In the previous blog I explained why do we need a theory of Stellar
Structure and Evolution when we can get information about stars by
just observing them. In this blog I will cover-
1. The basic assumptions of theory of Stellar Structure and
Evolution
2. Accuracy of
Assumptions of theory of Stellar Structure and Evolution
To understand above
mentioned aims more clearly, let’s first understand( answer these
questions – that I generally don’t see many writers stress about:
What are the
assumptions for a theory ?
Assumptions are
basically the foundation stones for a theory. These are taken as the
postulates that are generally assumed to be true throughout the
explanation.
Now, What is the
need for assumptions to a theory?
-First of all,
because they are reasonable – they offer observational or at least
mathematical verification.
— Secondly, Making
assumptions simplify our work in terms of crucial/critical
understanding and mathematics.
– Lastly, although
it is harsh but true: We only have limited information about things.
Thus, as long as these are not true in general, we choose not to
include them (esp while teaching) instead of including it with
ad-hoc suppositions that lack any observational verification.
Now that we know
what are assumptions and why do we need them for building a theory
let’s see what are the basic assumptions for the theory of Stellar
Structure and Evolution
1. Stars are
isolated in space – This is a fairly reasonable assumption for most
of the single stars in galaxies as this condition is satisfied to a
high degree – compare the distance of sun to its nearest star
Proxima Centauri. We are ignoring binary stars and stars in dense
clusters.
2.Stars are formed
with a homogeneous composition- it is again reasonable as the clouds
from which stars are formed are well – mixed.
3.Stars have no magnetic field – This is fully reasonable as for most of the stars magnetism plays a notable role only in phenomena related to surface of stars but in overall life cycle they don’t play any significant role.Stellar evolution is
fully determined by internal physical processes which take deep
inside the star near it’s core.
4. Stars are
rotating slowly- This one is a lot harder to justify as most of the
stars rotate at a considerable fraction of their critical
velocity(*1). Since we do not have a theory that shows how Stellar
interior rotates at the birth of the star and making this assumption
causes a huge mathematical simplification we are going to hold it.
5. Stars are in
mechanical equilibrium(*2) : Majority of stars are in such long lived
phases of their evolution that no structural changes are observed for
them for most period of times. This implies that there is no
noticeable acceleration and all the forces balance each other
perfectly.
For an isolated,
slowly rotating, homogeneous composition star with no magnetic field
these forces are gravity and pressure. Thus, all the stars are in
hydrostatic equilibrium.
All of the above
mentioned assumptions can be tested just by testing for accuracy of
hydrostatic and spherical symmetry assumptions.
Accuracy of
Hydrostatic Assumption –
Let’s first
understand the equation of Hydrostatic support using simplest
Newtonian dynamics
Balance between
gravity and pressure is called hydrostatic equilibrium.
For a given time t,
let’s consider a spherical mass shell with infinitesimal thickness δr
at a distance r from the centre. Mass of the element
δm at this distance is
δm= ρ(r)δs
δr
ρ(r)
= density at radius r
Outward force =
pressure exerted by stellar material on lower surface
P(r)δs
Inward force on mass
element= Pressure exerted by stellar material on upper surface and
gravitational attraction of all stellar material lying within r
P(r+δr)δs+
GM(r)δm/r2 =
P(r+δr)δs+ GM(r)ρ(r)δs δr/r2
For hydrostatic
equilibrium, inward force= outward force
P(r)δs=P(r+δr)δs+
GM(r)ρ(r)δs δr/r2
so,
P(r+δr)-P(r)=
GM(r)ρ(r)δr/r2
For infinitesimal
element:
P(r+δr)-P(r)/δr
= dP(r)/dr
Thus,
dP(r)/dr=-GM(r)ρ(r)/r2
which is the
equation of Hydrostatic support.
Accuracy of hydrostatic assumptionTo answer how valid
is that assumption let’s consider a situation where inward force and
outward force aren’t equal which gives rise to acceleration a.P(r+δr)δs+GM(r)ρ(r)δs
δr/r2 -P(r)δs
= ρ(r) δs
δr a
»dP(r)/dr
+GM(r)ρ(r)= ρ(r) a this is the generalized form of equation of hydrostatic support.Now consider there
is a resultant force on element with the sum being a small fraction
of gravitational term(β)
Inward acceleration
a = β.g
Spatial displacement
from rest after time t = d= 0.5.a.t2 =0.5.β.g.t2
If if allow star to
collapse or expand, by setting d=r we obtain
t=(2.r3/G.M.β)1/2
Assuming beta =1 we
obtain
t=(2.r3/G.M)1/2
This is the
dynamical time scale of the star.
Of course each mass
shell will be accelerated at different rate so this should be taken
as an average value for star to collapse at radius R.
Since average
density is we can also write this t to be
½.(G.ρ)1/2
For sun we obtain a
value of 1600 sec or about half an hour. Thus, any significant
departure from hydrostatic equilibrium should lead very quickly to an
observable phenomenon (sudden collapse or explosion of the star ) .
But age of sun is already
This is much smaller than the age of sun – 10^17 secs – by 14 orders of magnitude. Thus if this assumption have been wrong we would have noticed a significant collapse or explosion of sun much earlier.This assumption is
very much accurate.
Accuracy of
spherical symmetry assumption
Let’s see if average
rotation rate of stars is causing significant departure from
spherical symmetry.
Consider a star of
mass M and radius R with an element of mass δm
near the surface of the star.star of
mass M and radius R with an element of mass δm
near the surface rotating at angular speed w Gravity supplies the
extra centripetal force to make the object move around in a circular
path.Thus, for mass δm,
centripetal force = gravitational force
There will be a
departure from spherical symmetry if there will be any difference
between gravitational and centripetal force i.e.
(δmω2r)/(GMδm/r2)
<<1
or
ω2<<GM/r3
note
RHS of last equation is similar to t
>>
ω2<<2/t2
(ω=2π/T)
where T =rotation period
for spherical
symmetry to be valid T>>t
for example, for sun
t~2000 sec and T~21 month
Thus, for majority
of stars, departures from spherical symmetry can be ignored.
With this we
complete the basic assumptions of theory of Stellar Structure and
Evolution and see that these assumptions are qualified to build a
theory of Stellar Structure and Evolution.
*1.Definition of
Critical velocity : Stars have a maximum speed at which they can
spin. If stars exceed the critical rotation, the outward force caused
by their spinning will overcome the inward gravitational force that
keeps the star together.
If stars get to that
limit, they will begin to fly apart.
*2. Mechanical
equilibrium : A state of rest or unaccelerated motion in which sum of
all the forces acting on a particle is 0. In case of stars, this
state is reached when pressure forces are balanced by gravity. In
astronomy, this is called hydrostatic equilibrium.
