Transcript Astrophysics 14 - Black Holes

Black Holes
Astrophysics Lesson 14
Learning Objectives
To know: How to define the event horizon for a black
hole.
 How to calculate the Schwarzschild radius, RS,
for the event horizon of a black hole.
 To discuss the evidence for and the density of
the super massive black hole at the centre of the
Galaxy.
Homework
 Collecting - the mock EMPA.
 Reminder another mock EMPA on Monday.
 Homework – Q6-8, p180-181
Read p173-180 if you have the time – it’s
interesting stuff!
Recap
• What determines whether a black hole will form
in the first place?
•
•
•
•
What is the defining feature of a:Supernova
Neutron Stars
Black Holes
How can they be observed?
• Either from material falling into the black hole:• Gravitational potential energy  electromagnetic
radiation.
How can they be observed?
 Stars orbiting an invisible centre of mass.
This is what we observe at the centre of the
Milky Way galaxy.
Video clip…
Some definitions
• The Event Horizon:• This is defined as the boundary at which the escape
velocity is equal to the speed of light.
• The Schwarzschild Radius, RS:• This is defined as the radius of the event horizon.
• Anything that is within the Event Horizon of the black
hole cannot escape – not even light.
Energy Equations
• Supposed we have an object of mass, m on the
surface of a more massive object of mass M.
• How do we calculate the kinetic energy and
gravitational potential energy of mass m.
Energy Equations
• The kinetic energy of an object of mass, m:-
KE 
1
2
mv
2
• It’s gravitational potential energy on the surface of a
more massive object M is given by:-
GMm
GPE  R
• Think force x distance, where the force is Newton’s
law of gravity. At inifinity GPE = 0.
The Escape Velocity
• This is the velocity required for a less massive object of mass,
m, to completely escape the gravitational field (to infinity) of a
more massive object of mass M.
KE lost  GPE gained
• If m is taken to infinity, the difference in GPE is:-
 GMm  GMm
GPE gained  0 -  
R
 R 
• So the initial kinetic energy of m must be equal to:-
•
GMm
1 mv 2 
2
R
 make v the subject
The Escape Velocity
• So the escape velocity is given by:-
2GM
v
R
• At the boundary of the event horizon of a black hole,
R=RS, the Schwarzschild radius, and v = c, the speed
of light:•
2GM
c
RS
 Rearrange this for Rs
The Escape Velocity
• So the escape velocity is given by:-
2GM
v
R
• At the boundary of the event horizon of a black hole,
R=RS, the Schwarzschild radius, and v = c, the speed
of light:•
2GM
c
RS
 Rearrange this for Rs
Schwarzschild Radius
• This is defined as the radius of the event horizon of a
black hole.
2GM
RS  2
c
Density of a Black Hole
• Recall the equation for density:-
mass
density 
volume
M
M
 
3
4
V
3 R
• If we substitute our equation for RS into the equation:-
2GM
RS  2
c
• We can derive an equation for the density of a black hole.
Density of a Black Hole
• I get:6
3c

3
2
4 8G M
• Evaluate ρfor M= 10 solar masses.
• What value of M would give a density equal to
that of water? (1,000 kg m-3)
Density of a Black Hole
• Density is not constant, it is infinite at the
centre.