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Astro-7 Fall 1999


Review Problems, Midterm II (November 12)


1.
Conversion of hydrogen into helium transforms only 0.7% of the mass of the hydrogen into energy. Suppose you construct a fusion reactor which converts 10 kg of hydrogen into helium. How much energy (in joules) is released? Convert that energy to units of kilowatt-hour (a kilowatt-hour is the energy spent by using the power of 1 kilowatt during one hour, equal to $3.6\times 10^6$ J). A kilowatt-hour of electricity costs about 10 cents, as you know if you have ever looked at your electricity bill. If the energy from the fusion reactor can be converted to electricity, how much is the energy obtained from 10 kg of hydrogen worth?

2.
Knowing the luminosity of the Sun, $L_{\odot}=3.9\times 10^{26}$ W, and using the information from the previous problem, how many tons of hydrogen must the Sun fuse into helium every second to produce the power it radiates into space?

3.
During different stages of its life, the Sun "burns" different elements as fuel, either in its core or in a shell around the core.

(a)
What element is the Sun burning today? In what layer of the Sun do the reactions take place? Draw a cross section of the Sun indicating what's burning where.

(b)
After this current fuel supply runs out in the center of the Sun, will there be any other element that will burn in the center? What will the core of the Sun consist of after that? How will the Sun's color, size and luminosity change when this occurs?

(c)
The star Regulus is the brightest one in the constellation of Leo, and can easily be seen in Spring evenings. Like the Sun, Regulus is a main-sequence star. Regulus has a mass of 4 $M_{\odot}$ and a luminosity of $100 L_{\odot}$. What is the ratio of the duration of the life of Regulus in the main-sequence compared to the Sun's life? (Use proportionality).

(d)
If the Sun's lifetime on the main-sequence will be 1010 years, how long will Regulus be on the main-sequence? Assume that the luminosity of the stars can be considered to remain approximately constant over the main-sequence period.

4.
The Sun orbits the center of the galaxy at a speed of 220 km/s. The Sun is at a distance of 8 kpc (where $1\, {\rm kpc} = 1000 \,
{\rm pc}$) from the Galactic center.

(a)
Assuming the Sun is on a circular orbit, calculate the time it takes for the Sun to complete an orbit around the Galaxy.

(b)
The galaxy M31, the Andromeda galaxy, has the shape of a disk, similar to our own Galaxy the Milky Way. It covers a total angular size in the sky of 3 degrees. Its distance from us is 800 kpc. What is the physical size of the Andromeda galaxy?

(c)
The Andromeda galaxy is approximately in the direction along which the Sun is moving at present along its orbit around the Galactic center. The Andromeda Galaxy is moving toward the Milky Way at a velocity of 100 km/s relative to the center of the Milky Way. It will help you to draw a sketch of the orbit of the Sun around the Galactic center, indicating the present position of the Sun and the direction of its motion, and the position of the Andromeda galaxy and the direction of its motion relative to our Galaxy.

If we observe an absorption line in the spectrum of the Andromeda galaxy that has a laboratory wavelength $\lambda=3737\, {\rm\AA}$, to what wavelength will the line be shifted because of the Doppler effect? Think first about what is the relative velocity between the Sun and the Andromeda galaxy, and you can neglect the velocity of the Earth relative to the Sun.

5.
Sketch a cross-section of our Sun indicating where its energy is generated.
(a)
Indicate the region in the Sun where helium is most abundant.
(b)
In 5 billion years, the Sun will become a red giant star. How will its radius change? How will its surface temperature change?
(c)
When the Sun is a red giant, will it still be burning hydrogen into helium? If so, where will the hydrogen be burned? Will it burn any other elements?
(d)
At a later stage, nuclear fusion in the remnant of the Sun will cease. Will the radius of the Sun be much smaller, about the same, or much bigger than its present radius? What will be the most abundant elements in the Sun then? When and where were those elements created?

6.
A star with a mass equal to $30 M_{\odot}$ has just been formed.
(a)
At the beginning of its life, how is the star producing its energy?
(b)
Will the life of this star be much shorter, of similar duration, or much longer than the life of the Sun?
(c)
What will happen when this star dies?
(d)
What is the last element this star will have produced in its core before it dies? Why is this the last element it can produce?

7.
The parallax of a nearby star is 1/4 arcsec.
(a)
How far is the star (measured in parsecs)?
(b)
Suppose you find another star with the same luminosity but with a parallax of 1/8 arcsec. As viewed from the Earth, is this second star brighter or dimmer than the first? By what factor?
(c)
The nearest of these two stars has a surface temperature of 5000 K, and the furthest one has a surface temperature of 6000 K. Which star has the biggest radius? By what factor is it bigger than the radius of the other star?

8.
The star Capella (one of the brightest stars you can see in Winter evenings) has the same surface temperature as the Sun. The star $\alpha$ Cen A, at a distance of 1.33 parsecs from us, is very similar to the Sun and has the same luminosity and the same surface temperature as the Sun. The flux we observe from Capella and from $\alpha$ Cen A is about the same. However, Capella is ten times further away than $\alpha$ Cen A.
(a)
How much bigger is the radius of Capella than the radius of $\alpha$ Cen A?
(b)
How much bigger is the radius of Capella than the radius of the Sun?
(c)
Knowing the flux from the Sun as observed from Earth, $F_{\odot}=1370\, {\rm W}/{\rm m}^2$, what is the flux we observe from $\alpha$ Cen A and from Capella?
(d)
Imagine that you take a solid ball of radius equal to 1 inch = 2.5 cm, and heat it to the surface temperature of the Sun, T=5800 K. Assuming that the emission of light from this ball is well approximated by a blackbody, how distant should the ball be from you so that the flux you observe from it is the same as the flux from the star $\alpha$ Cen A or from Capella?

9.
The Solar wind consists of particles from the Sun's atmosphere streaming out into space. When they leave the Sun, what velocity must these particles be moving at in order to escape from the Sun's gravitational pull?

10.
In our usual experiences, Newton's laws are adequate to describe situations involving motion and/or gravity, but in certain circumstances Special Relativity or General Relativity are necessary. For each of the following situations, does Newton's theory provide the correct answer, or is it necessary to use special relativity, or is it necessary to use general relativity?
(a)
In a future Mars mission, a spacecraft travels to Mars and comes back to Earth, taking two years to complete the trip.
(b)
A superadvanced spacecraft leaves the Earth and goes to visit Mars, completing the trip in one hour.
(c)
The superadvanced spacecraft travels to the nearest star, $\alpha$ Cen, taking 10 years to get there.
(d)
Black holes are black.
(e)
The light emitted from the surface of a neutron star into space is redshifted by a large factor.
(f)
Particle physicists like to accelerate particles to near the speed of light, then crash them together, using accelerators built in the Earth.
(g)
Mercury's orbit is not quite a closed ellipse, but the orbit precesses
(h)
The Sun orbits the center of the Milky Way at about 220 km/s.
(i)
A grapefruit and a grape dropped from the leaning tower of Pisa land at the same time, startling the tourists below.
(j)
An apple falls from a tree.
(k)
When a star is seen very close to the Sun, its position is slightly altered because light is deflected due to the gravity of the Sun.



 
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Jordi Miralda-Escude
1999-11-08