Question

If the Sun were to become a black hole but maintain all of its mass, what would be its Schwarzschild radius? Given this and the fact that Earth’s average distance from the Sun is about 150 million km, would the gravity of the black hole pull Earth into it? Why or why not?

Answer 1

Part -1

We are assuming that we keep the Sun's mass the same but reduce the Sun's radius such that the escape velocity becomes greater than the speed of light (300,000 km/s = 3.0 x 10⁸ m/s). We solve the escape velocity equation for R:

    v_escape = (2GM/R)^0.5

    R = 2GM/v²_escape

    R = 2 x 6.67 x 10^-11 N m² kg^-3 x 1.99 x 10³⁰ kg / (3.0 x 10⁸ m/s)²

    R = 2.95 x 10³ m = 2.95 km

Part -2

This question was meant to imply that the mass of the Sun was unchanged, thus we shrunk the Sun to a radius of 3.0 km. In other words, you know the "size of the black hole." Certainly, the escape velocity from the surface of this tiny object would be 300,000 km/s. However, at the distance of the Earth, nothing has changed. The force of gravity depends on the distance between the centers of the two objects (unchanged) and the masses of the two objects (unchanged). The center of the sun is still 150 million km distant, the mass of the Earth and Sun are unchanged. Earth would continue to orbit the new Sun as if nothing happened. But it would get very, very dark and very, very cold. And lifeless.

Hope this explanation will help you.

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