Download ph607-15-test2ans

PH607/15/test2
UNIVERSITY OF KENT
PH607: GALAXIES
Test 2, Wednesday Week 21, 09:00
This is a 45 minute test. Please solve ALL questions.
1. The diagram below describes the extent of the convective regions inside stars as a
function of stellar mass. The ‘cloud/hatched’ areas indicate convective regions. The
convection instability criterion is written as
dT æ g -1 ö T dP
>ç
÷
dr è g ø P dr
(a) Define P, T and  .
.
[6 marks]
(b) State how and in what type of conditions the convection instability criterion can be
satisfied. [12 marks]
(c) State what quantity is represented by m/M on the y-axis. [4 marks]
(d) State the details of the convection region distribution inside stars in terms of
different stellar mass ranges, according to the following diagram. [8 marks]
]
2
2. a) State how the radiation pressure Prad is related to the temperature T of a star. [10 marks]
(b) Prove that the radiation pressure gradient, in terms of the mass m(r) of an interior sphere
of radius r inside a star, can be expressed in terms of the relevant temperature gradient as
dPrad 4 3 dT
 aT
.
dm
3
dm
[10 marks]
b) The total pressure gradient is described by the hydrostatic equilibrium equation
dP
Gm
dT
3k L

, and the temperature gradient is expressed as
.
=4
dm
dm
64p 2 acr 4T 3
4r
Prove that the maximum luminosity of a main sequence star is L edd 
4 c GM

, which is
called the Eddington luminosity.
[10 marks]
3. Sketch the rotation curve of our Galaxy. Label the axes and provide appropriate values on
the two axes.
[20 marks]
Describe how the enclosed gravitational mass, M(r), varies with radius r at very small and
very large radii. State how both the need for a central black hole and extended Dark Matter
can then be deduced.
[20 marks]
3
Solutions
Q1. a) T and P are the temperature and pressure at a radial distance r from the centre of the
star, and  is the ratio of specific heats under constant pressure and constant volume. [6 marks]
(b)The criteria for convection to occur can be satisfied in two ways;
either the ratio of specific heats is close to unity
or the temperature gradient is very steep.
The former occurs in cool layers, where the gas is partially ionized,
while the latter occurs in very hot layers where the energy generation rate is high.
Where, or whether, such layers occur in a particular star depends on the mass of the star, as
shown in the figure.
(c) m/M is the the fraction of mass, m(r), within r ( as a logarithmic function of stellar mass,
M).
(d) In stars with masses below about 1.3M, the surface layers are cool enough to be partially
ionised and they are hence unstable to convection.
The structure of solar-type stars then consists of a hot radiative core, at the centre of which the
nuclear energy is produced, surrounded by a cool convective envelope.
The envelope is shallow for stars near the upper-mass limit but becomes gradually deeper for
lower-mass stars, and stars with masses less than about 0.3 M are believed to be fully
convective.
The stars of higher mass than a few M can have a convective core which contains the energygenerating regions,
and a radiative envelope, since they Have a central temperature high enough for the CNO cycle
to operate, and the CNO cycle is much more sensitive to temperature than the proton-proton chain
and requires a much steeper temperature gradient, which is unstable to convection
1
2. a) The radiation pressure is 𝑃𝑟𝑎𝑑 = 3 𝑎𝑇 4
(b)
dP dm 4 3
dP
4
dT
dPrad 4 3
 aT  rad  aT 3
 aT  rad
dT
3
dm dT 3
dm
3
dm
(c) Divide the hydrostatic equilibrium equation
dPrad
16 aT 3 r 4 dT

dP
3Gm dm
(2)
(1)
dP
Gm

into (1)
dm
4r 4
[3 marks]
turn over
4
substitute
dPrad
L
dT
3k L


into
(2)
=dP
4cGm
dm
64p 2 acr 4T 3
(3)
Since P=Pgas+Prad and dPgas/dr <0 and dPrad/dr <0
dPrad/dP < 1, so  L < 4cGm L 
at the surface m=M,
4cGm

m M
4cGm
(4)

4cGM

 Ledd

[3 marks]
3*2 =
3*2 =
2
[6 marks ]
[6 marks]
[marks]
Therefore: L < Ledd is required when the star is in radiation equilibrium
3.
Shape of observed rotation curve
x-axis with correct unit
y-axis with correct unit
Within a parsec, the velocity of dispersion increases rapidly with decreasing radius, consistent
with a point mass close to the centre.


The approximately constant rotation speed implies that the mass is proportional to r.
Although the enclosed mass, M(r), continues to grow apparently without limit, the
enclosed luminosity, L(r), tends to a finite limit as we reach the edge of the
luminous material in the galaxy.
5

There must therefore be significant amounts of dark matter which continue to
contribute to M(r) out to very large radii.
turn over