Difference between revisions of "File:Space is BIG.jpg"

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==Angular size of a pinhead==
==Angular size of a pinhead==
θ_Moon = 0.5°, which is roughly the size of a dime held at arm's length:
θ_Moon = 0.5°, which is roughly the size of a large pea (8 mm or 0.314 in) held at arm's length:


(2) 0.5° = inverse tangent(x / 36 inches)  
(2) 0.5° = arc tangent(D / 36 inches)  


x = 0.314 inches
D = 0.314 inches


By this equation, we get that the angular size of a pin is:
By this equation, we get that the angular size of a pin is:


inverse tangent(0.0590551 inches / 36 inches) = θ<sub>pin</sub>
arc tangent(0.0590551 inches / 36 inches) = θ<sub>pin</sub>
:Where a pin head is 1.5 mm = 0.06 in. wide
:Where a pin head is 1.5 mm = 0.06 in. wide


θ_pin = 0.094° = 338 arcsec
θ_pin = 0.095° = 344 arcsec
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d = D * 206,265 / θ<sub>pin</sub>
d = D * 206,265 / θ<sub>pin</sub>


d = 120 km * 206,265 / 338 arcsec
d = 120 km * 206,265 / 344 arcsec


d = 73,152 km
d = 71,953 km
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θ<sub>Death Star</sub> = 120 km x 206,265 arcsec / 385,000 km
θ<sub>Death Star</sub> = 120 km x 206,265 arcsec / 385,000 km


θ<sub>Death Star</sub> = 64 arcseconds = 1 arcminute
θ<sub>Death Star</sub> = 64 arcseconds ≈ 1 arcminute
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Latest revision as of 10:34, 20 January 2017

Calculations

(1) Small-Angle Formula: D = θ d / 206,265 arcseconds, where the number of arcseconds in a circle divided by 2π = 206,265

D = linear size of an object

θ = angular size of the object, in arcseconds

1° = 60 arcminutes = 3600 arcseconds

d = distance to the object

Angular size of a pinhead

θ_Moon = 0.5°, which is roughly the size of a large pea (8 mm or 0.314 in) held at arm's length:

(2) 0.5° = arc tangent(D / 36 inches)

D = 0.314 inches

By this equation, we get that the angular size of a pin is:

arc tangent(0.0590551 inches / 36 inches) = θpin

Where a pin head is 1.5 mm = 0.06 in. wide

θ_pin = 0.095° = 344 arcsec

Calculate distance for pinhead-sized Death Star

(1) D = (θpin * d) / 206,265 arcsec

d = D * 206,265 / θpin

d = 120 km * 206,265 / 344 arcsec

d = 71,953 km

Calculate size at Moon's distance from Earth

(1) D = (θDeath Star x d) / 206,265 arcsec

θDeath Star = D x 206,265 arcsec / d

θDeath Star = 120 km x 206,265 arcsec / 385,000 km

θDeath Star = 64 arcseconds ≈ 1 arcminute

Calculate distance to appear as large as the Moon

(1) D = (θMoon x d) / 206,265 arcsec

d = 120 km * 206,265 arcsec / 1860 arcsec

d = 13,307 km = 8,264 mi, which is roughly the distance of a direct flight from New York to Hyderabad, India.

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