Why we are able to see Satellites in the night sky?

As there are so many websites and apps that give the map of the satellite’s orbit and predict at which time it is visible for your location, I searched for one of those and it gave me the timings of the satellites which I can see from my location. They use a gyroscope sensor of our smartphone and help us to map the night sky. That night I didn’t see anything in the sky that is moving. I was a bit disappointed and thought to myself that the light pollution in my city is a bit higher to see any small satellite. But the solution for it is a bigger satellite and International Space Station (ISS) is the only satellite that I know is big. So I looked for the ISS timings for my location. This time I saw a twinkling star-like thing moving across the night sky, I was very satisfied seeing it. I observed it for a few days and it didn’t always disappear at the horizon. Most of the time it disappeared in the middle of the sky. I didn’t care about this behavior much but recently, I saw one of the claims of flat earther that, if the earth is a sphere how can we see the satellites in the night sky when they are on the dark side of the earth.

This claim got my attention because even the moon being a satellite is at a distance of 384,400 km from the earth while ISS is at a distance of just 408 Km. Being so close to earth how did it even get the sunlight while it is in the shadow region of the Earth? Is the claim true? Of course not. The earth was proved to be a sphere by many other experiments. So I started to calculate how this is happening. I wanted to know at what angle sunlight hits the earth. If we considered they were close the shadow region formed is of very less volume but actually they are approximately at a distance of 151.94 million km. In short, the light is coming from infinity. Light rays coming from infinity are considered to be parallel.

International Space Station (ISS) orbits the earth at an altitude of 408 km which a lot less compared to the radius of the earth which is 6,371 km. The orbital radius, earth’s radius, and the light rays make a right angle triangle and an angle Θ at the center of Earth. Considering the β angle of the ISS orbit to be zero (β angle is the angle made by a plane with the sunlight) so that it receives sunlight for half of its orbit. With the angle Θ the satellite receives the last portion of light before going into the shadow of the earth.

cos Θ = Radius of Earth / Radius of the Orbit

= r1 / r2

Radius of Earth (r1) = 6,371 km

Radius of the Orbit (r2) = 6,371 + 408 = 6,779 km

cos Θ = 6,371 ÷ 6,779

cos Θ = 0.9398 

Θ = 19.97˚ → Approximately 20˚

When the International Space Station (ISS) passes through this 20˚ area it is able to reflect the light and we can see it. But I wanted to know what is the last or first timing of day to see the ISS. To find this I have to know at what angle the earth receives the last light reflected by the ISS at 20˚. Another triangle is drawn at the 20˚ point on its orbit to the Earth’s surface. As the earth radius and orbital radius remain constant it gives the same angle as 20˚. So adding them up gives 40˚. Places located at this angle only can see the ISS. If we divide the earth into 360˚, Every one hour makes a 15˚ angle.

Last or first time for viewing ISS = (40/15) + (time at sunset) (or) (40/15) – (time at sunrise) hrs

→ 2.6 + (time at sunset) hrs (or) 2.6 – (time at sunrise) hrs

→ 2 hrs 36 min + (time at sunset) hrs (or) 2 hrs 36 min – (time at sunrise) hrs

If the time at sunset is exactly 6:00 PM then you can only see it before (6 +2 hrs 36 min) 8:36 PM theoretically.

Now I wanted to know how long can we see it or how long does it stay in that region of 20˚.To know this we need the length of the arc at 20˚ angle, the radius of the orbit, and the speed at which it is revolving around the earth.

Length of an arc = radius * angle (in radians)

20˚ in radians = 0.349

→ 6,779 * 0.349

Length of an arc =2,365.871 km

Now, time taken by the satellite to cover this arc =

(length of the arc ÷ speed of the satellite * 60) min

Speed of ISS is 7.66 km/s

Time = (2,365.871 ÷ 7.66 * 60) min

→5.14 min

ISS is only available to see for those locations that are 40˚ from the shadow region of the earth and it is only visible for a maximum of 5.14 min( If your location doesn’t have any hills or valleys and if you can see the horizon clearly) and the last or first time of the day you can see ISS is 2 hrs 36 min after sunset or before sunrise. These results are only of an orbit of zero β angle, but ISS β angles range from +75.1˚ to -75.1˚. At these extremes, ISS receives sunlight for its whole orbit. In this orbit, we can view the ISS for 10.28 min (In zero β angle condition also it is visible for 10.28 min but this time is separated into half but the shadow of Earth). This concludes that flat Earther’s claim is wrong and the curvature of the earth can also allow satellites to be visible in the shadow cast by the earth.

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