Because it's Friday: a video tour of the International Space Station

This video tour of the International Space Station from NASA commander Sunny Williams (via Andrew Sullivan) is just amazing:

 

I loved, loved watching this -- it made me feel like I was six again, when I wanted to be an astronaut. I hope NASA does more videos like this to inspire more boys and girls to be scientists and aspire one day to float as effortly in Space. Do check it out this weekend if you don't have time right now, seriously it's worth it.

The bit of the video where Commander Williams is in the cupola and watching the surface of Earth glide below made me wonder: just how much of the Earth can you actually see from the ISS? Google searching didn't reveal the answer, but it's just a simple bit of trigonometry, right? (Cue me spending far too long scratching diagrams and equations on paper...) We just need to find the height of the viewable spherical cap, which if my math is right is just h*r/(h+r), where h is the orbit height and r is the diameter of the Earth. (Don't you just love it when all the terms in a complex formula cancel out, and you're left with something simple?) I wrote an R function to calculate the proportion of the surface that would be visible:

visible.surface <- function(r=6378,h=370) {
  # r - radius of planet (default: radius of earth in km)
  # h - height of satellite (default: mean orbit alt of ISS)
 
  ## find "x", depth of sperical cap
  x <- h*r/(h+r)
 
  ## ratio of spherical cap, 
  ##   2*pi*r*x, https://en.wikipedia.org/wiki/Spherical_cap
  ## to surface area of sphere
  ##   4*pi*r^2
  ## returned as a percentage
  x / (2*r) * 100
  }

The Earth has a radius of 6478 kilometers, and the ISS is in a low-Earth orbit that averages about 370km above the surface. Plugging those numbers in, we surprisingly find that only 3% of the Earth's surface is visible from the ISS at any one time!

> visible.surface(6378, 370)
[1] 2.741553

Running some other calculations, 

• From the top of the world's tallest building, the 830m Burj Khalifa, you can see 0.0065% of the Earth's surface (about 33,000 square kilometers)
• Astronauts on the Space Shuttle (orbiting at 390km) could see at most 2.9% of the surface at a time
• GPS satellites (orbiting at 20,200km) have a view of 38% of the earth's surface
• The famous "Blue Marble" photograph shows only 43.8% of the Earth's surface
• An astronaut on the moon can see only 49.2% of the Earth's surface (and conversely, we on Earth see 49.8% of the Moon's surface)

So it turns out you have to get a long, long way away from Earth until you can get close to seeing a complete hemisphere!

Comments

[DMac]

You'd never see a full hemisphere - it's the limit as your distance approaches infinitely far away!

[David Smith]

Exactly, Danny -- I wanted to get a sense of how far you needed to get away before the you could see "most" of a hemisphere. I think I always thought the ISS orbited higher than it does, which is why the 3% number was surprising to me. Also the fact that the Blue Marble doesn't nearly show half of the earth was a surprise.

[Matt G]

Thanks for posting. This was fantastic!

[Marty Swartz]

Hm, can we turn this around? At a given instant, what is the radius of ISS viewability on earth?

I've subscribed to SpotTheStation.NASA.gov in order to receive notifications of ISS visibility. Remarkably, (at least to me,) we've had overflights on three consecutive nights this week, along significantly different tracks each night. This set me to puzzling over how LARGE an area is in view at a given point in time. Any thoughts?