1.)
a.)
Proxima Centauri is the nearest star in the sky. It is
"only" 1.3 parsecs (i.e. 4.2 light years) away. Using the
parallax angle formula, calculate the parallax angle for
observing Proxima Centauri.
Answer:
d (in parsecs) = 1 / p (in arcseconds)
So, p (in arcseconds) = 1 / d (in parsecs)
So, p = 0.77 arcseconds
b.)
Another star, Lalande 21185, is approximately twice
as far away as Proxima Centauri. For the sake of argument,
let us treat it as if it were exactly twice as far away
as Proxima Centauri. Is its parallax angle half as large
or twice as large as that of Proxima Centauri?
Answer: Its parallax angle is half as large as that of Proxima Centauri
2.) Sirius is the brightest star in the night sky, but
if we moved it 10 times further away it would look
a.) just as bright as it is now
b.) 1/10 as bright
c.) 1/100 as bright
d.) 1/1000 as bright
Answer: "c", 1/100 as bright.
The reason is because the
apparent brightness = luminosity / (4 pi distance2).
Moving Sirius 10 times further away is the same as increasing
its distance by a factor of 10.
In that case,
its distance2
would increase by a factor of 100 and so its apparent brightness
would decrease by a factor of 100.
3.)
a.) What is a binary?
Answer: "binary" is the shortened name for "binary star system"
and a binary star system is two stars that are
orbiting each other.
b.) What are the differences between visual binaries, eclipsing binaries,
and spectroscopic binaries?
Answer: The difference between each of these types of binaries is related
to how we detect them (which, in some cases is related to their orientation
on the sky).
Visual binaries
are the most obvious. Each of the stars in a visual binary is close
to the other; the angle between them is small.
We compare the stars' positions on photos or maps taken at different
times. If both stars move around a common center of mass, then we know
that they are a binary. Note that it is easiest to see such
motion in cases where the plane of the binary's orbit is perpendicular
(or nearly perpendicular) to our line of sight.
The plane in which eclipsing binaries
orbit is parallel to
our line of sight. From our point of view,
each star passes behind the other star
once per orbit and each star passes in front of the other star
once per orbit.
When one star is in front
of the other star, it blocks some of the light from the more
distant star. The fraction of light that is blocked depends
on the sizes of the stars; a small star can only block a fraction
of the light from a larger star.
We study these stars by noting how the total brightness
of the binary changes with time.
The plane in which spectroscopic binaries
orbit can be along our line of sight
(like the eclipsing binary) or
somewhat tilted, but it is not
perpendicular to our line of sight. During each star's orbit around
the center of mass,
it moves closer and then further from the Earth.
While it is moving closer to us, its light is blue-shifted and
while it is moving further away, its light is red-shifted. We study
these stars by noting the Doppler shifts in
their spectra as they go through their orbits.
4.) Suppose that you want to calculate the mass of a binary
(i.e. you want the sum of the masses of the stars,
but don't necessary
need to know the individual masses)
Describe how you propose to learn the sum of the masses. You can use a
hypothetical binary, whose observable properties you
dream up, as long
as they are realistic. Note, there are a lot of steps in the explanation,
so be sure to list them all
(i.e. you will need a lot of information,
so be sure to consider everything you need and how you plan to
determine it).
For brevity, you don't need to describe the steps in great detail.
Answer:
Use Newton's version of Kepler's 3rd Law
p2 = (4 pi2 a3)/
(G (M1 + M2))
In order to solve for M1 + M2,
we will need to determine the period (p) and the semi-major axis (a)
and look up G.
If we observe an eclipsing binary, then we can measure the period.
To do so, we would measure the apparent brightness of the binary at
some point in time, then repeat the measurement many times, tracking
how the brightness varies with time. We would track the binary long
enough and at short enough time intervals to see a 2 dip pattern
repeat regularly. The time it takes for the pattern to repeat
is the period.
Now, we need the semi-major axis. We can learn this by taking
spectroscopic observations of the binary. While one of the stars
is approaching us in its orbit, its light
will be Doppler shifted to the blue and while it is moving away
from us, its light will be Doppler shifted to the red.
With a good spectrometer and fast enough stars, we would be
able to see the maximum approach and receding wavelength shifts
for both stars.
