For a futuristic spacecraft, a solar sail was used as a means of propulsion. Assume the spacecraft had a mass of 360 kg, the sail area (perpendicular to the Sun) was 1 million square meters, and the spacecraft was 0.5 AU (astronomical units) from the Sun. Remember than 1 AU is the Earth to Sun distance. Assume the reflectivity of the sail was 1.0. What is the acceleration (in m/s2) of the spacecraft away from the Sun?

a = 5.037 10⁻²  m / s²

Explanation:

For this exercise we will use the concept of radiation pressure for a reflective surface

           P = S / c

Where c is the speed of light and S is the pointing vector which is equal to the intensity of the radiation

The definition of Pressure is

          P = F / A

          F = P A

Newton's second law gives the relationship between force and acceleration

            F = m a

We substitute

           P A = m a

           a = I  A / c  m

Let's look for the radiation intensity, for this we use that the intensity in the Earth's orbit is 1360 W / m².

We must look for the intensity in the orbit would want 0.5 AU, for this we use the definition of instance

           I = P / A

Where P is the power of the sun's emission that is constant

         P = I A

         I₁ A₁ = I₂ A₂

Where index 1 is for Earth and 2 for the orbit at 05 AU

he area of a sphere is

          A = 4 π R²

           I₂ = I₁ A₁ / A₂

           I₂ = I₁ R₁² / R₂²

           I₂ = 1360 (150 10⁶/75 10⁶)²

           I₂ = 5440 W

This is the radiation at the point of interest

Let's calculate

           a = 5440/3 10⁸   10⁶/360

           a = 5.037 10⁻²  m / s²

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