χi
=
e
Ni
ne Z i ⎜⎝ h 2 ⎟⎠
kT
.
(4.2)
Zi+1 and Zi are partition function for (i+1)-times and i- times ionized atom. χ i is the energy required to ionize a
hydrogen atom from its ground state. (from i-times to (i+1)-times ionized). Figure 4.3 is showing temperature
dependence of ratio between ionized hydrogen (H II) and all hydrogen atoms. When we consider both,
Boltzmann and Saha equations, we get for the share of atoms in first excited state the result shown in figure 4.4.
It is clearly seen that the share of atoms in first excited state (these atoms are source for Balmer series, see sec.
3.1) is highest at around 10,000 K.
Figure 4.2: N 2
( N1 + N 2 )
for hydrogen from
Figure 4.3: N II N for hydrogen from Saha equation [2]
Boltzmann equation (4.2) [2]
Figure 4.4: N 2 N for hydrogen from Boltzmann and Saha equation. Notice the similarity with figure 4.5(a) [2]
For more intuitive picture let's take a closer look at Balmer absorption lines produced by hydrogen atoms.
These lines are produced by hydrogen atoms whose electrons are in the second energy level. If the surface of a
star is as cool as the sun or cooler, there are few violent collisions between atoms to excite the electrons, and
most atoms have their electrons in the ground state. As a result, we should expect to find weak Balmer lines in
the spectra of cool stars.
9
In the surface layers of stars hotter than about 20,000 K, however, there are many violent collisions between
atoms, exciting electrons to higher energy levels or knocking the electrons completely out of most atoms. That
is, most atoms become ionized. Therefore, few atoms have electrons in the second energy level to form Balmer
absorption lines, and we should expect hot stars, like cool stars, to have weak Balmer absorption lines.
At an intermediate temperature, roughly 10,000 K, the collisions have the correct amount of energy to excite
large numbers of electrons into the second energy level. With many atoms excited to the second level, the gas
absorbs Balmer wavelength photons well and thus produces strong Balmer lines. (see Fig. 4.5 (a)) [4]
Looking at the Fig. 4.5 (a) we can see that we cannot uniquely determine the temperature if we know the
strength of the Balmer lines. A star with Balmer lines of certain strength might have either of two temperatures,
one high and one low. Luckily, the same process affects the spectral lines of other elements, but the temperature
at which their lines reach maximum strength is different for each element. If we add a number of these elements
to our graph, we get a powerful tool for finding the temperature of stars (see Fig. 4.5 (b)). Of course, the same
discussion as above applies also for Paschen lines, thus making determination of temperature with this system
possible even in the GAIA range.
Figure. 4.5: (a) The strength of the Balmer lines in a stellar spectrum depends on the temperature of the star
(b) A graph of the principal features in stellar spectra can be used as the basis for determining temperature[4]
In addition, eq. (3.9) and (3.11) give us thermal and pressure broadening dependent on temperature. The
evolution of spectral lines with temperature is shown in Fig. 4.6 (page 13). It is clearly seen that metallic lines
that are strong at low temperatures eventually dim away until only hydrogen and helium remain.
4.4 Evolution of lines with pressure (gravity)
Let's take a look why it is appropriate to use log g (logarithm of gravity) as one of the parameters of the star.
Equation
dp
= −ρ g
(4.3)
dr
tells us how the pressure is dropping from center to surface of star. We define optical depth as
dτ = −κρ dr ,
(4.4)
where κ is an absorption coefficient. Optical depth is a depth at which the layer we are observing is located. The
average depth to which we can observe the star, is approximately at τ = 2 3. This is where the temperature is
equal to effective temperature. We can combine both equations to get
dp g
=
(4.5)
dτ κ
We can see that changes in pressure with optical depth are only dependant on gravity and absorption coefficient.
We can assume that g is a constant inside a photosphere, which is a valid assumption considering the thickness
of photosphere (500 km) regarding to the Sun's radius (700.000 km).
