1276: Angular Size

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Angular Size
If the celestial sphere were mapped to the Earth's surface, astronomy would get a LOT easier; you'd just need a magnifying glass.
Title text: If the celestial sphere were mapped to the Earth's surface, astronomy would get a LOT easier; you'd just need a magnifying glass.

[edit] Explanation

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 yd.), a ping pong table is 274 centimeters (9 ft.) long, a laptop is about 35 centimeters across (13.75 in.), the tilde symbol on a keyboard is about 5 millimeters (197 mils) long, 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 Sun and the Moon, and also the open cluster Messier 25, have approximately the same size (around 0.5 degrees of arc) when seen from the Earth.
  • Phobos and Deimos are the moons of Mars. Eris is another dwarf planet. R Doradus and Betelgeuse are giant stars, respectively around 180 and 640 light-years away. R Doradus is the star with the largest apparent diameter (other than the sun, of course).

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.):

Panel Object Distance Diameter Size at Earth sphere
Panel 1 Sun 149,600,000 km (9.296×10^7 mi.) 1,392,684 km (865,373.7 mi.) 59.3 km (36.8 mi.)
Moon Semi-major axis 384,399 km (238,854 mi.) 3,476.28 km (2,160.06 mi.) 57.6 km (35.8 mi.)
Moon Perigee 363,295 km (225,741 mi.) 3,476.28 km (2160.06 mi.) 60.9 km (37.8 mi.)
Messier 25 2,100 light-years
19.867 × 10^15 km (12.158×10^15 mi.)
19 light-years
179.753 × 10^12 km (111.693×10^12 mi.)
57.6 km (35.8 mi.)
Panel 2 Mercury closest approach 92 million km (57 million mi.) 4,879.4 km (3,031.9 mi.) 0.34 km (370yd.)
Venus closest approach 41 million km (25 million mi.) 12,103.6 km (7,520.83 mi.) 1.9 km (1.2 mi.)
Mars closest approach 56 million km (35 million mi.) 6,792.4 km (4,220.6 mi.) 0.77 km (840yd.)
Jupiter 778,547,200 km (4.8377×10^8 mi.) 139,822 km (86,881.4 mi.) 1.14 km (1,250yd.)
Saturn 1,433,449,370 km (1.5676×10^12 mi.) 120,536 km (74,897.6 mi.) 0.54 km (590yd.)
Uranus 2,876,679,082 km (1.7875×10^9 mi.) 51,118 km (31,763 mi.) 0.11 km (120yd.)
Neptune 4,503,443,661 km (2.7983×10^9 mi.) 49,528 km (30,775 mi.) 0.07 km (230 ft.)
Panel 3 Io 778,547,200 km (483.77×10^6 mi.) 3,643 km (2,264 mi.) 29.8 m (74.8 ft.)
Europa 778,547,200 km (483.77×10^6 mi.) 3,122 km (1,940 mi.) 25.5 m (83.7 ft.)
Ganymede 778,547,200 km (483.77×10^6 mi.) 5,262 km (3,270 mi.) 43.1 m (141 ft.)
Callisto 778,547,200 km (483.77×10^6 mi.) 4,821 km (2,996 mi.) 39.5 m (130 ft.)
Titan 1,433,449,370 km (890.7×10^6 mi.) 5,150 km (3,200 mi.) 22.9 m (75.1 ft.)
Triton 4,503,443,661 km (2.7983×10^9 mi.) 2,705.2 km (1,680.9 mi.) 3.8 m (12 ft.)
Ceres Perihelion 380,995,855 km (236.74×10^6 mi.) 974.6 km (605.6 mi.) 16.3 m (53.5 ft.)
Pluto Perihelion 4,437,000,000 km (2.757×10^9 mi.) 2,306 km (1,433 mi.) 3.3 m (11 ft.)
