In this insert, we outline the simplest approach to
star
modeling
in order to understand how it is done
and how models
are fitted to observations.
The fit verifies the
models
insofar as they are realistic: i.e., contain the right physical structures
and the right physics theories.
Verified models
then allow us to know things about
star which
CANNOT be known by direct NOR indirect observations
(which are both discussed in
IAL 19: Star Basics I: Introduction).
As preview/review of some of the topics of to come in
IALs
on stars,
see the Pleiades
open star cluster
in the figure below/above
(local link /
general link: pleiades.html).
In the sections following this one, we explicate further the ingredients of
star
modeling: i.e., sections
Star Shape and Structure,
Stellar Composition,
Luminosity, Flux, Photometry,
Distance and Stellar Parallax,
Luminosity Determination and Spectroscopic Parallax.
In the subsections below, we cover star
modeling itself:
The very basic qualitative
model of a star
is a large self-gravitating sphere of hot gas.
Going beyond this to quantitative star
modeling
necessarily includes the relevant
physics theories:
hydrostatics,
hydrodynamics,
nuclear physics,
quantum mechanics,
radiative transfer,
statistical mechanics,
thermodynamics,
and whatever other physics theory
is needed.
Going beyond the very basic qualitative
star
model
actually requires two quantitative models:
the (interior) stellar structure model
and the
stellar atmosphere model.
The stellar structure model
is entirely about what we CANNOT observe since it is all beneath the
opaque photosphere.
The reason for needing two
models is that
the scales
of stellar structure modeling
and
stellar atmosphere modeling
are so different that doing them in one
model is vastly impractical.
The two kinds of models
can be fitted together:
outer boundary conditions
of the stellar structure model
are the inner boundary conditions
of the stellar atmosphere model
and vice versa.
The two kinds of models
are connected as we discuss below
subsection Stellar Atmosphere Models.????
(Well we will one day when I write that up explicitly.???)
Since we are discussing very simple
modeling,
we ignore the complications of
stellar rotation,
stellar magnetic fields,
and
star-star
interactions which occur
close binaries.
Close binaries
show stellar evolution
NEVER seen for single
stars.
One can develop a simple
stellar structure model
for a main-sequence star
with only
mass and composition as
free parameters.
The stellar mass
range is ∼ 0.08--300 M_☉
(i.e., solar masses).
The composition
is usually the
solar composition
with metallicity (Z)
as a free parameter
for zero-age main sequence (ZAMS)
(i.e., the time right after
star formation).
As a star ages,
hydrogen burning
in the stellar core
converts hydrogen (H)
to helium-4 (He-4).
In the
post-main-sequence star phase,
there is complex layered
nuclear burning including
hydrogen burning,
helium burning,
and nuclear burning
of heavier elements.
The
post-main-sequence star phase
is discussed in
IAL 23: The Post-Main-Sequence Life of Stars
Note that the hypothetical
Population III stars
(see below section
Population I, II, and III Stars) that
formed at cosmic time ∼< 1 Gyr
may have had stellar masses
much larger than 300 M_☉
and have the
primordial cosmic composition (fiducial values by mass fraction:
0.75 H, 0.25 He-4, 0.001 D, 0.0001 He-3, 10**(-9) Li-7).
What a
stellar structure model
gives you is "runs" of quantities (i.e., their distribution with
radius coordinate).
The figure below
(local link /
general link: sun_model_interior.html)
is an example of
the "runs" for a
stellar structure model
of the Sun with
a brief discussion of how
stellar structure models
are calculated.
We discuss stellar structure models
further in
IAL 22: The Main Sequence Life of Stars:
Stellar Structure and Stellar Modeling.
The only SYNTHETIC direct observable calculable from
simple stellar structure models is
luminosity
(i.e., energy output per unit time:
i.e., power)---which
is typically given in
units of
solar luminosities L_☉ = 3.828*10**26 W.
SYNTHETIC luminosities
from
stellar structure models
can be compared to (observed)
luminosities
as test of the accuracy of the
stellar structure models.
But luminosity
is only a direct observable
when you know the distance to the
star
(see sections Distance and Stellar Parallax
and
Luminosity Determination and Spectroscopic Parallax),
can effectively integrate observed flux
over all wavelengths
(see section Luminosity, Flux, Photometry),
and can account for
extinction.
Determining and correcting for
extinction
is a major problem in
astronomy, but it's too intricate a subject for
IAL---and
so we will skirt it.
However, comparisons of
SYNTHETIC luminosities
and observed
luminosities
are only a very limited test.