10
Gravitational law tells us that
g=
GM
r2
(4.6)
So if we know star's mass M, we can calculate using (4.6) its size r. Instead of g we rather write its logarithm,
but we have to be careful on choosing the units, commonly used in astronomy, which are cm/s2. For the Sun, g
= 274 m/s2 = 27400 cm/s2 and log g = log 27400 = 4,44. The atmosphere is becoming increasingly more
opaque with higher log g, therefore the line strength slightly decreases (Fig. 4.7) (page 13). This is especially the
case for Paschen lines, which are moderately strong in stars with low log g, and are completely gone from stars
with high (Sun like) log g. This high sensitivity of Paschen lines to pressure is attributed to the following fact: as
opacity is lower with low pressure, we see deeper down in the photosphere (both in the optical and geometrical
sense); this means that we see hotter layers, where the Paschen transitions are more likely to be stimulated.
If we compare the effect of higher temperature and higher pressure, we can see that with temperature the
changes are significant, opposite to almost no changes seen with higher pressure, except for previously
mentioned hydrogen lines.
4.5 Evolution of lines with chemical composition [Fe/H]
Metallicity is defined as follows:
⎛ N Fe
⎡ Fe ⎤
⎢ H ⎥ = log10 ⎜ N
⎣ ⎦
⎝ H
⎞
⎛ N Fe ⎞
⎟ − log10 ⎜
⎟ .
⎠
⎝ NH ⎠
(4.7)
NFe is the number of atoms of iron, which for most stars is equal to number of atoms of all metal (in astronomy
the term "metal" is used for every element heavier than helium), NH is the number of hydrogen atoms and sign
stands for the Sun as a reference. So the star with metallicity [Fe/H] = 0 has the same share of metal as the
Sun, the one with metallicity [Fe/H] = 1 has ten times more metal than Sun etc. The evolution of the spectrum
with metallicity [Fe/H] is as expected (Fig. 4.8) (page 14), with lines gaining strength with increasing
metallicity. At very low metallicities ([Fe/H] 7 -2.0), it can be clearly seen that only hydrogen lines (Paschen 13
to 16) and Ca II lines remain. The first case is obvious given that hydrogen completely overwhelms the rest of
the elements; however, the survival of the Ca II triplet is by no means evident, but if we calculate the ionization
fractions for the first four stages of Ca (neutral + 3 ionized) out of Saha's equation (4.2), it is seen that Ca II
completely dominates at temperatures of several thousand degrees [3]. These lines can be expected to appear in a
wide variety of stars, even very metal-poor ones, therefore, they are a prime tool for the determination of, e.g.
their radial velocities by measuring their Doppler shift.
4.6 Evolution of lines with rotational velocity (vrot)
Equation (3.13) gives us the effect that rotating star has on spectra. Star's rotation is one of the reasons for
shallow and wide spectral lines (Fig. 4.9) (page 14). Comparing the results of line broadening for different
physical phenomena (see sec. 3.2) we can see that rotational broadening is quiet prominent feature of line
broadening and the result (3.14) is for slowly rotating star not to mention the effect on fast rotating stars. Notice
the similar effect of rotational broadening to the one caused by temperature dependency. This is a good example
of parameter degeneracy, where fast rotating star produces similar spectrum as slowly rotating star with higher
temperature (see more in conclusion).
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5. Conclusion
For future Gaia mission, which will survey more than 109 stars, the main goal is to find out as much as we can
for each star and with smallest possible errors. As it was shown in this seminar, the best way to determine stars
parameters is to compare synthetic spectra with observed ones and find the best match for each star and thus its
physical parameters. Combining parameters mentioned in this seminar with the parallax and interstellar
extinction, the luminosity, radius and mass of the star can also be determined. The goal is to find automated
methods, which must be more robust and sophisticated than those used to date for stellar classification and
parameterization.
