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DATA Supernova! (a) Equation (16.30) can be written as fr=fs(1vc)12(1+vc)12

where c is the speed of light in vacuum3.0×108m/s. Most objects move much slower than this (v/c is very small), so calculations made with Eq. (16.30) must be done carefully to avoid rounding errors. Use the binomial theorem to show that if v, Eq. (16.30) approximately reduces tofr=fs(1-vc). (b) The gas cloud known as the Crab Nebula can be seen with even a small telescope. It is the remnant of a supernova, a cataclysmic explosion of a star. (The explosion was seen on the earth on July 4,1054 C.E.) Its streamers glow with the characteristic red colour of heated hydrogen gas. In a laboratory on the earth, heated hydrogen produces red light with frequencyrole="math" localid="1668146026646" 4.568×1014Hz; the red light received from streamers in the Crab Nebula that are pointed toward the earth has frequency4.568×1014HzEstimate the speed with which the outer edges of the Crab Nebula are expanding. Assume that the speed of the centre of the nebula relative to the earth is negligible. (c) Assuming that the expansion speed of the Crab Nebula has been constant since the supernova that produced it, estimate the diameter of the Crab Nebula. Give your answer in meters and in light-years. (d) The angular diameter of the Crab Nebula as seen from the earth is about 5 arc-minutes(1arcmin160degree)Estimate the distance (in light-years) to the Crab Nebula, and estimate the year in which the supernova actually, took place.

Short Answer

Expert verified

a) The expression of the frequency of the sound measured by the receiver when v c isfs=1-vc b) The speed of gas cloud Crab Nebula is1.18×108m/sc c) The diameter of the supernova in terms of meters is 7.5×1016mand the diameter of the supernova in light year is 7.6ly d) The distance of the Crab Nebula from the earth is 10.45×103ly, and the explosion of the supernova actually took place before10.45×103ly

Step by step solution

01

Calculate the frequency of the sound

The speed of the light in vacuum is 3.0 x 108 m/s.

Formula to calculate the frequency of the sound measured by the receiver according to the Doppler effect isfr=fs1vc121+vc12 where, is the frequency of the light measured by the receiver.is the frequency of the light emitted by the source, v is the speed of the light in the medium, c is the speed of the light in vacuum.

According to binomial theorem, if then

Applying binomial theorem in the equation (l) whento find,

fr=fs1vc121+vc12=fs112vc1+12vc=fs112vc2=fs112(2)vc=fs1vc

Thus, the expression of the frequency of the sound measured by the receiver in the case of v ֏s is =fs1vc

02

Calculate the speed of gas cloud Crab Nebula

The frequency of the red light emitted by the Nebula is4.568×1014Hz, the frequency of the light received in earth is4.568×1014Hz, the speed of the light in vacuum is3.0×108m/s

Formula to calculate the frequency of the sound measured by the receiver forfrom the solution of part

fr=fs1vc4.568×1014Hz=4.568×1014Hz1v3.0×108m/sv3.0×108m/s=14.568×1014Hz4.568×1014HzV=1.18×108m/s

Therefore, the speed of gas cloud Crab Nebula is 1.18×108m/s

03

STEP 3 Calculate the diameter of the supernova

The supernova is occurred 960 years ago from the year 2015 and the speed of the gas of the supernova is-1.18×106m/s Formula to calculate the diameter of the supernova is, D = yv where,D is the diameter of the supernova, v is the number of years that the supernova were occurred, v is the speed of the supernova gas.

D=960×1.18×106m/s=3.52×1016m/s

Convert the diameter of the supernova from meter to light years is, Dy=Dm×1ly9.46×1015m

Substitute 3.52×1016m/s for in the above equation to find Dly

Dly=3.52×1016m/s×1ly9.46×1015m=7.6ly

Therefore, the diameter of the supernova in terms of meters is 7.5×1016mand the diameter of the supernova in light year is 7.6ly

04

Calculate the distance of the Nebula from the earth surface

Formula iss=22xDijN1a were,s is the distance of the Nebula from the earth.

DlyNdiameter of the Nebula, a is the arc-minutes. Substitute the values in the above equation

s=22xDijN1as=22x(7.6ly)15arcminutes×1deg360×60arcmin=10.45×103ly

Therefore, the distance of the Crab Nebula from the earth is 10.45×103ly, and the explosion of the supernova actually took place before10.45×103ly

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