Stars
of the same luminosities
may in general have very different
phases (main sequence,
post-main-sequence,
masses,
and stellar structure.
Neither of the prime
free parameters of
stellar structure models
(i.e.,
stellar mass
and stellar composition)
are direct observables in general.
So they CANNOT be set by direct observation.
In fact, the only SYNTHETIC direct observable calculable from
simple stellar structure models is
luminosity
(i.e., energy output per unit time:
i.e., power)---which
is typically given in
units of
solar luminosities L_☉ = 3.828*10**26 W.
Determining and correcting for
extinction
is a major problem in
astronomy, but it's too intricate a subject for
IAL---and
so we will skirt it.
But in general we do NOT have
stellar mass
and often NOT luminosity.
What we directly observe for
stars are
photometry
(broad
wavelength band
measurements of flux:
see section Luminosity, Flux, Photometry below),
spectroscopy
(narrow
wavelength band
measurements of flux),
and, for sufficiently near stars, distance.
Spectroscopy
gives more detailed information than photometry,
but is harder to obtain to the same level of accuracy and for distant
stars NOT obtainable at all.
With sufficient
photometry
and spectroscopy
we can model
the stellar atmosphere:
i.e., create
a model
of the
stellar atmosphere.
Adjusting the
free parameters
of the model
to fit the
photometry
and spectroscopy
gives us values for those
free parameters.
The values are as good as the
photometry,
spectroscopy,
and modeling allow.
From the above, we can obtain three fitted
free parameters:
stellar atmosphere
(or nearly equivalently metallicity Z),
gravitational field g,
and effective temperature.
Alas, the there are 3 unknowns M, R, and L
for 2 equations: the ones for
gravitational field g,
and effective temperature.
So we cannot solve for M, R, and L separately without more information.
If we had any 2 of M, R, and L, and some estimate of core composition,
then stellar structure model
could be fitted and we would understand the
star insofar as simple
modeling allows.
However, as discussed in section
Stellar Structure Models and Observables
we usually do NOT have
stellar mass
or luminosity.
And stellar radius
is known to accuracy/precision
only for the Sun.
A few red supergiants
can be barely resolved and have their
radii determined: e.g., Betelgeuse
with R = 887(203) R_☉ ≅ 4 AU
(see solar radius R_☉ = 6.957*10**5 km = 109.1 R_eq_⊕ = 4.650*10**(-3) AU;
Dolan et al. 2016).
So are we stuck?
No.
Distances (and therefore luminosities)
can be obtained for relatively nearby stars
by stellar parallax
(see section Distance and Stellar Parallax below).
Distances by stellar parallax
were originally only obtainable to
very nearby stars
within a few
parsecs
in the 19th century,
(see Wikipedia: Stellar parallax:
19th and 20th centuries),
but with advancing technology, smaller
stellar parallaxes,
and so greater distances have been obtained progressively.
As of 2018, the
Gaia spacecraft (mission 2013--2025?)
has provided us with
accurate/precise
stellar parallax
to distances up to 8 kpc
(see Wikipedia:
Gaia spacecraft: Objectives)) which is about the
distance to center of the Milky Way.????
Also as mentioned above in section
Stellar Structure Models and Observables,
stellar mass
is a direct observable for some
binary star systems.
The upshot is that
we can nowadays fit
stellar models
(both stellar structure models and
stellar atmosphere models)
to vastly many stars
and thereby understand them
and know their parameters insofar as our modeling is sufficiently realistic.
We can go well beyond the simple modeling described here and include
stellar rotation,
stellar magnetic fields,
and
star-star
interactions which occur
close binaries.
As previewed in IAL 8: The Sun
and as discussed at greater length in
IAL 20: Star Basics II,
stars
can be classified by
spectral type.
The full
spectral type classification
(which includes its
luminosity class)
very full characterizes
a star.
The spectral type classification
is just empirical: it is a direct observable.
Now all stars
of the same full
spectral type classification
are very much alike
and so all have the same
stellar model
(both stellar structure model
stellar atmosphere model)
insofar as they are alike.
So we do NOT have to model every
star
we want to understand.
We just have to model
stars of all
spectral types.
And this has been done.
Now there remain imperfections in
spectral type classification
and modeling of stars,
but there is continually work to reduce those imperfections.
The current status is that we understand
main-sequence stars
in their bulk properties and evolution very well.
Pre-main-sequence stars
and
post-main-sequence stars
are more difficult to model, and so are less well understood, especially quantitatively.
