Question

In Example 34.6, we imagined equipping 1950DA, an asteroid on a collision course with the Earth, with a solar sail in hopes of ejecting it from the solar system. We found that the enormous size required for the solar sail makes the plan impossible at this time. Of course, there is no need to eject such an object from the solar system; we only need to change the orbit. A much more pressing problem is Apophis, a 300-m asteroid that may be on a collision course with the Earth and is due to come by on April 13, 2029. It is unlikely to hit the Earth on that pass, but it will return again in 2036. If Apophis passes through a 600-m keyhole on its 2029 pass, it is expected to hit the Earth in 2036, causing great damage. There are plans to deflect Apophis when it comes by in 2029. For example, we could hit it with a 10- to 150-kg impactor accelerated by a solar sail. The impactor is launched from the Earth to start orbiting the Sun in the same direction as the Earth and Apophis. The idea is to use a solar sail to accelerate the impactor so that it reverses direction and collides head-on with Apophis at 80–90 km/s and thereby keeps Apophis out of the keyhole. Consider the momentum in the impactor's orbit (see figure below) when the solar sail makes an angle of 

? = 49°

 with the tangent to its orbit. Current solar sails may be about 40 m on a side, but the hope is to construct some that are about 160 m on a side. Estimate the impactor's tangential acceleration when it is about 1 AU from the Sun. Keep in mind that the sail is neither a perfect absorber nor a perfect reflector, and a heavier impactor would presumably be equipped with a larger sail. Don't be surprised by what may seem like a very small acceleration. (Assume the only forces acting on the impactor are the gravitational attraction of the Sun and the force from radiation on its solar sail. Estimate the radiation pressure to be 

P = 1.5

I

c

 and use 

A cos(?)

 as the area that is exposed to the Sun's radiation, where I is the intensity of the radiation, c is the speed of light, and A is the area of the sail. 

1 AU = 1.50 ✕ 10¹¹ m.)

 m/s²