An online calculators to calculate the percentage of the area of the earth (assumed to be a sphere) covered by a satellite, given its altitude and angle of elevation, is presented.
In the figure below a satellite at point \( S \), in orbit around the earth, can only cover part of the earth that has the shape of spherical cap.
The figure below shows a two dimentional representation of a satellite at an altitude \( H \) above the surface of the earth. \( \theta \) is the angle of coverage and \( \gamma \) is the angle of elevation of the satellite , measured with respect to the tangent to the earth.
The area of the spherical cap is given by
\[ \displaystyle \text{A}_{cap} = 2\pi R h \]
The ratio \( f \) of the \( \displaystyle \text{A}_{cap} \) and the total area of the earth is given by
\[ f = \dfrac{2\pi R h}{4 \pi R^2} = \dfrac{1}{2} \dfrac{h}{R} \]
where \( R \approx 6378 \) km is the radius of the earth and \( h \) is the height of the spherical cap (in red).
\( \quad \cos(\alpha) = \dfrac{\overline{OD}}{\overline{OA}} = \dfrac{R-h}{R} = 1 - \dfrac{h}{R} \)
hence
\( \quad \dfrac{h}{R} = 1 - \cos(\alpha) \)
Substitute \( \dfrac{h}{R} \) in the formula for \( f \) above to obtain
\[ f = \dfrac{1}{2} (1 - \cos(\alpha)) \quad \quad (I)\]
We now need to find \( \alpha \)
\( \quad \alpha + \gamma + \theta + 90^{\circ} = 180^{\circ} \)
Use the sine law in triangle \( OAS \) to write
\( \quad \dfrac{\sin(\gamma+90^{\circ})}{ \overline{OS} } = \dfrac{\sin(\theta)}{ \overline{OA}} \quad \quad (II) \)
Note that
\( \quad \sin(\gamma+90^{\circ}) = \cos(\gamma) \)
\( \quad \theta = 90^{\circ}-\alpha - \gamma \)
\( \quad \sin(\theta) = \sin(90^{\circ}-\alpha - \gamma ) = \cos(\alpha + \gamma) \)
\( \quad \overline{OS} = H + R \)
\( \quad \overline{OA} = R \)
Substitute in equation (II) and rewrite it as
\( \quad \dfrac{\cos(\gamma)}{ H+R } = \dfrac{\cos(\alpha + \gamma)}{ \overline{R}} \)
Use cross product to obtain
\( \quad \cos(\alpha + \gamma) = \dfrac{R}{H+R} \cos(\gamma) \)
Take \( arccos \) of both sides
\( \quad \alpha + \gamma = \arccos (\dfrac{R}{H+R} \cos(\gamma)) \)
\[ \alpha = \arccos (\dfrac{R}{H+R} \cos(\gamma)) - \gamma \quad \quad (III) \]
This calculator uses formulae \( I \) and \( III \) to calculate the percentage \( f \) of the area of the earth covered by the satellite at an altitude \( H \) and angle of elevation \( \gamma \)
Enter the altitude \( H \) of the satellite as a positive real number. Enter the angle of elevation \( \gamma \) whose value is in the range \( [0 \; , \; 90^{\circ}] \) and press "calculate".
The output is the percent of the area of the suface of the earth that is covered by the satellite.
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