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The Schwarzschild radius RBH for an object of mass M is defined as: Rbh = (2GM)/(c^2)

Where c is the speed of light and G is the universal gravitational constant. RBH gives the radius of the event horizon of a black hole with mass M (in other words, it gives the radius to which some amount of mass M would need to be compressed in order to form a black hole).

1) The mass of the Sun is about 1.99 × 1030 kg. What would be the radius of a black hole with this mass?

2)The mass of Mars is about 6.42 × 1023 kg. What would be the radius of a black hole with this mass?

3) Suppose you want to make a black hole that is roughly the size of an atom (take RBH = 1.30 × 10-10 m). What would be the mass M of such a black hole?

Respuesta :

cryssatemp

Answers:

1) 2951.39 m

2) 0.000952 m

3) [tex]8.765(10)^{16}kg[/tex]

Explanation:

1) We know the Schwarzschild radius [tex]RBH[/tex] is given by the following equation:

[tex]RBH=\frac{2GM}{c^{2}}[/tex]   (1)

Where:

[tex]G=6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}[/tex] is the Universal Gravitational Constant

[tex]M[/tex] the mass of the black hole

[tex]c=3(10)^{8}m/s[/tex] is the speed of light

Now, if we have a black hole with the mass of the Sun ([tex]M_{Sun}=1.99(10)^{30}kg[/tex]), its radius will be:

[tex]RBH_{Sun}=\frac{2GM_{Sun}}{c^{2}}[/tex]   (2)

[tex]RBH_{Sun}=\frac{2(6.674(10)^{-11}\frac{m^{3}}{kgs^{2}})(1.99(10)^{30}kg)}{{(3(10)^{8}m/s)}^{2}}[/tex]   (3)

[tex]RBH_{Sun}=2951.39m[/tex]   (4) This is the radius of the black hole with the mass of the Sun

2) On the other hand, if the black hole has the mass of Mars ([tex]M_{Mars}=6.42(10)^{23}kg[/tex]), its radius will be:

[tex]RBH_{Mars}=\frac{2GM_{Mars}}{c^{2}}[/tex]   (5)

[tex]RBH_{Mars}=\frac{2(6.674(10)^{-11}\frac{m^{3}}{kgs^{2}})(6.42(10)^{23}kg)}{{(3(10)^{8}m/s)}^{2}}[/tex]   (6)

[tex]RBH_{Mars}=0.000952m[/tex]   (7) This is the radius of the black hole with the mass of Mars

3) In this case, we have to isolate [tex]M[/tex] from (1):

[tex]M=\frac{RBH c^{2}}{2G}[/tex]   (8)

Where [tex]RBH=1.30(10)^{-10}m[/tex]

Solving (8) with the known values:

[tex]M=\frac{(1.30(10)^{-10}m)(3(10)^{8}m/s)^{2}}{2(6.674(10)^{-11}m^{3}/kgs^{2}}[/tex]   (9)

[tex]M=8.765(10)^{16}kg[/tex]   (10) This is the mass of the black hole

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