Review Questions – Stars
1. (a) Determine the distance to a star with parallax 30.0 mas. (b) Determine the parallax for a star at distance of 40.0 pc.
2. A certain astronomical observatory has equipment capable of measuring stellar positions to the nearest 0.1 arc second. At what distance would a star’s apparent position shift in one direction on the celestial sphere by this amount?
3. A certain star has proper motion 25 arc sec per year and is located at a distance of 5.0 ly. Determine its transverse velocity.
4. Two stars have equal proper motion but star B is twice the distance as star A. How does the transverse velocity of the two stars compare?
5. Two stars have equal transverse velocity but star D is ten times the distance as star C. How do the proper motions of these two stars compare?
6. Star F has twice the redshift as star E – what does this indicate about the two stars?
7. Astronomers are sometimes tasked with detecting and assessing “impact threats” – objects in the universe that might hit the Earth. No star has been detected that is predicted to hit the Earth. BUT – what would be some specific observations of a star that would indicate a likely collision course with Earth and/or the Sun?
8. A certain star has three times the temperature of the Sun and three times its diameter. What is its luminosity? Judging by an HR Diagram, what type of star is it?
9. The Sun is type G main sequence with temperature 5800 K and absolute magnitude 4.8. Compare those numbers to a type M star on the main sequence with temperature 2800 K and absolute magnitude 14.8. (a) Based on its absolute magnitude what is the luminosity of the type M star? (b) Based on its temperature and its luminosity what is the diameter of the type M star? (c) Why do astronomers call this kind of star a red dwarf?
10. White dwarfs are interesting stars to think about. Some of these stars are basically the same size as the Earth (1/100th the size of the Sun) but twice as hot as the Sun. (a) Determine the luminosity of such a star. (b) What is different about this kind of dwarf that makes it white instead of red? (c) What makes the luminosity of a white dwarf fundamentally different from that of all other types of stars?
11. A graph of apparent brightness versus time is sometimes called a star’s light curve. A typical light curve may show two dips in brightness – a bigger dip and a smaller dip. The two dips occur over and over in precisely the same amount of time. This type of light curve indicates a binary star with A and B component stars. (a) Explain what causes the two dips. (b) What effect does the diameter of the A star have on the width and duration of the dips in brightness? Why? (c) What aspect(s) of the light curve are used to calculate the mass of the star(s)? Explain.
12. Suppose there is a spectroscopic binary consisting of a type A star and a type M star. In this type of binary it is not possible to resolve the two stars with a telescope and the orbit of the system is tilted such that there is no eclipse occurring from our perspective and line of sight. However, the existence of the two stars is detectable by the spectrum of light, which will contain lines resulting from each star. (a) Which star would produce the strongest hydrogen lines? (b) One of these stars is likely quite a bit more massive than the other – which one? Explain. (c) The lines from which of these two stars would have the greatest amount of Doppler shift? Explain. (d) When the lines from the A star are redshifted the lines from the B star are blueshifted and vice versa – explain why. (e) Using only observations of the combined spectrum, astronomers can determine mass or diameter or both? Explain.
13. Besides eclipsing binaries and spectroscopic binaries what other method do astronomers use to observe binary star systems? Explain why this method is useful only for a relatively small number of star systems.
14. Why (in the world) do astronomers describe dimmer stars as having greater magnitude? How did this wacky system get started?
15. The asteroid Vesta at its brightest has apparent visual magnitude m = 5.2. (a) Would it be visible to the naked eye? (b) How many times brighter in appearance is Polaris (m = 2.0, only a “medium bright” star)?
16. Sometimes people mistakenly think that the North Star is the brightest star in the sky. The actual brightest in appearance is Sirius at magnitude m = –1.46. How many times brighter is Sirius than Polaris?
17. Speaking of Polaris, to the naked eye it appears to be a single magnitude 2.0 star. However using telescopes astronomers discovered it to be a triple star system, the dimmer of which are magnitudes m = 8.7 and m = 9.2. These stars orbit one another and so are essentially the same distance from Earth. The absolute magnitude of the brightest of the three stars, Polaris Aa, is M = –3.6. (a) Given the apparent magnitudes of Polaris B and Polaris Ab, why is the apparent magnitude of Polaris Aa essentially the same as the apparent magnitude of the three stars combined? (b) How much brighter is the star Deneb, m = 1.25, M = –8.38 than Polaris Aa – both in apprearance and in reality (intrinsically)? (c) What would be the absolute magnitudes of the smaller stars? Hint: equal difference in either type of magnitude means an equal multiple of brightness.
18. Stars beyond a certain distance from Earth always have an apparent magnitude greater than absolute magnitude, whereas stars closer than this same distance always have an apparent magnitude less than absolute magnitude. What is this distance and why?
19. A certain star has a B–V index of –0.1. (a) Through which type of filter, B or V does it appear brighter? (b) If it has an apparent magnitude of mB = 3.0 through the B filter what is its apparent magnitude mV through the V filter? (c) Using this information and an HR diagram, can you determine or estimate the spectral type? If so what is it? If not, why not? How about the luminosity class?
