Universe: Stars And Galaxies
Universe: Stars And Galaxies
6th Edition
ISBN: 9781319115098
Author: Roger Freedman, Robert Geller, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 20, Problem 32Q
To determine

To discuss:

The effects on Earth if the Sun is suddenly replaced by a 1M white dwarf.

The appearance of this white dwarf when viewed from the Earth.

Whetherit can be seen by naked eye or not.

Whether the surface temperature of the Earth would remain the same or not.

Expert Solution & Answer
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Explanation of Solution

Given data:

Mass of the newly formed white dwarf=1MSurface temperature of the newly formed white dwarf=100,000KSurface temperature of the Sun=5,800KSurface temperature of the Earth=300KRadius of the Sun=696,000kmRadius of the Earth=6,378km

Formula used:

Luminosity can be found by using the formula

L=4πσT4R2(1)

Where,

L=Luminosityσ=StefanBoltzmann constantT=surface temperatureR=Radius

Stefan-Boltzmannlaw

E=σT4

Where,

E=Energy,σ=StefanBoltzmann constantT=surface temperature

Calculation:

For the Sun

L=4πσT4R2(2)

Where

L=Luminosity of the Sunσ=StefanBoltzmann constantT=surface temperature of the SunR=Radius of the Sun

(1)(2)LL=T4T4R2R2LL= ( 100,000 )4 ( 5,800 )4 ( 6,378 )2 ( 696,000 )2L=7.4L

The newly formed white dwarf would be 7.4times more luminous than the Sun.

Let E1 be the energy radiated by the Earth’s surface today and E2 be the energy radiated by the Earth’s surface if the Sun was suddenly replaced by 1M white dwarf.

Using the formula,

At present dayE1=σT14(1)When the sun was suddenly replaced by white dwarfE2=σT24(2)

(1)(2)E1E2=T14T24

E 17.4E1= ( 300 )4T24T24=3004×7.4T24=5.99×1010T2=494.8KT2=2210C

Conclusion:

Even though white dwarfs said to be cooler, newly formed white dwarfs are much hotter and luminous than the Sun. Since the core temperature increases rapidly due to the Helium fusion, a typical newly formed white dwarf’s surface temperature is estimated to be about 100,000K. Average size of a white dwarf is estimated to be the size of Earth. Since the mass of the newly found white dwarf is 1M, the gravitational attraction would be same as the Sun. Therefore, the Earth’s orbit would not be changed.

Even the present-day Sun cannot be viewed safely with unaided eye. Therefore, it is not safe to see a white dwarf by the naked eye. Besides, the surface temperature would be 2210C. No life will retain on Earth at this temperature.

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