To make this easy on ourselves, let us chose a binary in which the
stars are on circular orbits. Then, the semi-major axis is just
the radius (r) and the velocities are constant during the orbits.
We would convert the Doppler shifts into velocities. Note that
the velocity found this way is the line of sight component of the
velocity vector. So, we should see the maximum positive velocity when
the star is going straight away from us. We would take this
value to the be the star's speed. Speed = distance traveled/time required.
In one orbit, the distance traveled = 2 x pi x r and the time required
= the period. So, speed = 2 x pi x r / p. So, r = speed x p / (2 pi).
We would solve for r. Now, we know p and r and we have a = r.
We just substitute p and a into Newton's version of Kepler's 3rd law
and solve for M1 + M2.
5.) Which one of these statements is true? (Note that if you
"directly measure" something, that means
that you use a ruler, thermometer, or other measurement tool
to measure the object. It is the opposite
of calculating the answer from other information that you know.)
a.) Astronomers find the distances to nearby stars
by directly measuring how much their angle on the sky
varies throughout
the year and then calculating the distance from that information
b.) Astronomers find the apparent brightness of stars
by directly measuring their luminosities and
temperatures and then
calculating their apparent brightnesses from that information
c.) Astronomers find the temperatures of stars by directly
measuring how much their angle on the sky
varies throughout
the year and then calculating the temperature from that information
d.) Astronomers find the masses of stars by directly measuring their
luminosities and calculating their masses
from that information
Answer: "a" is true. This is how parallax measurements work.
"b" is false because you cannot directly measure a star's luminosity. In fact, "b" is backwards. You can directly measure a star's apparent brightness and from
that information (and the distance) determine the luminosity.
"c" is false because you find the temperature from the star's spectrum,
not from how its angle on the sky varies.
"d" is false because you cannot directly measure the star's luminosity.
6.) As you go through the stellar sequence (OBAFGKM), what
characteristic of the star changes and how?
Answer: the star's spectrum (and apparent color) changes.
During the sequence, OBAFGKM, the peak of the emission shifts
from shorter wavelengths to longer wavelengths.
O stars are brightest
in UV light, so they look blue to our eyes. M stars are brightest
in IR light, so they look red to our eyes.
7.) Table of stars and questions from textbook:
Star Name..........Absolute magnitude..........Apparent magnitude
..........Spectral Type..........Luminosity Class
Aldebaran..........-0.2
..................................+0.9
....................................K5
.........................III
Alpha Centauri A...+4.4
..............................0.0
....................................G2
.........................V
Antares..............-4.5
..................................+0.9
....................................M1
.........................I
Canopus............-3.1
...................................-0.7
....................................F0
..........................II
Fomalhaut.........+2.0
.................................+1.2
....................................A3
.........................V
Regulas.............-0.6
..................................+1.4
....................................B7
.........................V
Sirius.................+1.4
..................................-1.4
....................................A1
.........................V
Spica.................-3.6
.............................. ....+0.9
...................................B1
.........................V
a.) Which star appears brightest in our sky?
Answer: Sirius
b.) Which star appears faintest in our sky?
Answer: Regulas
c.) Which star has the greatest luminosity?
Answer: Antares
d.) Which star has the least luminosity?
Answer: Alpha Centauri A
e.) Which star has the highest surface temperature?
Answer: Spica
f.) Which star has the lowest surface temperature?
Answer: Antares
g.) Which star is most similar to the Sun?
Answer: Alpha Centauri A
h.) Which star is a red supergiant? (use Table 15.2 to decode the Luminosity Class)
Answer: Antares
i.) Which star has the largest radius? (use Table 15.2 to decode the Luminosity Class)
Answer: Antares
j.) Which stars have finished burning hydrogen in their cores? (use Table 15.2 to decode the Luminosity Class)
Answer: Aldebaran, Antares, Canopus
k.) Among the main-sequence stars listed (use Table 15.2 to decode the Luminosity Class), which one is the most massive?
Answer: Spica
l.) Among the main-sequence stars listed (use Table 15.2 to decode the Luminosity Class), which one has the longest lifetime?
Answer: Alpha Centauri A
8.)
a.) What material do main sequence stars burn and where in
the stars is it burned?
