1. Escape velocity formula

The escape velocity from the surface of a planet is

$v_e = \sqrt{\frac{2GM}{R}}$

where:

• $G$ = gravitational constant
• $M$ = mass of Earth
• $R$ = radius of Earth

2. Effect of compression

If Earth is compressed, its mass $M$ remains the same. So,

$v_e \propto \frac{1}{\sqrt{R}}$

3. Condition given

The escape velocity is increased by a factor of $10$:

$v_e' = 10 v_e$

Using proportionality,

$\frac{v_e'}{v_e} = \sqrt{\frac{R}{R'}}$

So,

$10 = \sqrt{\frac{R}{R'}}$

Squaring both sides,

$100 = \frac{R}{R'}$

Hence,

$R' = \frac{R}{100}$

4. Substitute Earth's present radius

Given:

$R = 6400\text{ km}$

Therefore,

$R' = \frac{6400}{100} = 64\text{ km}$

5. Final answer

The Earth must be compressed to a radius of

$\boxed{64\text{ km}}$