Earth Map in the Sky — Star In A Star
See the Earth in the starry sky
Stars help us find our way. Stars are like a giant map in the sky that tells us where we are on the surface of the Earth. Sailors use them as a “map” to navigate the world. For thousands of years, the stars were our best markers of latitude and longitude.
Landforms as Constellations — it’s reversed
We can map something new onto the stars: locations on the Earth! We can create an exciting new set of “constellations” out of the shapes of the continents on the Earth.
Zenith Stars — Everyone has them
We live on a sphere so we see half of the sky (a hemisphere) at any one moment. It’s easy to imagine half the Earth mapped onto half the sky.
Wherever you are on the Earth, when you look straight up (toward your zenith), you might see one star, but there are a bunch of other stars within view. All of the stars you see in the sky are directly overhead some other place on the Earth. Every place on Earth has their own set of stars directly overhead — their “zenith stars.”
Look up at any star in the night sky; that star is directly over some place on Earth. There are hundreds of “faraway zeniths” up there.
World Zeniths — See an Earth Map in the Sky
Every star maps down to a location on Earth and every location on Earth maps up to a star.
If you live in the western hemisphere, you can look up and “see” the land borders of the North American and South American continents visible, projected into the sky like a giant painting on a curved ceiling. You can see even more landforms in the sky — you can see the entire western half of the Earth projected in the sky.
Visualize Countries in the Sky
We can see country outlines in the sky. The key is to imagine yourself at the center of the Earth looking out into space “through” a translucent Earth surface.
Here is a good way to visualize these countries-in-the-sky even when you are on the surface. Look straight up and try to “see” a map-like image of your location at the zenith.
When I do this, I see southern New York state, Long Island jutting out into the water like a long pier, and the wide Hudson River emptying past New York City. Eastward is the dark expanse of the Atlantic Ocean and low on the eastern horizon are the countries of Europe and West Africa.
Westward in the sky, I can see the outline of the west coast of the US. Then, there is a big blank space of the Pacific Ocean and a spot near the western horizon that is Hawaii.
The Map of the Earth in the Sky is Reversed
There is one odd thing about the map as you see it in the sky… it’s reversed — as if seen in a mirror! This happens because we project the map lines outward into space toward the stars. When we look at the map this way it’s as if we are “inside” the Earth looking outward.
The map of the USA covers about 58˚ of sky from east-to-west. 58˚ is about 2x pinky-to-thumb (spread out all your fingers of both hands and touch thumbs). Your left pinky tip should be on your zenith. If you are in New York or somewhere on the east coast, the right pinky tip will indicate the approximate western edge of the USA.
Physical Astronomy — Stars Map to Places on Earth
Learn to see the zenith map in your sky using this Physical Astronomy technique.
Exercise 1: face south and point high in the sky.
Face south. Then, reach both hands straight up over your head and point above your head with both pointer fingers. You are pointing at your zenith. Now, bring both arms down until they are pointing one due east and one due west. You are pointing at two points in the sky that are zeniths for someone else.
When I do this exercise in New York, my left hand (the eastern) points at a spot in the starry sky that is the zenith star for someone in the country of Nigeria in West Africa. This is a location on the globe that is 6 time zones east. My right hand (the western) points at a spot in the sky that is the zenith for someone in the island state of Hawaii in the middle of the Pacific Ocean. This location is 6 time zones west of New York.
So, when I look at the eastern horizon sky I am looking at the starry sky that is already directly above a place 6 time zones ahead of me. I am looking at someone else’s zenith stars.
Exercise 2: Repeat ex. 1. But this time, face East.
Face east, point up. Now, bring your arms down and point toward the north and the south directions. This time your right hand points south and your left hand points north. Your right hand points at a spot in the sky that is over the city of Cuzco, Peru (the closest city to Machu Pichu) and your left hand points to sky that is over Yekaterinburg, Russia — the 4th largest city in Russia.
Secret! See a Star That Another Human Can’t
If you do this physical astronomy exercise immediately after sunset, the eastern and southern zenith locations are in night, but the western and northern sky points are over Earth locations that still have daytime.