Answer: Main-sequence stars burn hydrogen in their cores.
b.) Do white dwarf stars burn material (by fusion)?
If so, where in the stars is it burned and if not,
why are they able to give off energy in the form of photons?
Answer: No, white dwarfs do not burn material by fusion.
At the time that a star evolves to become a white dwarf,
it is hot. So, it has a lot of thermal energy. Over time,
it converts this energy into photons which it emits.
9.) There are many more low-mass main sequence
stars than high-mass main sequence stars.
One reason for this is that more low mass stars are born. What
is the other reason?
Answer: High mass stars burn their mass much faster than
low mass stars. (The reason is that their greater masses
compress their cores to greater temperatures and so they
burn material faster. Even though high mass stars are born
with more material to use as fuel, their burning rate is so great that
they exhaust their fuel before the low mass stars
exhaust their more modest supply of fuel.) So, high mass stars
die off much earlier than low mass stars. As a result,
there are fewer high mass stars in existence.
10.)
a.) What observable property of a Cepheid Variable star varies?
Answer: Its apparent brightness
b.) Why does it vary?
Answer: In Cepheid variable stars, energy isn't able to pass
through the outer layers of the star at the 'right' rate, i.e.
the same rate that energy is released in the core of the star.
As a result, the outer layers of the star go through a cycle in
which they swell and then collapse, swell and then collapse, over
and over.
When the outer layers are overly puffed up, they let
energy pass through too easily. This causes the star to look
very bright and causes the outer layers to cool. The outer layers
then collapse and become compressed.
This causes the star's outer layers to emit too little of the
star's energy and absorb too much of it. Thus the
star looks dimmer. After the outer layers absorb energy, they
puff up again, setting the star on the path to repeating the
cycle.
c.) What other property of the star can be determined from the variation?
Answer: It is possible to determine the luminosity from the
period because all Cepheid variable stars that have the same period
also have the same luminosity. It is also possible to calculate
the distance to Cepheid variable stars from their luminosities and
their apparent brightnesses.
In order for astronomers to "calibrate" this system, they
first had to determine the
the luminosities of some Cepheid variable stars from observations
and calculations. This can be done by
chosing some Cepheids that are near enough
to Earth for their distances to be measured by parallax or some
other method, measuring
their distances and apparent brightnesses, and then
calculating their luminosities from luminosity = 4 pi distance2
x apparent brightness. Astronomers found that
there is a mathematical trend between
the period and the luminosity.
As a result of that work, astronomers can determine the distances
to other Cepheids. To do that, they
measure the Cepheid's period, use it to look up the luminosity
of Cepheid's that have the same period (or use the mathematical
relationship between period and luminosity to calculate the
period), measure the
star's apparent brightness,
then calculate the distance from
distance = square root of (luminosity / (4 pi x apparent brightness)).
11.) Regarding globular clusters and open clusters:
a.) Which are older?
Answer: globular clusters
b.) Which type is often found in the Galaxy's halo?
Answer: globular clusters
c.) Which has more stars in it?
Answer: globular clusters
d.) Could our Galaxy contain a globular cluster that contains A stars?
Answer: no, globular clusters are too old. Their A stars have already
died off.
e.) Could our Galaxy contain an open cluster that
no longer contains K stars?
Answer: no, K stars should live for greater than or approximately
equal to 100 billion years. The universe is only 14 billion years
old. So, all the K stars that were ever born should still be
"alive", as long as something strange hasn't happened to them.
(Note that in the following chapter, we discuss a way
to modify a star after it is born. The modification can change its
lifetime.)
f.) Bonus question: in which type of cluster are the stars more
attracted to their cluster than to the galaxy?
Answer: globular cluster
12.)
Suppose that you are studying a cluster of stars.
Most of the stars in the cluster are main sequence stars.
But, of the main sequence stars, there are no O, B, or A stars, even
though there are F, G, K, and M main sequence stars.
How old is the cluster?
a.) less than 10 million (107) years
b.) between 10 million and 100 million (108) years
c.) between 100 million and a billion (109) years
d.) between a billion and 10 billion (1010) years
e.) between 10 billion and 100 billion (1011) years
f.) more than 100 billion years
Answer: d, the cluster is between a billion and 10 billion years
old.