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Lund Observatory
LUND
August 2011
ASTB01 INTRODUCTION TO ASTROPHYSICS (7.5 ECTS CREDITS)
Exercises
1
Suppose that two closely spaced objects are observed with the 10.4 m telescope
(GTC) at La Palma at visual wavelengths. How large must a radio telescope,
observing at a wavelength of 10 cm, be, in order to have the same resolving
power? The effect of the Earth’s atmosphere can be neglected.
2
A star has been observed to have the apparent visual magnitude V. Later
observations show that the star in fact is a binary with two identical components.
What is apparent visual magnitude of one of these components?
3
Calculate the flux density ratio between a star of magnitude V = 27 and the Sun,
which has the apparent visual magnitude V = -26.8.
4.
Calculate the total velocity (space motion) of Sirius (α Canis Majoris) relative to
the Sun using data from Table C.17.
5.
For a star the following data has been observed In the UBV system:
V = 12.25, B-V = + 1.05, U-B = 0.00.
The spectral class is B2 III. Stars belonging to this class can be assumed to have
intrinsic colours, (B-V)0 = -0.25, and absolute magnitudes, MV = -3.0. Calculate
the distance to the star.
6.
Use the data in Table C.13 for the moon Phobos for calculating the mass of the
planet Mars.
7.
The absolute bolometric magnitude of Capella (α Aurigae) is estimated to be -0.6
and its effective temperature is 4400 K. Corresponding values for the Sun is +4.7
and 5800 K. Determine the luminosity and radius of Capella with the luminosity
and radius of the Sun, respectively, as units.
8.
How large will the linear diameter of the planet Venus be on a CCD camera
attached to the 5 m telescope at Mount Palomar? At the time of the observation
the angular diameter of the planet is 0.70’. The aperture ratio of the telescope is
f/3.3
9.
An eclipsing binary with components A and B normally has the magnitude 12.2.
When the brightest of the two components, A, is eclipsed by B, the eclipse is total,
and the magnitude of the binary drops to 14.2. Determine the magnitude of
component A.
10.
The intrinsic colour of the sun is (B-V)0 = 0.62. However, the sunlight we receive
is affected by the passage of the Earth’s atmosphere. Determine the maximum
and minimum value during the year of the observed colour index, B-V, of the Sun
when it passes the meridian in Lund. The extinction coefficient in the visual is 0.15
and 0.27 in blue.
1
11.
Barnard’s star (see Table C.17) has the largest proper motion, µ = 10.3’’/year.
Assuming that its space velocity is constant, what will its maximum proper motion
be?
12.
For five stars one has observed the following data for spectral class, luminosity
class, V magnitude and B-V colour index.
Sirius
V
B-V
A1V
- 1.47
+ 0.01
Two components of
a binary
A
B
A1V
B0V
+ 9.51
+ 4.23
+0.11
- 0.19
Two stars in a cluster
C
D
B0V
B0V
+ 9.40
+ 10.00
+ 0.35
+ 0.55
The trigonometric parallax for Sirius has been determined to 0.375’’. Stars of the
same spectral and luminosity class are supposed to have the same absolute
magnitudes and intrinsic colour indices. The interstellar reddening of the two
binary components is assumed to be the same. The two cluster stars are
imbedded in clouds of interstellar matter. The distance to the stars C and D is
assumed to be the same. The extinction coefficient for interstellar matter is
supposed to be constant everywhere. Calculate:
a) The absolute magnitude MV for a B0V star
b) The distance to the binary
c) The distance to the cluster containing stars C and D.
13.
For a star with apparent visual rnagnitude V=3.08 one has observed a radial
velocity of -30 km/s and a proper motion of 0.444’’/year. Its absolute magnitude is
known to be: MV= +1.2. The extinction in the visual is assumed to be AV= 0.82.
Calculate the space velocity of the star.
14.
The star HD 22091 is a binary, consisting of two identical components of spectral
class A3V. The period of the binary is 568.22 years. The maximum angular
distance between the components is 0.85’’ and the minimum distance is 0.45’’.
The orbital plane is perpendicular to the line-of-sight. The parallax of the binary
has been measured to be 0.006’’. Calculate from this a value for the mass of an
A3V star (expressed in solar masses).
15.
a) Calculate the temperature of the Sun from Wien’s displacement law if the
intensity maximum for the solar radiation is 470 nm.
b) At the wavelength of 430 nm the intensity of the Sun is 1.14 times higher than
at 550 nm. Calculate the colour temperature of the Sun using Wien’s
approximation of Planck’s law.
c) Calculate the effective temperature of the Sun from the solar constant
1.368⋅103 W/m2. The distance from the Earth to the Sun is 1.496⋅108 km and the
radius of the Sun is 6.96⋅105 km.
16.
a) Calculate the distance to the star Deneb (α Cygni), which has an absolute
magnitude of MV = -7.2 and an apparent magnitude V = 1.25 assuming space to
be extinction free.
2
b) Calculate the distance to Deneb assuming an interstellar extinction of 0.8 mag.
per kpc.
17.
A G5 star has the apparent visual magnitude V = 9.85 and the colour index B-V =
1.15. Determine if the star is a supergiant, a giant or a main-sequence star
(dwarf). The visual extinction is three times the colour excess B-V. In thr direction
of the star the visual extinction, AV, can be assumed to be given by AV = a ⋅ r,
where r is the distance to the star and a = 0.003 magnitudes per parsec.
For G5 stars it can be assumed
MV
(B-V)0
Supergiant
-3.8
0.85
Giant
1.0
0.79
Dwarf
5.0
0.68
18.
The supernova remanant, the Crab nebula (Messier 1) has nowadays a radius of
approximately 3 arc minutes. It expands with a speedreod 0.21 arc seceonds per
year. One has measured a radial velocity of 1300 km/s for the nebular gas in
relation to the central pulsar. We assume a symmetrical expansion.
a. How far away is the crab nebula?
b. How long ago did the supernova occur according to these measurements?
c. How bright was it then, if a supernova of this kind has a maximum
absolute magnitude of -18?
19.
The luminosity of the Sun is 3.86. .1026 W. Assume that the luminosity is constant
over the main sequence evolution of the Sun. We can assume that the (initial)
hydrogen abundance is X = 0.73 and that each helium nucleus formed weighs 0.7
% less than the four protons, from which it was formed.
a) How much mass does the Sun loose per second due to hydrogen burning?
b) How much of the hydrogen in the Sun has been transformed into helium up til
now if the age of the Sun is 4.5 ⋅ 109 years?
20.
Observation of the spectrum of a galaxy revealed that a group of lines with rest
wavelengths around 420 nm were redshifted with 2.347 nm. Calculate the radial
velocity and distance to the galaxy. One can assume that the Hubble constant is
H0 = 70 km s-1 Mpc-1.
3
Answers to exercises ASTB01:
1. 1.9 ⋅106 m
2. V +0.75
3. 3.3 ⋅1021
4. 18 km/s
5. 1.9 ⋅103 pc
11. 25.4’’
12. a) -3.9, b) 1.9 kpc, c) 360 pc
13. 46 km/s
14. 2.0 M
15. a) 6100kK, b) 6300K, c) 5780K
6. 6.43 ⋅ 1023 kg
7. 130 L, 20 R
8. 3.4 mm
9. 12.4
10. 1.25 and 0.75
16. a) 490 pc, b) 420 pc
17. Giant
18. a) 1.3 kpc, b) 860 years, c) -7.4
19. a) 4.3 ⋅109 kg, b) 6 %
43. 1675 km s-1; 24 Mpc
4