# Astronomy

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What is the size, i.e. the radius, of a black hole with a mass equal to that of Earth (MEarth=6×1024kg)? Give your answer to one significant figure.

• Astronomy -

R = 2 M in natural units. To convert to SI units, you have to insert c and G in here, that's most easily done as follows.

In Si units M^2 G/R has the dimensions of an energy and Mc^2 also has the dimensions of an energy, the ratio of the two is thus dimensionless:

M G/(R c^2) = dimensionless

So, we need to multiply M by G/c^2 to convert it to a lentgth in SI units. The equation in natural units must this be changed into:

R = 2 M G/c^2

You can also argue that since c = G = 1 in natural units, yo are free to write the natural units equations in this form first as it is strictly the same as R = 2 M but you then have an equation that in SI units is dimensionally correct and you can then insert the mass in kilograms in there with for c and G the SI units values to obtain R in meters.

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