Universe
Universe
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ISBN: 9781319039448
Author: Robert Geller, Roger Freedman, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 19, Problem 50Q
To determine

The time period for which the Sun will be a red giant. Assuming that its luminosity is because of fusion of remaining hydrogen at its core and will be 2000 times greater than the present time.

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Answer to Problem 50Q

Solution:

35 million years.

Explanation of Solution

Introduction:

Rate of consumption of hydrogen can be defined as the mass of hydrogen combusted per second to produce energy and helium.

Rate of hydrogen combustion = mass of hydrogen combustedtime in seconds

The rate of hydrogen consumption is the same hence, the rate for its main sequence lifetime and red giant phase can be equated.

Explanation:

Obtain the value of main sequence lifetime of the Sun from the refered figure 19-9, as 1.2×1010 years.

Understand that the amount of hydrogen consumed by the Sun per second is 600 millioin tons of hydrogen, which in kilogram is 6×1011 kg.

Since, there are 3.16×107 seconds in a year.

Thus, calculate the amount of hydrogen consumed by the Sun in 12 billion years.

M12billion=6×1011 kg/s(12×109 yr(3.16×107 s1 yr))=2.275×1029 kg2.28×1029 kg

Consider, the standrad value for the mass of the Sun as, 1.99×1030 kg.

Understand that the mass of hydrogen present in the present-Sun, assuming 74% of the Sun is hydrogen. Thus,

Mp=(1.99×1030 kg)(74%)=(1.99×1030 kg)(74100)=(1.99×1030 kg)(0.74)=1.47×1030 kg

Here, Mp is the mass of the present-Sun.

Assume, the Sun is 5 billion years old. Then, the hydrogen converted in 5 billion years is given by,

M5billion=6×1011 kg/s(5×109 yr(3.16×107 s1 yr))=9.48×1028 kg0.09×1030 kg

Thus, the initial mass of hydrogen present in the Sun was,

Mtotal=Mp+M5billion=1.47×1030 kg+0.09×1030 kg=1.56×1030 kg

Subtract the mass of hydrogen, consumed throughout the main-sequence life, from the total mass of hydrogen present in the Sun to obtain the mass of hydrogen remaining in the Sun. So, the mass of hydrogen remaining after main-sequence lifetime is over.

Mr=MtotalM12billion=1.56×1030 kg0.228×1030 kg=1.332×1030 kg

To calculate time form a proportion, use the unitary method.

Assume that the rate, by which the Sun consumes hydrogen, is constant throughout its main-sequence lifetime.

Since, the time required to finish 2.28×1029 kg of hydrogen is 12×109 years.

Thus, the time required to finish 1.332×1030kg of hydrogen will be ‘t’ years, which is calculared as,

2.28×1029 kg12×109 years=1.332×1030 kgt yearst=(12×109)(1.33×1030)2.28×1029 years=7×109 years

At 2000 times, the rate of consumption remaining life would be,

tnew=7×1010 years2000=35 million years

Conclusion:

The time, for the red giant phase to finish the conversion of hydrogen, is 35 million years.

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