This document provides an introduction to astrophysics and the physics of stars and black holes. It discusses key topics like: - Astrophysics deals with the nature of heavenly bodies like stars and galaxies. - The brightness and classification of stars depends on factors like their mass, temperature, and distance from Earth. - A Hertzsprung-Russell diagram shows the relationship between stars' luminosities and temperatures. - Stars evolve over their lifetime, from birth in nebulae to their eventual death as remnants like neutron stars or black holes. - Black holes have such strong gravity that not even light can escape, so they can only be studied indirectly through their accretion disks and effects on nearby gases.

1. PRESENTED BY
PUJITA DAS
APC/PG(S4)/15/PHYS/10

2. Introduction to astrophysics
• Astrophysics is the branch of astronomy that
deals with the physics of the universe,
especially with "the nature of the heavenly
bodies, rather than their positions or motions
in space". Among the objects studied are the
Sun, other stars, galaxies, extra solar
planets, the interstellar medium and the
cosmic microwave background.

3. Brightness of stars
• Measure of star’s
brightness is called
magnitude and depends
on:
• Mass of the star
• Distance from earth
• Star’s temperature
• Apparent magnitude(m)-
a measure of how bright
the star appears to be on
earth
• Absolute magnitude(M)-
measure of how bright a
star would be if all the
stars were at same
distance from earth i.e ,
10 pc

4. Classification of stars
• Stars are divided on the
basis of spectral class
corresponding to
different surface
temperatures.
• Spectral classes on the
basis of decreasing
temperatures are :

5. Hertzsprung Russell Diagram
• H–R
diagram shows
the relation
between
luminosities
versus the
classification and
temperatures of
stars.

6. Life cycle of star
Life of all stars begin in the same way
 Interstellar medium
 Nebula
 Protostar
 Main sequence star

16. SUPER RED GIANTS Occurs when stars of large mass runs out of
hydrogen – starts cooling and expanding at the same time. Centre
shrinks and atmosphere swells.

18. Core of massive star that are 1.5 -4 solar mass end up as
NEUTRON STAR after supernova explosion.

19. Core of massive star more that 8 solar mass becomes BLACK
HOLE after supernova. It is so Massive and dense that even
light cannot escape its gravity.

20. Black Hole
• A black hole is a region of space in which the
gravitational field is so powerful that nothing,
including electromagnetic radiation such as visible
light, can escape its pull - a kind of bottomless pit in
space-time.
• Since light cannot escape black hole we cannot study
black hole directly.
• Nature provides us with a means to indirectly study
the black hole , that is through the process of gas
falling on to the black hole , a process called
• ‘ accretion’.

21. Accretion and accretion disc
• Accretion is a process
of a growth of a
massive object by
gravitationally
attracting and
collecting of
additional material.
• Disc forms because in
falling matter has
angular momentum.

22. Gas dynamics
• All accreting matter in universe is in gaseous form.
• The constituent particles, usually free electrons and
various species of ions, interact directly only by
collisions, rather than by more complicated short-
range forces.
• regard the gas as a continuous fluid, having
velocity v, temperature T and density ρ defined at
each point.
• We then study the behavior of these and other fluid
variables as functions of position and time by
imposing the laws of conservation of mass ,
momentum and energy.

23. Gas dynamics equation
• Conservation of mass is ensured by
continuity equation :
• Conservation of momentum is ensured by
Euler equation :
• Conservation of energy can be written in
terms of polytropic equation:

24. Under stationary condition
• Considering spherical polar coordinate (r,θ,φ).
• Fluid variables are independent of (θ,φ)
• Continuity equation :
• Euler equation :
1
𝑟2
𝑑
𝑑𝑟
𝑟2
𝜌𝑣

 𝑣
𝑑𝑣
𝑑𝑟
+
1
𝜌
𝑑𝑃
𝑑𝑟
+
𝐺𝑀
𝑟2 = 0 

25. 𝜌𝑣𝑟2
= −
𝑀
4𝜋
; a constant where 𝑀 is the mass accretion rate.
Now, 𝑃 = 𝐾𝜌 𝛾
,the polytropic equation of state.
𝐶 𝑠
2
=
𝑑𝑃
𝑑𝜌
Thus, Euler equation becomes
𝑣
𝑑𝑣
𝑑𝑟
+
1
𝛾 −1
𝑑 𝐶 𝑠
2
𝑑𝑟
+
𝐺𝑀
𝑟2 = 0
⇒
𝑑
𝑑𝑟
𝑣
2
2
+
𝐶 𝑠
2
𝛾 −1
−
𝐺𝑀
𝑟
= 0
We obtain the Bernoulli integral,
𝑣2
2
+
𝐶 𝑠
2
𝛾−1
−
𝐺𝑀
𝑟
= 𝜀 , a constant called specific energy.