20. Using apparent magnitude, spectral type, and luminosity class, one can determine the distance to a star based on the HR diagram. (a) What do astronomers call this technique? Use this method to determine the approximate distance for the following stars: (b) B4V, m = 6.0; (c) m = 6.0, M0II; (d) m = 15, M0II; (e) m = 15, M0V
Answers
1.
a. 33 pc
b. 0.025″ or 25 mas
2. 20 pc (parallactic angle is half of the overall shift, so p = 0.05″)
3. 180 km/s
4. Star B has twice the transverse velocity as star A.
5. The proper motion of star C will be ten times greater than that of star D.
6. Both stars are moving away from us but star F has a radial velocity twice as great as the radial velocity of star E.
7. A star on a collision course with us would have a negative radial velocity and zero transverse velocity. This would be observable by a certain amount of blueshift in the spectrum of its light (shortening of wavelength) and by a complete lack of proper motion.
8. L = 729 L⨀, most likely a type BV – a bluish-white main sequence star.
9.
a. L = 0.0001
L⨀
b. R = 0.043 R⨀
c. Because of its relatively low temperature it has a reddish color and it is
relatively small compared to other stars (this one would be only about 5 times
the diameter of the Earth).
10.
a. L = 0.0016
L⨀
b. It is white because it has a much higher temperature than a red dwarf even
though the two types of stars are roughly the same size.
c. A white dwarf’s luminosity is not a result of fusion reactions of any sort,
but rather simply the “leftover” heat of the “dying” star that produced it.
11.
a. The two dips
are caused by mutual eclipses when the B star passes in front of the A star and
when the B star passes behind the A star. In either situation the total amount
of light detected is decreased and causes the dips.
b. If the A star has a greater and greater diameter it will tend to make the
width (i.e. duration) of the dip greater because it will take more time
for the B star to pass in front (or behind) going from one side of the A star
to the other.
c. The amount of time for the pattern of dips to repeat is the orbital period
of the binary system. Using Newton’s law of Universal Gravitation it is possible
to determine the mass of the system – the greater the mass, the stronger the
gravity, the faster the stars orbit and the shorter the period.
12.
a. The A star
would have the strongest hydrogen signature – originally the spectral types
were ranked alphabetically by strength of hydrogen lines and A is before M!
b. The A star is likely more massive. The most common types of stars are main
sequence stars and if the two stars are on the main sequence then the A star is
definitely greater mass – more massive stars are larger, hotter, and more
luminous.
c. The lines of the M type star would be shifted more assuming it is much less
mass than the A type. Because of this, the M star would be orbiting the A star
and moving at a higher velocity. Greater velocity would result in a greater
amount of Doppler shift. The A star has great mass and great inertia and
therefore does not move much and hence has smaller resulting Doppler shift.
d. Stars orbiting one another always move in opposite directions, so whenever
one of the stars moving away from us and redshifted the other is moving toward
us and blueshifted.
e. The shifting spectra can be observed over time and the period of the orbit
can be determined by the duration of the cycle. From this the mass can be
determined. The diameter of the stars does not have a direct effect on spectra.
(However, based on the width of spectral lines, binary stars or otherwise, astronomers
can indirectly judge diameter from the luminosity class – the more giant the
star the more narrow its lines.)
13. For some binary star systems it is possible to simply resolve the two stars and observe that they revolve around one another as time passes. This is referred to as a “visual binary”. Because it is necessary to resolve the separation of the stars this is easiest to do for stars that are relatively nearby – binaries at greater distances would require resolution better than existing telescopes in order to use this method.
14. The magnitude system originated as a ranking of brightness where number 1 was brightest and 2 was next brightest and so on. Therefore bigger numbers actually mean dimmer.
15.
a. A good rule
of thumb is that stars brighter than magnitude 6 are visible to the naked eye.
Therefore Vesta should theoretically be visible to the naked eye at times of
its maximum brightness (5.2 being just a little brighter than magnitude 6).
However it would be challenging to see and you would need a very dark
location.
b. Polaris shines steadily at a brightness about 19 times greater in
appearance than that of Vesta’s maximum apparent brightness, which only occurs
every other year or so.
16. Sirius is about 24 times brighter in the nighttime sky than the North Star, Polaris.
17.
a. The apparent
magnitude of Aa is 2.0 because the other stars are hundreads of times dimmer
and adding their light to that of Polaris Aa would have a negligible effect on
its perceived brightness.
b. Deneb appears 2.0 times brighter and in reality is 82 times brighter.
c. M = 3.1 and M = 3.6, respectively
(work: 8.7 – 2 = 6.7, –3.6 + 6.7 = 3.1; 9.2 – 2 = 7.2, –3.6 + 7.2 = 3.6)
18. The absolute magnitude of a star is defined as being equal to the apparent magnitude that the star would have if it were 10 pc distant. Therefore a star that is exactly 10 pc away would have an absolute magnitude exactly equal to its apparent magnitude. If it is farther than 10 pc away it appears dimmer at its true distance and therefore m > M. If it is closer than 10 pc then it appears brighter at its true distance and m < M.
19.
a. It appears
brighter through the B filter than the V filter.
b. mV = 2.9
c. Would seem to be a B8 or thereabouts. Luminosity class cannot be determined
without more information. Apparent magnitude is not a measure of true
luminosity since it is affected by distance. And the B–V index
reveals nothing about line width or diameter of the star.
20.
a. This
technique is called spectroscopic parallax.
b. 400 pc
c. 400 pc
d. 25000 pc
e. 100 pc