Panel 4 Phobos 56 million km (35 million mi.) 26.8 × 22.4 × 18.4 km (16.7 × 13.9 × 11.4 mi.) 3.05 m (10.0 ft.)
Deimos 56 million km (35 million mi.) 15 × 12.2 × 10.4 km (9.3 × 7.58 × 6.46 mi.) 1.71 m (5 ft.7in.)
Eris Perihelion 5.723 × 10^9 km (3.556×10^9 mi.) 2,326 km (1,445 mi.) 2.59 m (8 ft.6in.)
Eris Aphelion 14.602 × 10^9 km (9.0733×10^9 mi.) 2,326 km (1,445 mi.) 1.01 m (3 ft.3.8in.)
Betelguse 643 ± 146 light-years
max. 7.464 × 10^15 km (4.638×10^15 mi.)
950–1,200 solar radii
max. 1.671 × 10^9 km (1.038×10^9 mi.)
1.43 m (4 ft.8.3in.)
R Doradus 178 ± 10 light-years
max. 1.778 × 10^15 km (1.105×10^15 mi.)
370 ± 50 solar radii
max. 0.515 × 10^9 km (320×10^6 mi.)
1.85 m (6 ft.0.8in.)
Panel 5 4942 Munroe 2.2 AU (1.2 AU closest to earth, 179.4 × 10^6 km or 111.5×10^6 mi.) 9–10 km (5.6-6.2 mi.) 35.5 cm (14.0in.)
Proxima Centauri 4.243 light-years
40.142 × 10^12 km (24.943×10^12 mi.)
0.141 solar radii
0.196 × 10^6 km (122×10^3 mi.)
3.11 cm (1.22in.)
Barnard's Star 5.980 light-years
56.574 × 10^12 km (32.668×10^12)
0.196 solar radii
0.272 × 10^6 km (169×10^3)
3.06 cm (1.20in.)
Alpha Centauri B 4.366 light-years
41.305 × 10^12 km (25.666×10^12 mi.)
0.865 solar radii
1.204 × 10^6 km (748×10^3 mi.)
18.6 cm (7.32in.)
Sirius 8.6 light-years
81.362 × 10^12 km (50.556×10^12 mi.)
1.711 solar radii
2.382 × 10^6 km (1.480×10^6 mi.)
18.7 cm (7.36in.)
Alpha Centauri A 4.366 light-years
41.305 × 10^12 km (25.666×10^12 mi.)
1.227 solar radii
1.708 × 10^6 km (1.061×10^6 mi.)
26.3 cm (10.4in.)
Panel 6 HD 189733 b 63 light-years
596.024 × 10^12 km (370.352×10^12 mi.)
1.138 Jupiter radii
159,117 km (98,870.7 mi.)
1.7 mm (67 mils)
Gliese 581 g 20.3 light-years
192.052 × 10^12 km (119.336×10^12 mi.)
1.3 to 2.0 Earth radii
max. 25,484 km (15,835 mi.)
0.85 mm (33 mils)
Gliese 581 (Parent star) 20.3 light-years
192.052 × 10^12 km (119.336×10^12 mi.)
0.29 solar radii
201,695 km (125,411 mi.)
1.34 cm (528 mils)
Black Hole at the center of our Galaxy 25,900 light-years
245.032 × 10^15 km (152,260×10^15 mi.)
Mass 4.31 × 10^6:
12.684 × 10^6 km (7.8815×10^6 mi.)
0.33 mm (13 mils)
Gliese 667 Cc 22.7 light-years
214.757 × 10^12 km (133.444×10^12 mi.)
2.0 Earth radii
25,484 km (15,835 mi.)
0.76 mm (30 mils)
Gliese 667 (Parent star) 22.7 light-years
214.757 × 10^12 km (133.444×10^12 mi.)
0.42 solar radii
584,927 km (363,457 mi.)
1.74 cm (685 mils)
HD 20794 c (Parent star) 20 light-years
189.214 × 10^12 km (117.572×10^12)
0.92 solar radii
1,281,269 km (796,143.6 mi.)
4.31 cm (1.70in.)
Tau Ceti C (Parent star) 11.905 light-years
112.629 × 10^12 km (69.9844×10^12 mi.)
0.793 solar radii
1,104,398 km (686,241.1 mi.)
6.25 cm (2.46in.)
Koi-1686.01 (Parent star) 1033.8 light-years
9.780 × 10^15 km (6.077×10^15 mi.)
0.52 solar radii
724,195 km (449,994 mi.)
0.47 mm (19 mils)
Panel 7 Voyager probes (Voyager 1 at 126.10 AU) 18.86×10^9 km (11.72×10^9 mi.) 20 meters (66 ft.) (with antennas) 6.76 µm (266 millionths of an in.)

