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Math help!
DEFAULT Moon and Corset
clwilson2006 wrote in little_details
OK, Here is the problem. I'm told that the horizon is approximately 3 miles away, I know this is dependent on height of the person and curve of the earth and what not. I can find numerous formula on the internet.

What I need to know. Say my character is six foot tall, and looking across a desert landscape, I need the horizon to be TWO miles away, so is the planet he is standing on bigger or smaller than earth? I just cannot wrap my head around it. Math and science are not my thing, when I google things like "how far is the horizon" I get formula for Earth, not as if he was on another planet.

help?

Smaller. You can look for Mars or Moon horizonts for example.

if you want the horizon to be a smaller distance away, decrease the size of your planet. if you want it further away, increase the planet size.

Smaller. A bigger planet would have gentler curve, causing the horizon to be further away. A smaller planet curves away sooner.

It has to be smaller. that's not even a math problem, that's a logic problem. </p>

And since I haven't read the comments yet, I bet someone has already beaten me to this answer.


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Black lab mix and Chihuahua. They think she's his mother.

Think of it this way. If you're standing on a flat, infinite plane, the horizon will be infinitely far away. If you're standing on a basketball-sized object, the horizon will be very close indeed. So, the larger the planet, the farther the horizon. Keep in mind that things beyond the horizon can still be seen, though, because they themselves stand up above the horizon.

Bear in mind as well that if the horizon is two miles away for a six foot man, then he can see the top of the head of another six foot man up to four miles away.

horizon while standing on a small and large planet

Now with (a slightly exaggerated) illustration.

The horizon is where the tangent of your sight line touches the planet. So as you can see in my wonderful paint picture, a smaller planet makes the horizon closer. I could go further into the math of it all, but have a feeling that it's not that important to you. But doing some quick and dirty math (with some abbreviations and hoping I did everything correctly) I get that the new planets radius should be about 4/9ths or about half of earths.

Yay, applied geometry!

The elevation of the desert will also influence the distance your person can see. Just because the ground looks flat doesn't mean there isn't a very slight slope; most people won't notice a 5o incline outdoors.

Even if your world is 4/9ths of Earth, it may have a similar gravity depending on its density and composition.

The planet's volume would be much smaller than 4/9ths - around a tenth, if my math is right - so it'd need to do some interesting things with composition to be close to Earth's gravity.

Yep, 9% of Earth volume.

At 4/9ths Earth's radius, assuming it has a similar composition, gravity would be 4/9ths Earth-normal. (Volume and hence mass scale with the r-cubed, but gravity scales with mass on r-squared, so the net effect is that under constant composition, gravity scales with r.) So, noticeably lighter than Earth, but you're not going to go floating off into space.

To get earthlike gravity at that mass, you'd need a composition about as dense as lead. That seems to be possible (there are plenty of things more dense than lead) but I don't think it's common among rocky planets.

That *is* a wonderful paint picture.

All of the above. If you want to get technical, here's what google brought me:

"The distance to the horizon can be calculated, in metric units, using the formula d = sqrt(h(2r+h)), where sqrt is the square root, h is the height of the observer (or rather his eyes) above sea level and r is the radius of the planet. All units are in meters. The distance (d) you get out of this formula is the distance in direct line from the observer's eyes to the horizon. It is not the distance along the curvature of the planet. The difference between these two figures is, however, very small for moderate values of h (i.e., values below 100 kilometers)."

I am intrigued.

Lol, also--I looked up all that info writing my pirate AU. :) But it sounds like you've got it.

If you know anyone who is or has been in the services, ask them - they always need to know how far they can see and of course it varies according to how high up you are. My stepdad (ex-Marines) had a very simple formula, and it's annoyed me ever since that I can't remember it!

I don't think the formula the Marines use would involve the radius of the planet, though, which is what the OP's interested in.

*reads question more carefully* Damn. Fair point! I'd still love to know that formula though, because it was very simple indeed.

“The square root of your height (or elevation) in feet times 1.2 equals the distance to your horizon in miles.

They teach it to the Army as well - Grin.

So, given that most people over ?forty? / ?fifty? still think in feet and miles anyway, all you have to remember is "Distance equals one point two root height". Nice. Thank you!

Although of course all the stupid maps have shown contour lines in metres for years...

Edited at 2012-05-16 11:33 am (UTC)