26. From continuity eq, 𝑣𝑟2 𝑑𝜌
𝑑𝑟
+ 𝜌
𝑑 (𝑣𝑟 2 )
𝑑𝑟
= 0
Or,
1
𝜌
𝑑𝜌
𝑑𝑟
= −
1
𝑣𝑟 2
𝑑 (𝑣𝑟 2 )
𝑑𝑟
After rearranging the terms we get
𝑑𝑣
𝑑𝑟
=
𝑣(
2𝐶 𝑠
2
𝑟
−
𝐺𝑀
𝑟 2 )
𝑣2 − 𝐶 𝑠
2
Thus, critical point condition
𝑣 𝑐 = 𝐶 𝑠𝑐
and 𝐶 𝑠𝑐
2
=
𝐺𝑀
2𝑟 𝑐
Putting these conditions into specific energy expression we
get,
∴ 𝑟𝑐 =
𝐺𝑀
𝜀
[
5 − 3𝛾
4 𝛾 − 1
]

27. Results
• i,e
Large r, the gravitational pull of the black
hole is weak and the flow is subsonic.
• i.e small r,
As one moves to smaller r, the inflow velocity
increases to become supersonic and the gas is
effectively in free fall.
𝑣2
< 𝐶 𝑆0
2
, 𝑟 > 𝑟𝑐
𝑣2
> 𝐶𝑆0
2
, 𝑟 < 𝑟𝑐

28. Under non stationary condition
Given a gas with, a velocity field v, density ρ and
temperature T,all the equation of gas dynamics are defined
as a functions of position r and time t.
Continuity equation is,
𝑑𝜌
𝑑𝑡
+
1
𝑟 2
𝜕 (𝜌𝑣 𝑟 2 )
𝜕𝑟
= 0. .....(1)
Euler equation is,
𝜕𝑣
𝜕𝑡
+ 𝑣
𝑑𝑣
𝑑𝑟
+
1
𝜌
𝑑𝑃
𝑑𝑟
+ 𝜙 𝑟 = 0 .....(2)
𝜙 𝑟 = −
𝐺𝑀
𝑟
Again, 𝑃 = 𝐾𝜌 𝛾
, polytropic equation

29. 𝐶 𝑠
2
=
𝜕𝑃
𝜕𝜌
= 𝛾𝐾𝜌 𝛾 −1
Let us introduce,
𝜌 𝑟, 𝑡 = 𝜌0 𝑟 + 𝜌 𝑟, 𝑡
𝑣 𝑟, 𝑡 = 𝑣0 𝑟 + 𝑣(𝑟, 𝑡)
𝑃 𝑟, 𝑡 = 𝑃0 𝑟 + 𝑃(𝑟, 𝑡)
𝑓 𝑟, 𝑡 = 𝑓0 𝑟 + 𝑓(𝑟, 𝑡)
Where 𝑓 = 𝜌𝑣𝑟2
.......(3)
where the primed quantities are small & higher order
can be neglected
From (1) and (2),keeping linear terms in variation
only,one gets,
𝜕 𝑣
𝜕𝑡
+
𝜕 (𝑣0 𝑣+
𝐶 𝑠0
2 𝜌
𝜌 0
)
𝜕𝑟
= 0

30. Where 𝐶 𝑠0
2
=
𝑑 𝑃0
𝑑 𝜌0
Also,
𝑓
𝑓0
=
𝜌
𝜌0
+
𝑣
𝑣0
Solving the above equations we get
𝑣0
𝑓0
𝜕2
𝑓
𝜕𝑡2
+
𝜕(
𝑣0
2
𝑓0
𝜕𝑓
𝜕𝑟
)
𝜕𝑡
+
𝜕(
𝑣0
2
𝑓0
𝜕𝑓
𝜕𝑡
)
𝜕𝑟
+
𝜕
𝑣0
𝑓0
(𝑣0
2
−𝑐 𝑠0
2
)
𝜕𝑓
𝜕𝑟
𝜕𝑟
=0
..(4)
i.e, 𝜕 𝜇 𝑓 𝜇𝜈
𝜕𝜈 𝑓= 0
where, 𝑓 𝜇𝜈
=
𝑣0
𝑓0
1 𝑣0
𝑣0 𝑣0
2
− 𝐶 𝑠0
2
equation (4) is wave equation with wave speed 𝑪 𝒔𝟎
𝟐

31. Results
• This implies that small perturbations about
hydrostatic equilibrium propagate through
the gas as sound waves with speed
• Since is the speed at which pressure
disturbances travel through the gas, it limits
the rapidity with which the gas can respond
to pressure changes.
𝐶𝑠0
𝐶𝑠0

32. • For supersonic flow : where the gas moves
with |v| > , then the gas cannot respond
on the flow time L/|v| < L / ,so pressure
gradients have little effect on the flow.
• For subsonic flow : At the other extreme, for
subsonic flow with |v| < , the gas can
adjust in less than the flow time , so to a first
approximation the gas behaves as if in
hydrostatic equilibrium.
• Where L is the size of the region of the gas.
• L / is the response time.
𝐶 𝑠0
𝐶𝑠0
𝐶𝑠0
𝐶 𝑠0

33. THANK YOU