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We often see photos of Earth from space, but it is rarely clear what percentage of Earth's hemisphere is actually visible from any particular orbit.

• Is there a graphic or diagram that would show the percentage of Earth's hemisphere that is visible from low Earth orbit? • high Earth orbit? • geosynchronous orbit? • lunar orbit? • 1 million miles? • the Sun?

• Alternatively, is there a graphic or diagram that would show the percentage of a mathematical hemisphere "seen" from various points at multiples of a sphere's radii?

• Is there a website with interactive graphics that can calculate the above?

• I am a fan of space exploration (and accurate graphics). My working knowledge of math and geometry is limited.

William Elliot
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Jon Alexandr
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    Assuming the surface of the earth is a sphere, or more realistically? – kimchi lover May 22 '20 at 22:52
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    $\frac{h/2}{r+h}$ of the surface of a sphere with radius $r$ can be seen when viewed from an altitude of $h$. I'll write this up. – robjohn May 22 '20 at 23:03
  • $\frac{h}{r+h}$ of the surface of a hemisphere of a sphere with radius $r$ can be seen when viewed from an altitude of $h$. I've written this up. – robjohn May 23 '20 at 03:58

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Suppose we are at an altitude of $h$ above a sphere with radius $r$. $\triangle ABD\cong\triangle DCB$; therefore, $$ \frac{BC}{BD}=\frac{BD}{AD} $$ Since $AD=h+r$ and $BD=r$, we get $BC=\frac{r^2}{h+r}$. Subtracting from $r$ gives the width of the cap to be $\frac{hr}{h+r}$. The surface area of any cap or band on a sphere is the width of the cap or band times the circumference of the sphere: $2\pi r\frac{hr}{h+r}$. Since the surface area of the hemisphere is $2\pi r^2$, the portion of the hemisphere area in the cap is $$ \bbox[5px,border:2px solid #C0A000]{\frac{h}{h+r}} $$

enter image description here

robjohn
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