Escape velocity.

Step 1 of 5

Energy to leave

Throw a ball up, and it comes back. Throw it faster, and it goes higher before it comes back. Is there a speed at which it never comes back? Gravity weakens with distance, so the work it does to stop the ball all the way out is finite too: from a distance $r$⁠ from the centre of a body of mass $M$⁠ out to infinity it is $\dfrac{G M m}{r}$⁠. This is the gravitational potential energy, $U = -\dfrac{G M m}{r}$$U =$$-\dfrac{G M m}{r}$⁠: negative, and zero only very far away.

A body escapes if its kinetic energy is enough to cover the whole of that well:

$\tfrac12 m v^2 = \frac{G M m}{r} \quad\Longrightarrow\quad v_{\text{escape}} = \sqrt{\frac{2 G M}{r}}$$\tfrac12 m v^2 =$$\frac{G M m}{r} \quad\Longrightarrow\quad v_{\text{escape}} =$$\sqrt{\frac{2 G M}{r}}$⁠

This is the escape speed. In the Scene a planet goes round the Sun on a circle of radius 1 AU, like the Earth.

Step 2 of 5

From the Earth

Compare it with the speed of a circular orbit from the last Unit, $v_{\text{orbit}} = \sqrt{\dfrac{GM}{r}}$$v_{\text{orbit}} =$$\sqrt{\dfrac{GM}{r}}$⁠. The only difference is the factor $2$⁠ inside the root:

$v_{\text{escape}} = \sqrt{2}\; v_{\text{orbit}}$$v_{\text{escape}} =$$\sqrt{2}\; v_{\text{orbit}}$⁠

A body on a circular orbit is already moving at about $71\%$⁠ of the escape speed. To get away, it has to be given the rest.

Step 3 of 5

From the Sun

The Earth itself moves round the Sun at about 29.8 km/s. A probe leaving the Earth already carries that speed with it. To leave the Solar System altogether, it has to reach the escape speed from the Sun at the Earth’s distance. Eccentricity and Size of the orbit are locked on the Earth’s circle.

Step 4 of 5

Stretched orbits

On an elliptical orbit the speed at perihelion is greater than the circular speed there, so it is closer to the escape speed. Conservation of energy gives a simple ratio:

$\frac{v_{\text{perihelion}}}{v_{\text{escape}}} = \sqrt{\frac{1 + e}{2}}$$\frac{v_{\text{perihelion}}}{v_{\text{escape}}} =$$\sqrt{\frac{1 + e}{2}}$⁠

At $e = 0$⁠ this is $\sqrt{1/2} \approx 0.71$$\sqrt{1/2} \approx$$0.71$⁠, and as $e$⁠ approaches $1$⁠ the ratio approaches $1$⁠. At exactly $e = 1$⁠ the orbit is no longer a closed ellipse but a parabola, and the body does not come back. Size of the orbit is locked.

Step 5 of 5

The Moon

A comet on an orbit like this one, $e = 0.85$⁠, passes the Sun at almost the escape speed, and at its far end it is almost standing still. The escape speed depends only on $M$⁠ and $r$⁠, so a small body is easy to leave. On the Moon, whose mass is about $1\%$⁠ of the Earth’s, the Apollo astronauts took off again with one small engine.