Question

One of the newest major telescopes in the world, the LAMOST telescope in China measures the wavelength spectra of 4000 objects simultaneously. The spectrum of a star tells us what the star is composed of and it tells us how it is moving relative to us. One of the missions of LAMOST is to observe motions of stars in the Milky Way to understand the evolution of our galaxy.
The objective mirror diameter of LAMOST is 4.9 meters with a focal length of 20 meters. The optical output is fed into 4000 optical fibers, each one of which is set in the focal plane of the telescope to accept light from one object, such as a star or galaxy. The output from each optical fiber is fed into a spectrograph or spectrometer to measure the spectrum.
There are many design elements of the LAMOST telescope that are related to topics from Physics II. You will work through several of them.
a)   One of the important spectral lines of hydrogen is the longest wavelength Balmer line, in the red with wavelength 656.3 nm. Most of the detectors for LAMOST are CCD arrays using the semiconductor silicon. The detection method involves optical excitation of an electron across the band gap from the top of the valence band to the bottom of the conduction band. If the photon energy is smaller than the band gap, no signal is seen. Do photons from the red line Balmer transition have enough energy to excite a carrier across the band gap of Si? (You will need to look up the bandgap of silicon.) Explain.
b)   List the range in wavelength, photon energy, and frequency ranges for each of the following types of electromagnetic radiation: gamma rays, x-rays, visible light, microwaves, and radio waves.
c)   The Sun emits about 4 x 1026 W in all wavelengths from the radio to gamma rays. Estimate the intensity of the radiation (in W/m2) incident on the Earth using an approximate orbital distance of the Earth from the Sun of 1.5 x 108 km.
d)   Estimate the flux of photons on the Earth (in photons/(s-m2)) from the Sun assuming that all of the incident radiation has a wavelength of 600 nm.
e)   Estimate the intensity at the Earth (in W/m2) from a similar star at the distant edge of the Milky Way, about 105 light-years away. (Assume that the only reason the intensity changes is distance (no absorption.))
f)   Estimate the power entering the LAMOST telescope from the distant star in part (d). (Assume a telescope objective aperture of 4.9 m diameter which collects all of the light incident on it.)
g)   Assuming that all of the light from the star has a wavelength of 656.3 nm, estimate the number of photons per second from the star in part (d) collected by the LAMOST telescope.
h)   Assume that the size of the image of a star is given by the diameter of the central spot of the diffraction pattern of the star. Assuming a wavelength of 656.3 nm and using a focal length of 20 m, estimate the diameter of this image. How does it compare with the typical multimode fiber optic core diameter of 50 micrometers?
i)   Assuming a distant object could be observed at a wavelength of 650 nm. How far apart would two stars on the far side of the galaxy have to be for a perfect 4.9 meter telescope to just be able to resolve their images as two distinct stars?
j)   List at least four real-world issues that would degrade the ability of such a ground-based telescope to observe and/or resolve distant stars.

Answer 1

(a)

Band Gap of Si is 1.14 eV

The energy transferred by 653 nm Balmer line is

E=hc/=3.04 x 10-19 J

1 J = 6.242 x 1018 eV

so,

E=18.98 x 10-1 = 1.9 eV

Hence, we can see E is greater than Band gap of Si and hence it can put the electron in conduction band.

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(b)

Gamma Rays

Wavelength: Less than 10-11m

Photon Energy: use hc/ to find it (should be in  keV)

Frequency:(use c= frequency x wavelength, where c is speed of light to find frequency)( should be around 1018 Hz)

X-Rays

Wavelength: 10-11 to 10-8 m

Photon Energy

Frequency:

Visible

Wavength:0.4 to 0.7 micrometer

(find other quantities using expression above)

Microwave

Wavelength: 10 raised to -3 to -1 meters

(find other quantities using the expression above)

Radio

Wavelength: 10 raised to -1 and higher

(find other quantities using expression above)

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(c)

divide the intensity sent out by sun by the surface area of the imaginary sphere subtended by the orbital distance between sun and earth

i.e.

where r is the distance between sun and earth

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(d)

Power density = flux x Energy of each photon

Energy of each photon = hc/ ( where denominator is 600 nm, dont forget to convert this to meter)

Power density was found in previous questions in W/m2

Using this find flux

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(e)

Use the same expression as used in (c)

only the value of r changes to be as given in the part (e)

1 light year is 9.5 x 1015 m

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(f)

We know the power density as calculated in (b) multiply it with the area of the telescope aperture

i.e. where d=4.9 m

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(g)

As

multiply flux found in (d) with the area found in (f) to find number of photons / second

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(h)

Read about Airy disk

s = 0.01745 f

f is the focal length and s is called plate scale

0,017... is the number of radians in a degree

size of image

i=s

where is the angular size of the object in sky in degrees (angular size of sun in sky in degrees is 0.5 degrees)

Use this find the answer

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(i)

Resolving Power = D/(1.22 )

D is the size of aperture

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