homework and exercises - What would become of the earth's orbit if half of the sun's mass were suddenly to disappear?

I shall assume that the sun is at rest, so before the half mass disappear, the orbit of earth is an ellipse.

Answer

Earth will escape from the solar system (It no longer in a)

IF you remember the Virial Theorem in order for a planet to have an orbit the average potential energy and average kinetic energy of a planet must have a relationship of $$ 2K_{avg} = -U_{avg}$$

Where $U$ is a gravitational potential energy, which is given by $$U_{avg} = -G\frac{M_{\text{sun}}m_{\text{planet}}}{R_{\text{planet}}}$$

If you reduce the mass of the sun by a half, potential energy will be half too.

Thus, now, $$ 2K_{avg} > -\frac{1}{2}U_{avg}=-U_{avg,new}$$

In such case, the velocity of the planet is the escape velocity of the new sun. The planet will fly away from the sun to somewhere else and becomes a massive interstellar junk.