To determine the mass per unit area of a solar sail that will maintain a balance between the gravitational pull of the Sun and the pressure exerted by sunlight, we need to analyze the forces acting on the sail. The gravitational force pulling the sail towards the Sun must equal the radiation pressure pushing it away. Let's break this down step by step.

Understanding the Forces

1. **Gravitational Force**: The gravitational force acting on the sail can be calculated using Newton's law of gravitation. The formula is:

F_gravity = G * (M_sun * m_sail) / r^2

Where:

• G is the gravitational constant, approximately 6.674 × 10^-11 N(m/kg)².
• M_sun is the mass of the Sun, about 1.989 × 10³⁰ kg.
• m_sail is the mass of the sail, which we can express as the product of the mass per unit area (σ) and the area (A).
• r is the distance from the Sun to the sail, which is approximately 1.5 × 10¹¹ m (the mean distance of the Earth from the Sun).

2. **Radiation Pressure**: The pressure exerted by sunlight on the sail can be calculated using the formula:

P_radiation = I / c

Where:

• I is the intensity of sunlight at the distance of the Earth, approximately 1361 W/m².
• c is the speed of light, about 3 × 10⁸ m/s.

The force due to radiation pressure can then be calculated as:

F_radiation = P_radiation * A

Setting the Forces Equal

For the sail to remain stationary, these two forces must be equal:

F_gravity = F_radiation

Substituting the expressions we derived:

G * (M_sun * σ * A) / r^2 = (I / c) * A

We can cancel the area (A) from both sides, as long as A is not zero:

G * (M_sun * σ) / r^2 = I / c

Solving for Mass per Unit Area

Now, we can rearrange this equation to solve for σ:

σ = (I / c) * (r^2 / (G * M_sun))

Substituting the known values:

σ = (1361 W/m² / (3 × 10⁸ m/s)) * ((1.5 × 10¹¹ m)² / (6.674 × 10^-11 N(m/kg)² * 1.989 × 10³⁰ kg))

Calculating Each Component

1. Calculate the radiation pressure:

P_radiation = 1361 / (3 × 10⁸) ≈ 4.537 × 10^-6 N/m²

2. Calculate the gravitational force component:

G * M_sun ≈ 6.674 × 10^-11 * 1.989 × 10³⁰ ≈ 1.327 × 10²⁰ m³/s²

3. Now plug these values into the equation for σ:

σ = (4.537 × 10^-6) * ((1.5 × 10¹¹)² / (1.327 × 10²⁰))

4. Calculate the area term:

(1.5 × 10¹¹)² = 2.25 × 10²² m²

5. Now substitute back:

σ = (4.537 × 10^-6) * (2.25 × 10²² / 1.327 × 10²⁰)

σ ≈ (4.537 × 10^-6) * (16.95) ≈ 7.686 × 10^-5 kg/m²

Final Conversion to mg/m²

To express this in mg/m², we convert kilograms to milligrams (1 kg = 1,000,000 mg):

σ ≈ 7.686 × 10^-5 kg/m² * 1,000,000 mg/kg ≈ 76.86 mg/m²

Rounding this to the nearest integer gives us:

Mass per unit area of the sail ≈ 77 mg/m²

This means that for the solar sail to maintain its position without being pulled in or blown away by the Sun, it should have a mass per unit area of approximately 77 mg/m².

Approved