How far is the horizon?

While trying to find a possible location to spot the comet PANSTARRS (C/2011 L4), an interesting question came up. Would we actually get a better (and by better I mean practically better) chance of viewing if we were on a rooftop? Given that the comet is between 5-10 degrees above the Sun, there is only a small viewing window. How much better would this get if you were on a rooftop?

The question actually converted into something else, how far can you see and how much better does it get if you climb up high?

The calculation is pretty simple if you assume the Earth to be a perfect sphere. Although it is an ellipsoid, let's just assume it to be a sphere for the sake of calculation. In the diagram above,

• r is the radius of the earth
• h is the height of the observer (or height to eye level)
• d is the direct distance from eye of the observer to the horizon and 
• D is the actual distance (i.e. on the surface of earth)

Since the line with the distance d is a tangent to the sphere, the radius and it form a right angle. So with Pythagorean theory, we get 

d² = (r+h)² - r²

Mean radius of earth is about 6371km, so assuming height to eye level is about 1.5m, we get

d² = (6371001.5)² - 6371000²

 This is where it gets better, the difference between the two numbers is so small that it should be almost negligible right? Well the answer is both yes and no. The answer for d in the above equation is 4371.872. Compared to the radius of Earth, this looks small but 4.3km is something right?

Now let's suppose the guy goes to the fourth floor rooftop. Assuming each floor is about 8ft(2.4m), the value of h here is 1.5+9.6 = apprx. 11m. Now we get the distance d as, wait for it, 11.839km. Pretty good yeah?

But this is not the answer we are after, we don't need d, we need D, the real distance between us and the horizon. The reality is that because the angle t (marked in the figure above) is so small, d is almost the same as D. But I wanted to make sure and did the math anyway, and they are equal to four decimal places.

So here are some interesting points:
On a spherical  Earth

• If you were standing at sea shore, you would be seeing about 4.3km
• A child on your shoulders might see about 4.7km (assuming his eye level is about a foot above yours)
• If you are running a marathon, you would've reached the horizon 9 times.
• If you were on Burj Kalifa (the tallest building on Earth)
• If you were on the observatory, you could see up to 76km. But it is still less than 0.2% of the circumference of the Earth (theoretically calculated)
• If you were on the very tip of it, you could see up to 103km. About 0.25% of the circumference.
• If you were on top of Mount Everest, you could see up to 335km, still less than 1% of the circumference.
• From something closer to home, if you were on top of Piduruthalagala, the tallest mountain in Sri Lanka, you could see upto about 175kms (so you could, in theory, see the ocean).

But this is all good if the Earth was a perfect sphere and was without air (atmospheric refraction has a reasonable affect on the horizon and the Earth is not really a sphere. So none of those numbers are really real :)

As for the original question, whether you would have a better viewing opportunity when you are up on a rooftop, the answer is both yes and no. Since the comet is close to the sun, you have to wait till the sun goes down. When you climb higher up, further you see and the sun actually gets above the ocean. So you effectively get only the same viewing window.