[edit] Transcript

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".]

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What is the meaning of "football field" in panel #2? --Kevang (talk) 04:50, 11 October 2013 (UTC)

I was wondering the same thing. Probably misplaced text. Irino. (talk) 05:49, 11 October 2013 (UTC)
It does seem to be misplaced, but if that's the only glitch, this is the only panel without a unique reference object. "20 football pitches long" isn't all that easy to grasp. jameslucas (" " / +) 09:09, 11 October 2013 (UTC)
The image is fixed by Randall. I did an update here.--Dgbrt (talk) 11:28, 11 October 2013 (UTC)
Well, that's a letdown. I'm surprised Randall didn't use Heathrow. jameslucas (" " / +) 13:42, 11 October 2013 (UTC)

I haven't done any lookups or maths to check these, but give the size of these as "stars" in the sky, everything from panel 2 onwards seems to me to be an order of magnitude or two too large. Mark Hurd (talk) 05:17, 11 October 2013 (UTC)

Not really. You see the stars and planets as points because their angular size is lower than your eyes' resolution. They have measurable (or, in case of really distant or small objects, computable) angular sizes. For stars etc. these angular sizes are really small - but Earth is quite big, so if you cut a portion of a sphere the radius of Earth corresponding to these small solid angles, you get sizable areas. I haven't checked Randall's math, but I'd rather believe his results. If it is non-intuitive for you consider the Sun and Moon example - when observed by naked eye, the Moon looks for you as being the size of a dime held up in your hand - and yet it's shadow during an eclipse covers quite an area of Earth's surface. It is true that sizes of some of these "footprints" are quite surprising compared to other ones. 08:55, 11 October 2013 (UTC)
Definitely surprising. I'll put faith in Randall doing his math correctly, but still needed to check on a couple of these because they did elicit a "What? No. Really? Can't be." reaction. Using the formula described in the Explanation above, for Venus I get 12742 km (Earth radius) * 12104 km (Venus diameter) / 38000000 (shortest distance to Venus) = 2.03 km.
Hard to picture that something that is such a small dot in the sky is actually directly over such a large patch of ground. But there you are. 17:11, 11 October 2013 (UTC)

The question that sprung to my mind was, which distance is he using for the planets and asteroids, since those vary hugely depending on where objects are relative to each other along their orbits. Is he going with closest approach, maybe? Or the distance that we happen to be at just this instant? --Rmharman (talk) 21:42, 11 October 2013 (UTC)
I just checked for Deimos, and got to ~50 mio km, so that´d be the closest approach. --Wilberforce (talk) 13:24, 13 October 2013 (UTC)

Does someone know how to use LaTeX formulas? And if so, can they translate my formula into something more pleasing to the eye? Irino. (talk) 05:49, 11 October 2013 (UTC)

According to the wikipedia page, the M25 is 117 miles long. That sounds more like "37 miles across" to me. Kaa-ching (talk) 08:46, 11 October 2013 (UTC)

it originally stated 15 miles, someone has fixed it now. Thanks! Kaa-ching (talk) 11:35, 11 October 2013 (UTC)

