Angular Size
Angular Size
The size of the part of Earth's surface directly under various space objects [Several images are shown, of space objects of differing size and at different distances from Earth, illustrating the differing sizes of their "shadows" as mapped onto Earth's surface viewed from the center of the Earth.]
[The first image merely sets the stage: Earth is a full circle, with two figures — Cueball and Megan — standing on it; a small space object casts a "shadow" on Megan, while a slightly larger object, though proportionally farther away, casts just about the same size shadow next to Cueball.]
[The second image is a map showing London, with the Thames running through it, and a ring highway running around it labeled "London's M25 Orbital Freeway". A grey circle is overlaid, just about the diameter of the M25; it is labeled "The Sun and The Moon (about the same size)".]
[The third image has a small grey rectangle in one corner labeled "Soccer field" for comparison. The image is dominated by four large, grey circles, considerably larger than the soccer field, labeled "Saturn", "Mars", "Jupiter", and "Venus", with Mars slightly larger than Saturn, Jupiter approximately twice the diameter of Saturn, and Venus approximately three times the diameter of Saturn. Smaller circles are labeled "Mercury", "Uranus", and "Neptune", with Mercury still somewhat larger than the soccer field, Uranus about its size, and Neptune slightly smaller.]
[The fourth image has the soccer field blown up to take up much of the view; its center circle, goal areas, and corner kick areas are visible. Labeled grey circles of various circles are again overlain: Callisto and Ganymede are about as large as one half of the field; Io, Titan, and Europa are somewhat smaller than half the field; and Ceres, Triton, and Pluto are much smaller (all three together would probably fit into the soccer field's center circle).]
[In the fifth image, there is a different grey rectangle, this one labeled "Ping Pong table", with a few balls and paddles visible. An irregular ovoid labeled "Phobos" is about the size of the table, as is a circle labeled "R. Doradus". An irregular circular shape labeled "Deimos" is about the size of half the table; a circle labeled "Betelgeuse" is a little smaller, and a circle labeled "Eris" is a little smaller, though still comfortably filling most of half of the table.]
[In the sixth image, a light grey image of laptop computer keyboard and screen is shown, viewed from directly above. An irregular shape labeled "4942 Munroe" is slightly larger than the laptop, while circles labeled "Alpha Centauri A", "Sirius", and "Alpha Centauri B" form a descending series somewhat smaller than it. Circles labeled "Proxima Centauri" and "Barnard's Star" are considerably smaller than the laptop: Proxima Centauri would fit on the trackpad, while Barnard's Star covers perhaps four keys on the keyboard.]
[In the seventh image we see a greatly zoomed-in shape which is identified as the "Tilde on laptop keyboard". A circle labeled "HD 189733 b (permadeath)" is almost as large as the tilde is wide; a circle labeled "Tau Ceti C (giant dog planet)" is somewhat smaller. Circles labeled "Gleise 581 g (jelly-filled planet)", "Gleise 667 C c (PILF)", and "HD 20794 c (moonchild)" are all 1/2 to 1/3 as wide as the tilde is wide. A smaller circle labeled "Event horizon of the black hole at the center of our galaxy" fits comfortably within the tilde's stroke width. A very small dot off in one corner (much smaller than the tilde or anything else in the image) is labeled "KOI-1686.01 (emergency backup earth)".]
[Finally, in the eighth image, the size comparison is to the grey outlines of four bacilliform bacteria labeled "E. coli". The outlines of two interstellar probes are shown, with circular main housings and protruding instruments and antennae. They are labeled "Voyager 1" and "Voyager 2".]
The exact dimensions of a professional soccer field are not precisely defined. In international matches they must be between 64 and 75m wide and between 100 and 110m long. Therefore soccer fields aren't well suited as a precise unit of measurement. However, they are commonly used to give people a rough understanding of an areas size. According to the angular diameter of the referenced celestial objects, Randall did his calculations using dimensions of 105m × 68m.
A similar model of representing the angular diameter of celestial objects as portions of the Earth's surface seen from its center was described in Isaac Asimov's May 1961 essay "Heaven on Earth".

This comic is a comparison of the angular diameters (or apparent diameter) of various celestial objects at the surface of the earth relative to a vertex at the center of the Earth as diagrammed in the opening panel. The objects' scales are compared to actual objects on earth. Each size given is for the object at its closest approach to earth.
London's M25 motorway is around 60 km (35 miles) across, a soccer field is about 100 meters long (109 yards), a ping pong table is 274 centimeters long (9 feet), a laptop is about 35 centimeters across (13.75 inches), the tilde symbol on a keyboard is about 5 millimeters long (197 mils), and a cell of E. coli is about 2 microns long (78.75 millionths of an inch).