A general reason is that
pre-main-sequence stars
and
post-main-sequence stars
evolve more rapidly than main-sequence stars,
and that just makes them harder to model.
Also in the case
of pre-main-sequence stars,
they are embedded in
star forming regions
which are opaque in the
visible band (fiducial range 0.4--0.7 μm
= 4000--7000 Å)
and this makes them harder to undersand observationally.
And in the case of
post-main-sequence stars,
they are subject to explosive events
(core helium flashes,
thermal pulses
(AKA helium shell flashes),
and for stars > ∼ 8
M_☉,
supernova explosions)
which are hard to model because they are so complex.
See the figure below
(local link /
general link: stellar_evolution_overview.html)
for an overview of the
stellar evolution of
a star
of less than ∼
8
M_☉.
File: Star file:
star_modeling.html.
Modeling Stars
The
modeling
of stars
in general is very complex.
There is an important exception in the case of the
Sun.
We can observe solar neutrinos
that are produced by the nuclear burning
in the Sun's core.
Neutrinos are so unreactive that
the Sun is nearly transparent to them.
Our detectors are nearly transparent to them too, but we detect enough that
give important information about the Sun
and neutrino physics.
However, we CANNOT neutrinos from
any other star than the
Sun.
We will NOT further expand on
neutrinos here.
The stellar atmosphere model
(which is from just below the
photosphere outward)
is about lots that we do NOT see too, but
we do get direct observations of
photometry
and spectroscopy.
For the Sun, we get a lot
more information about
solar atmosphere,
than for any other star
because we are so close.
As always the we are so close to the
Sun is special case because of its closeness.
Extinction is
the amount of absorption and scattering of
EMR by the
interstellar medium (ISM)
along the line of sight
from the star to the
observer.
In fact, in the modern age,
distance
(from the via stellar parallax
Gaia spacecraft (mission 2013--2025?))
and extinction
are much less of problems than in the past.
Stellar atmosphere models
allow you to calculate SYNTHETIC
spectra and
photometry
But luminosity
is only a direct observable
when you know the distance to the
star
(see sections Distance and Stellar Parallax
and
Luminosity Determination and Spectroscopic Parallax),
can effectively integrate observed flux
over all wavelengths
(see section Luminosity, Flux, Photometry),
and can account for
extinction.
Extinction is
the amount of absorption and scattering of
EMR by the
interstellar medium (ISM)
along the line of sight
from the star to the
observer.
Now observed luminosity
and stellar mass
(if we had them) are enough (if we can assume composition)
to constrain
simple stellar structure models,
and so tell us what
stars of thoses masses and luminosities
are like.
Actually, stellar mass
is a direct observable for some
binary star systems.
Such systems are important tests of
stellar structure modeling.
The upshot of the above is that in order verify
our understanding of stars
based on stellar structure models
we need another kind of
model
from which we can calculate sufficient sythetic observables to fit to actual
observables to verify and constrain our
stellar structure models.
That model
is the
stellar atmosphere model
which we discuss below in the subsections
Stellar Atmosphere Models
and Spectral Types and Stellar Models.
Like Stellar structure models,
stellar atmosphere models
must be calculated using
numerical methods
on the computer.
There are NO
analytic solutions,
except for highly simplified
cases like the
grey atmosphere.
Such simplified cases are very useful in understanding
stellar atmospheres
and in testing
computer codes,
but do NOT have realistic behaviors, except in a very approximate way sometimes.
For simple stellar atmosphere modeling,
the free parameters
(which are determined by fits to observed
photometry
and spectroscopy) are usually:
g = GM/R**2 ,
where
gravitational constant G = 6.67430(15)*10**(-11) (MKS units),
M is stellar mass
and
R is stellar radius
(i.e., photospheric radius).
But note the modeling only gives g, NOT M and R separately.
T_eff = (F/σ)**(1/4) = [(L/(4πR**2))/σ]**(1/4) ,
F is flux,
σ is the
Stefan-Boltzmann constant
σ = (5.670374 19 ...)*10**(-8) W/*m**2/K**4 (exact)
(see NIST:
Fundamental Physical Constants --- Complete Listing 2018 CODATA adjustment)),
L is luminosity,
and
R is stellar radius
(i.e., photospheric radius).
Note effective temperature
is the temperature
the star would have if
it radiated like an exact
blackbody radiator of
radius R.
Stars
do NOT radiate like
exact
blackbody radiators.
Nevertheless,
effective temperature
is a good characteristic or sort-of average
temperature
for their photosphere.