This means that you can see the star that is at their zenith, but they cannot see that star. For example, Seattle still has 3 hours of sunlight left in their day so stars are invisible behind blue sky. The city of Yekaterinburg is on the opposite side of the world and just after sunset in New York it faces the Sun and has a bright daytime sky!
We are on the night time side of the Earth and we can see the current zenith stars of Seattle and Yekaterinburg — but people who live in these cities cannot see them! They have to wait to rotate to the night time side of the Earth to see stars.
Weird, right?
The Math — How High Up is the Zenith Map?
Project an imaginary map of the Earth into the sky. The map has to be the correct size so that when it is viewed from a distance it “covers” the same distances.
If a map is too close, it is just the same size as the territory. So, we have to choose the correct distance to project the zeniths. As the zenith map “projector screen” moves away from the Earth we see more of the borders of the Earth. But, at some point the distance of the map corresponds exactly to the faraway zeniths.
Our question is: “How far away from the Earth do you have to be so the landforms (like the continents) have an angular diameter that is equivalent to their “actual size” in the sky?” How far away does our imaginary zenith map USA (about 3000 miles wide) image have to be to cover 58 degrees of arc in the sky?
To answer this we need math — not difficult math, you learned it in 8th grade.
The Zenith Map Distance from Earth
The Earth is approximately 24,901 miles in circumference at the equator. If we can see half the sky from any point on the Earth, then we can “see” half the Earth projected onto the sky by the zenith map. That means that for 180˚ of sky we can “see” about 12,450 miles of the Earth’s surface projected into space. 12,450/180 = 69 miles. When 1 degree of arc spans 69 equatorial miles the image is “at” the correct distance.
1 Degree of Sky equals 69 Miles
So, at the equator every degree of sky covers about 69 miles in every direction. As you go towards the poles the longitude degrees (east and west) cover less and less zenith map distance, but the latitude degrees (north and south) always stretch 69 miles. Every 15˚ of sky equals about 1035 (69*15) miles.
The distance between your pointer finger and your pinky (when you hold your arm and hand stretched out in front of you) is 15˚ — so you are measuring about 1035 miles on Earth with that sky measurement. One pinky width is equal to 1˚, which is 69 miles of zenith map!
The Math — Inverse Tangent and Angular Diameter
There is a simple calculation that helps us determine how far away something needs to be to fill just 1˚ of the sky. Here we use just a tiny drop of trigonometry to discover the “tangent of 1 degree.”
The tangent of 1˚ is 0.017455. The inverse of something is when you divide 1 by the number you want to invert. So, the inverse of 0.017455 (1/0.017455) is 57.29. The inverse of the tangent of 1˚ helps us figure out the distance something has to be to appear to be 1 degree angular diameter.
This page explains how to calculate distance from a known angular size. “When an object’s distance is 57.29 times its size, it has an angular size of 1 degree.”
So, 57.29 * 69 miles = 3,953 miles away! This is how far away the “map” has to be to show you your hemisphere of the Earth map. 3,953 miles is higher than low Earth orbit (LEO) satellites (lower than 1200 miles); it’s closer than geosynchronous satellites (at about 23,000 miles); and it’s about 1/60 the way to the Moon.
So, imagine that the Earth map is projected onto a screen — an imaginary celestial sphere, shell-shaped — that is quite close to the Earth and encircles us. It shows us our Earthen landforms and the oceans beside, superimposed in the sky.
Summary
We live on a sphere. When we look at out night sky we are able to see stars low on our horizon that are visible directly above someone else — one-quarter the way around the around the world in all directions.
If you live within 6 time zones of someone that means that you share some “simultaneous sky.” Anyone living further than 6 time zones away sees a completely different sky — unless you can see circumpolar stars that dip under the North Star. That means that you can see countries past the North Pole and down the other side of the globe.
Your zenith is yours — it is unique and changing all the time. Not even someone standing right beside you shares your zenith. You can use this idea of the zenith stars to comprehend the vast and mysterious experience of life on a sphere.
References
A list of extreme geographic points in the USA — Wikipedia — https://en.wikipedia.org/wiki/List_of_extreme_points_of_the_United_States
Originally published at https://starinastar.com on December 2, 2019.
Daniel Cummings is the inventor of The Moon Hat — an object Forbes called “The Best Science Gift” in 2018.