A simple formula can be used to find the size on earth of a celestial object when the size of or distance to the object is known. This is done by taking the radius of the earth, multiplying by the diameter of the object, and dividing by the distance to the object from the center of the earth.
The space objects referenced in the panels are:
The title text states that astronomy would be much easier if the celestial sphere were mapped to the earth - like a giant globe . Due to the distance of the stars you would just need magnifying glass to see the areas representing distant stars instead of an expensive powerful telescope to see huge distances.
Approximate values for the mappings to the Earth sphere (based on mean Earth radius at 6,371.0 km or 3,958.8 mi.):

The size of the part of Earth's surface directly under various space objects [Several images are shown, of space objects of differing size and at different distances from Earth, illustrating the differing sizes of their "shadows" as mapped onto Earth's surface viewed from the center of the Earth.]
[The first image merely sets the stage: Earth is a full circle, with two figures — Cueball and Megan — standing on it; a small space object casts a "shadow" on Megan, while a slightly larger object, though proportionally farther away, casts just about the same size shadow next to Cueball.]
[The second image is a map showing London, with the Thames running through it, and a ring highway running around it labeled "London's M25 Orbital Freeway". A grey circle is overlaid, just about the diameter of the M25; it is labeled "The Sun and The Moon (about the same size)".]
[The third image has a small grey rectangle in one corner labeled "Soccer field" for comparison. The image is dominated by four large, grey circles, considerably larger than the soccer field, labeled "Saturn", "Mars", "Jupiter", and "Venus", with Mars slightly larger than Saturn, Jupiter approximately twice the diameter of Saturn, and Venus approximately three times the diameter of Saturn. Smaller circles are labeled "Mercury", "Uranus", and "Neptune", with Mercury still somewhat larger than the soccer field, Uranus about its size, and Neptune slightly smaller.]
[The fourth image has the soccer field blown up to take up much of the view; its center circle, goal areas, and corner kick areas are visible. Labeled grey circles of various circles are again overlain: Callisto and Ganymede are about as large as one half of the field; Io, Titan, and Europa are somewhat smaller than half the field; and Ceres, Triton, and Pluto are much smaller (all three together would probably fit into the soccer field's center circle).]
[In the fifth image, there is a different grey rectangle, this one labeled "Ping Pong table", with a few balls and paddles visible. An irregular ovoid labeled "Phobos" is about the size of the table, as is a circle labeled "R. Doradus". An irregular circular shape labeled "Deimos" is about the size of half the table; a circle labeled "Betelgeuse" is a little smaller, and a circle labeled "Eris" is a little smaller, though still comfortably filling most of half of the table.]
[In the sixth image, a light grey image of laptop computer keyboard and screen is shown, viewed from directly above. An irregular shape labeled "4942 Munroe" is slightly larger than the laptop, while circles labeled "Alpha Centauri A", "Sirius", and "Alpha Centauri B" form a descending series somewhat smaller than it. Circles labeled "Proxima Centauri" and "Barnard's Star" are considerably smaller than the laptop: Proxima Centauri would fit on the trackpad, while Barnard's Star covers perhaps four keys on the keyboard.]
[In the seventh image we see a greatly zoomed-in shape which is identified as the "Tilde on laptop keyboard". A circle labeled "HD 189733 b (permadeath)" is almost as large as the tilde is wide; a circle labeled "Tau Ceti C (giant dog planet)" is somewhat smaller. Circles labeled "Gleise 581 g (jelly-filled planet)", "Gleise 667 C c (PILF)", and "HD 20794 c (moonchild)" are all 1/2 to 1/3 as wide as the tilde is wide. A smaller circle labeled "Event horizon of the black hole at the center of our galaxy" fits comfortably within the tilde's stroke width. A very small dot off in one corner (much smaller than the tilde or anything else in the image) is labeled "KOI-1686.01 (emergency backup earth)".]
[Finally, in the eighth image, the size comparison is to the grey outlines of four bacilliform bacteria labeled "E. coli". The outlines of two interstellar probes are shown, with circular main housings and protruding instruments and antennae. They are labeled "Voyager 1" and "Voyager 2".]
The exact dimensions of a professional soccer field are not precisely defined. In international matches they must be between 64 and 75m wide and between 100 and 110m long. Therefore soccer fields aren't well suited as a precise unit of measurement. However, they are commonly used to give people a rough understanding of an areas size. According to the angular diameter of the referenced celestial objects, Randall did his calculations using dimensions of 105m × 68m.
A similar model of representing the angular diameter of celestial objects as portions of the Earth's surface seen from its center was described in Isaac Asimov's May 1961 essay "Heaven on Earth".