Neither the sun or moon, nor Messier 25 (declination -19°) can ever culminate in the zenith over London. :-( Admittedly, Townsville, Australia would be sort of overwhelmed by M25. -- 11:27, 11 October 2013 (UTC)

M25 is a reference to the highway that surrounds London, not the Messier object, which is probably nowhere near the angular size of the moon. 15:17, 11 October 2013 (UTC)
Why would Randall choose London, if it wasn´t for the obvious disambiguity of the name M25? ----Wilberforce (talk) 12:33, 13 October 2013 (UTC)

Does anyone know why the exoplanet "HD 189733 b" is labled as "Permadeath" ? Same question for the other weird names in the same pannel (the "tilde on keyboard" one) ? Jahvascriptmaniac (talk) 11:32, 11 October 2013 (UTC)

A reference to Exoplanet Names. Squornshellous Beta (talk) 12:08, 11 October 2013 (UTC)

If you were looking from the center of the earth, as the situation suggests, wouldn't the M25 be reversed, east-to-west, as you look at the sun and the moon?-- 16:09, 11 October 2013 (UTC)

I visualize it as looking down on Earth, with the "shadow" of the celestial object on top of the M25/soccer field/laptop/etc. 17:02, 11 October 2013 (UTC)

Has anyone noticed that Voyager 1 and 2 look like viruses? Kind of funny considering they're next to E. Coli... StelarCF (talk) 17:54, 11 October 2013 (UTC)

Yeah, I thought that too. It's a happy thought. Why, you ask? Well, with vastly diminished (or - in the course of time - zero) output from RTG power sources, they're like weakened (or inactivated) viruses - that we've sent out to the rest of the Universe, to any other intelligent lifeforms that may find them. What does that remind you of?
VACCINES! We've vaccinated aliens to the human condition :D To the Earthly condition even.
[PS - that's a happy thought because I choose to interpret from a cross-contamination standpoint. Which, in this case, allows them to observe us in our own locale, and establish our intrinsic nature - before a two-way interaction with us, in 'shared space' :P, and observing us through the medium of those interactions.] 18:46, 11 October 2013 (UTC)

Does "Explain XKCD" installation of MediaWiki has MathJax [plugin] installed for writing mathematics formulas? --JakubNarebski (talk) 08:15, 12 October 2013 (UTC)

Too bad he didn't do the Pleiades. I mean, instead of using the Vatican, he could have used something geeky: Bletchley Park or something (though that's probably not big enough). Homunq (talk) 14:11, 12 October 2013 (UTC)

Panel 6 (extrasolar planets)

My table doesn't really match the image. An earth sized Planet would be at some micrometer, their hosting stars are about some centimeter. Who is wrong? Me or Randall?--Dgbrt (talk) 19:49, 13 October 2013 (UTC)

Is it just me or is the laptop a MacBook Pro? Xyz (talk) 13:55, 14 October 2013 (UTC)

I'm a bit disappointed by the lack of extragalactical objects. Or did I miss something? Starblue (talk) 08:52, 17 October 2013 (UTC)

Moon shadow

If I understand correctly, the comic show the size of objects at the Earth surface. So, if the shadow of the Moon is projected on London, it will cover approximately all the space inside the M25 motorway ?
If so, why it is said that a total solar eclipse will normally cover a band of about 250 km wide (and not 60 km wide) on Earth ? 09:38, 21 October 2013 (UTC)

The projection is to the center of the earth, not w.r.t. the sun like the shadow of a solar eclipse. Starblue (talk) 09:56, 21 October 2013 (UTC)
Look at the first picture at this comic and compare it this to this one (left): Solar_eclipse_types.svg. You are just behind the moon at the surface of the earth, and when the moon is not close enough a total eclipse will not happen (right). All distances and also angular sizes belong to the surface but not the center of the Earth.--Dgbrt (talk) 18:54, 21 October 2013 (UTC)
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