Like we show earlier the spectral line is broadened by several physical phenomena and therefore when trying to
determine several physical parameters from a dataset there exists the problem of parameter degeneracy, i.e. two
different astrophysical parameters manifesting themselves in the same way in the spectrum in certain parts of the
astrophysical parameter space. An example of degeneracy is the effect of higher temperatures on metallicity
determination: in hot stars the metals are ionized leaving only very weak metal lines, making it difficult to
determine metallicity in O and B stars (temperature higher than 20.000 K) at low resolution. There is then the
danger that one could confuse metallicity and temperature characteristics. It is therefore essential that these
parameters are determined simultaneously. Clearly, for degenerate cases, a parameterization algorithm is
required which can give a range of possible parameters, and not just a single set. [6]
There is also a previously mentioned problem of determining radial velocity. The problem lies in the fact that
spectrum has a limited resolution (discrete values of λ at which flux intensity is observed) and therefore it is
difficult to determine the exact position of the minimum of specific spectral line within an error of about 10 Å,
which results in an error of radial velocity up to 4,4 km/s (see more in [2], pages 39 to 43). An important
parameter is also signal to noise ratio (S/N). If this ratio is too low (below 50) the errors can become quite large.
Most of the spectra recorded with GAIA will have low S/N ratio opposite to higher ratio for spectra recorded
with RAVE (Radial Velocity Experiment), which means that with RAVE measurements we will be able to learn
more about physics of stellar atmospheres.
Another problem with finding the best matching synthetic spectrum to the observed one is small differences in
synthetic spectra for spectra with similar parameters. For example, let’s take a look at Fig.4.7, where we can see
spectra for different values of log g. As it was already mentioned in sec. 4.4, there is not a big difference in those
spectra. One might falsely assume that changes are still quite significant (comparing the spectrum with log g = 1
with spectrum with log g = 4.5 we can see that there is a change of about 0.1 in flux). One must bear in mind that
the gravity of the star with higher log g is 4500 times higher than of the star with lower log g, which is a lot. If
we now try to determine log g within an error of 0.1, then we can imagine how small are the changes in flux for
spectra with similar log g. And if we also add let’s say small S/N ratio on top of that, we can see how quickly
one can make significant errors in determining wanted parameters.
Therefore there is still a lot of work to be done for the modern scientist to find the best (better) methods and
equipment to be able to determine stars parameters with as smallest errors as possible.
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Figure 4.6: Synthetic spectra for different values of T. Other parameters are log g = 4,5, vrot = 0,0 km/s and
Fe/H] = 0,0. [2]
Figure 4.7: Synthetic spectra for different values of log g. Other parameters are T = 5750 K, vrot = 0,0 km/s and
[Fe/H] = 0,0. [2]
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Figure 4.8: Synthetic spectra for different values of [Fe/H]. Other parameters are T = 5750 K, vrot = 0,0 km/s
and log g = 4,5. [2]
Figure 4.9: Synthetic spectra for different values of vrot. Other parameters are T = 5750 K, [Fe/H] = 0,0 and
log g = 4,5. [2]
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Literature
[1] Lawrence H. Aller, Atoms, Stars and Nebulae, revised edition, Hardward University
Press, Cambridge, Massachusetts, 1971, chapter 4.
[2] S. Zamuda, Določanje parametrov zvezd iz spektrov misije GAIA, diploma thesis,
University of Ljubljana, 2003.
[3] U. Jauregi, Simulation of Stellar Photospheres with Kurucz Models: Analysis of a Model
Grid and of Line Formation Depth. A Preliminary Study of the Peculiar Variable V838
Mon., diploma thesis, University of Ljubljana, 2002.
[4] M.A.Seeds, Horizons – Exploring the Universe, Wadsworth publishing company, 1998,
chapter 6.
[5] T. Zwitter, GAIA accuracy on radial velocities assessed from a synthetic spectra
database, Astronomy and Astrophysics, astro-ph/0202312 v2, feb. 2002.
[6] Bailer-Jones, C.A.L., Determination of Stellar Parameters with GAIA, Kluwer Academic
Publishers, 2002, astro-ph/0201014 p. 1-3
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