Answer:
A = 8.34 x 10^(5) m²
Explanation:
The intensity of the solar radiation is the average solar power per unit area. Thus,
I = P/4A = P/(4(πr²))
Where;
P is average solar power.
r is it's distance from centre of sun
A is area
Now, The rate at which the Sun emits energy has a standard value of 3.90 × 10^(26) W
Thus;
I = [3.90 × 10^(26)]/(4πr²)
I = [3 x 10^(25)]/r²
Now, the formula for radiation pressure is;
P_rad = 2I/c
Where c is speed of light and has a value of 3 x 10^(8) m/s
Thus,
P_rad = 2([3 x 10^(25)]/r²)/(3 x 10^(8))
P_rad = [2.07 x 10^(17)]/r² N/m²
Also, Radiation pressure on ship; P_rad = F/A
Where Force on ship and A is area.
Thus;
F = P_rad x A
So, F = [2.07 x 10^(17)•A]/r²
Now,
F_grav = GMm/r²
Where;
G is gravitational constant with a value = 6.67 x 10^(-11) Nm²/kg²
M is mass of sun with a value of 1.99 x 10^(30)
m is mass of ship and sail = 1300 kg
Thus, plugging in the relevant values to obtain;
F_grav = (6.67×10^(-11) × 1.99 x 10^(30) × 1300)/r²
F_grav = [17.255 x 10^(22)]/r²
Now, equating F to F_grav, we get;
[2.07 x 10^(17)•A]/r² = [17.255 x 10^(22)]/r²
r² will cancel out to give;
2.07 x 10^(17)•A= [17.255 x 10^(22)]
A = [17.255 x 10^(22)]/2.07 x 10^(17)
A = 8.34 x 10^(5) m²
