Earth is about 28,000 light years from the center of the Milky Way Galaxy. If a supernova explosion occurred approximately 14,000 light years away from us and released 2.0 × 10⁴⁶ J of energy in a sudden burst of light, how much energy would enter one of your pupils? Assume the diameter of a pupil to be about 4 mm. For comparison, our Sun radiates approximately 4 × 10²⁶J/s.)

Question

Quizplus · Accepted Answer

First, we need to calculate how much of the initial energy is reaching Earth. Since the explosion is 14,000 light years away, the energy will be spread out over the surface of an expanding sphere with a radius of 14,000 light years. The surface area of a sphere is given by:

4πr²

So, the surface area of the sphere in question is:

4π(14000 light years)² = 2.97 × 10¹⁷ km²

Now, we need to calculate what fraction of that surface area is intercepted by Earth. Earth is a tiny target compared to the size of the sphere, so we can approximate the fraction as the ratio of the area of Earth to the total surface area of the sphere. The area of a circle is given by:

πr²

So, the area of Earth is:

π(6371 km)² = 1.27 × 10⁸ km²

Therefore, the fraction of the surface area intercepted by Earth is:

(1.27 × 10⁸ km²) / (2.97 × 10¹⁷ km²) = 4.28 × 10⁻¹⁰

Finally, we can calculate how much of the initial energy reaches one of your pupils. The pupil has an area of:

π(2 mm)² = 1.26 × 10⁻⁵ m²

So, the energy that enters one pupil is:

(2.0 × 10⁴⁶ J) × (4.28 × 10⁻¹⁰) × (1.26 × 10⁻⁵ m²) = 1.1 J

Earth Is About 28,000 Light Years from the Center of the Milky