Download Problem Sheet for Introduction to Astrophysics

Work sheet one for Chapter Six and Seven (PH304)
1.
The definition of one parsec is the distance of the object when it subtends a parallax angle of
one arc second to the observer on the surface of Earth. Use the following diagram to prove
that: 1pc = 3.086X1013km.
2.
The parallax of a star is 1/10th of an arc second. What is its distance to the observer ?
3.
a) How much brighter will a star of 1st magnitude appear compared with one of 9th
magnitude?
b) What is the apparent magnitude of a star that appears 2.512 times less bright than the 0th
magnitude star Vega?
c) Two stars of identical luminosity are observed. The apparent brightness of the more
distant one is 10,000 times less than the brightness of the closer star.
i) What is the difference in their apparent magnitude?
ii) How many times further away is it?
4.
Two stars are at the same distance from an observer.
a) Show that the difference of their absolute magnitudes M1 and M2 can be related to the
ratio of their luminosities (L1 and L2) by the formulae
M
1
 M
2
 2 . 5 log
L2
L1
b) Letting one of the two stars be the Sun, prove that the relationship between a star’s
absolute magnitude M and its luminosity L can be derived as:
M
 M
sun
 2 . 5 log
L
L
sun
c) Based on the formula derived in b), show that a star’s apparent magnitude m is related
to its observed radiant flux F by the formula
m  M
sun
F
 2 . 5 log
F sun ,10 pc
(Fsun,10pc is the solar flux at a distance of 10pc)
d) Sirius has an absolute visual magnitude MV = 1.4 and a distance of 2.65 pc. The extinction
of a cloud between Sirius and the observer AV is 4mag. Find its apparent magnitude mV.
[hint for a): From
d
)
 m1  m 2  M 1  M 2
pc
L1
L
F1
F 


 ( ii )
2
F
2
L
4 R
2
M  m  5  5 log 10 (
F1
 100
F2
m
2
m1
5
 ( iii )
(i)
With i) , ii) and iii), it is not difficult to find the relation between the difference of
absolution magnitudes to the ratio of luminosities as requested. ]
5.
Two stars A and B are observed at the same distance from the observer and their
fluxes are 2563 and 1357 W/m2 respectively.
a) If the apparent magnitude of star A is - 10.5mag, what is the magnitude of star B?
b) Assuming the two stars are very close to each other so that it is difficult to distinguish
their individual apparent magnitudes, what is the apparent magnitude of the two stars
combined?
c) A star behind a dense massive cloud, at a distance 1kpc, has an absolute magnitude of
10mag. If the extinction of the foreground dust to the star is A = 4mag, what is the
apparent magnitude of the star?
6.
A distant F0 star is 6.3 times less bright than a nearer F0 star which has a parallax of 0.2
arcsec. What is its distance in parsec of the distant F0 star?
7.
A star has a luminosity of 8 × 10
temperature.
8.
The tidal force on a body is the difference in the forces acting on either side of the body.
For example, if you could stand on the edge (event horizon) of a black hole, the tidal
force would be the difference between the gravity of the black hole acting on your head
and on your feet. This is quite important – it’s the difference of the forces acting
between your feet and head which rips you apart while you are falling into a black hole!
and a diameter of 8 × 10
. Estimate its surface
a) If you could stand on the event horizon of a one-solar-mass black hole (M=1.991030
kg), what is the tidal force acting on you?
(Assume your weight is 70kg and your height is 2 m)
b) If you could stand on the event horizon of a 109 solar mass black hole, what is the tidal
force acting on you (the same conditions of you as in the part a) are assumed):
(Hint: First you have to find the Schwarzschild radius in each case in order to be able to
calculate the tidal force).
c) Compare the results from a) and b), discuss what you find.
.