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 I See No More Ships (Posted on 2006-01-29)
For an observer at height h above the surface of the Earth,

i) What area, A, of the Earth's surface is visible?
ii) At what altitude, h, does the curvature of the Earth become apparent?

You may assume one can detect, with the human eye, an angle of one degree between the two ends of a line (i.e. two tangents at either end of the visible horizon, appear to intersect at an angle of one degree), that the average human field of view is 180°, also that the Earth is a sphere of radius 6378 km, or you may provide your own figures for the calculations.

 No Solution Yet Submitted by Sir Percivale Rating: 4.5000 (2 votes)

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The first task in solving for A is to find out how far the horizon is, in angular measure.  The observer is r+h distant from the center of the earth, and his line of sight to the horizon is perpendicular to a radius of the earth, r, to that point on the horizon, thus forming a right triangle with one leg r and the hypotenuse r+h.

The angle of this triangle at the center of the earth is therefore arccos(r/(r+h)).  The web site http://www.rism.com/Trig/spherical_cap.htm (from Google search on Area of spherical cap), calls this angle alpha and gives a formula

A = 2 pi (1 - cos(alpha)) rho^2

for the area of the spherical cap, where rho is the radius of the sphere.  This makes the formula for the visible area

A = 2 pi (1 - r/(r+h)) rho^2

where rho = 6378 km and h must be expressed in km, giving an answer in km^2.

A table:

`  h      h        area       area  area/area of earth (km)   (mi)      km^2       mi^2   (fraction)  100    62.14   3945554   1523387 0.007718432  200   124.27   7771145   3000456 0.015202189  300   186.41  11482164   4433288 0.022461815  400   248.55  15083681   5823842 0.029507229  500   310.69  18580472   7173960 0.036347776  600   372.82  21977039   8485382 0.042992261  700   434.96  25277632   9759748 0.049448997  800   497.10  28486260  10998606 0.055725829  900   559.23  31606715  12203421 0.061830173 1000   621.37  34642581  13375575 0.067769043 2000  1242.74  61015270  23558127 0.119360229 3000  1864.11  81763585  31569097 0.159948816 4000  2485.48  98513381  38036229 0.192715359 5000  3106.86 112318934  43366583 0.219722271 6000  3728.23 123893828  47835674 0.242365487 7000  4349.60 133738284  51636640 0.261623561 8000  4970.97 142213363  54908886 0.278202810 9000  5592.34 149586207  57755558 0.29262582910000  6213.71 156058717  60254607 0.30528758111000  6835.08 161786318  62466046 0.31649211612000  7456.45 166890608  64436824 0.32647731013000  8077.83 171468086  66204198 0.33543193314000  8699.20 175596306  67798113 0.34350770415000  9320.57 179338315  69242910 0.35082795416000  9941.94 182745887  70558581 0.35749396717000 10563.31 185861940  71761696 0.36358970018000 11184.68 188722348  72866106 0.36918533119000 11806.05 191357332  73883479 0.37433998020000 12427.42 193792529  74823714 0.37910379921000 13048.80 196049832  75695263 0.38351961422000 13670.17 198148046  76505388 0.38762421623000 14291.54 200103418  77260362 0.39144938424000 14912.91 201930054  77965630 0.39502271425000 15534.28 203640262  78625945 0.39836828426000 16155.65 205244831  79245472 0.40150719627000 16777.02 206753253  79827877 0.40445802628000 17398.39 208173921  80376400 0.40723718729000 18019.76 209514275  80893914 0.40985923530000 18641.14 210780939  81382975 0.41233712731000 19262.51 211979826  81845869 0.41468243432000 19883.88 213116236  82284639 0.41690551933000 20505.25 214194928  82701124 0.41901569434000 21126.62 215220191  83096980 0.42102134835000 21747.99 216195897  83473702 0.42293005936000 22369.36 217125555  83832646 0.42474869037000 22990.73 218012351  84175039 0.42648347138000 23612.11 218859181  84502002 0.42814006939000 24233.48 219668687  84814554 0.42972365540000 24854.85 220443285  85113628 0.43123895041000 25476.22 221185183  85400077 0.43269027842000 26097.59 221896411  85674683 0.43408160743000 26718.96 222578832  85938167 0.43541658244000 27340.33 223234160  86191191 0.43669855945000 27961.70 223863978  86434365 0.437930632`

The criteria for part 2 indicate that the earth, from horizon to opposite point on horizon, should subtend, at the viewer's eye, 179 degrees or less in order for the curvature of the earth to register on that observer as apparently curved. (180 degrees by any measure would not register as curved as that great circle is the equivalent of a stright line, since there'd be no way of assigning it as being curved in one direction or the other.)

In that right triangle mentioned above, therefore, the angle at the observer would be 179/2 degrees and h would be r / sin(179/2 degrees) - r. This comes out to  .249 km, or .1509 miles or 797 feet.  This leads me to think that the threshhold is overly optimistic in its estimation of human perception.  797 feet is lower than the 86th floor observation deck on the Empire State Building.  From there, I can't honestly say I could notice that the earth was a sphere or that the horizon was curved.

If instead of the earth's subtending 179 degrees as being sufficiently small enough to see the curvature of the horizon, other figures are used, the following table shows h, in km, miles and feet:

`subtend h(km)   h(miles)  h(feet)179     0.24     0.15      797178.9   0.29     0.18      964178.8   0.35     0.22     1147178.7   0.41     0.26     1347178.6   0.48     0.30     1562178.5   0.55     0.34     1793178.4   0.62     0.39     2040178.3   0.70     0.44     2303178.2   0.79     0.49     2582178.1   0.88     0.54     2877178.0   0.97     0.60     3187177.9   1.07     0.67     3514177.8   1.18     0.73     3857177.7   1.28     0.80     4216177.6   1.40     0.87     4590177.5   1.52     0.94     4981177.4   1.64     1.02     5387177.3   1.77     1.10     5810177.2   1.90     1.18     6248177.1   2.04     1.27     6703177     2.19     1.36     7173175     6.08     3.78    19935173    11.92     7.41    39103171    19.72    12.25    64705169    29.50    18.33    96781167    41.26    25.64   135380165    55.04    34.20   180563163    70.83    44.01   232398161    88.69    55.11   290967159   108.62    67.49   356362157   130.66    81.19   428685155   154.85    96.22   508054153   181.23   112.61   594594151   209.84   130.39   688446149   240.72   149.58   789763147   273.93   170.21   898714145   309.52   192.33  1015481143   347.55   215.96  1140262141   388.09   241.15  1273271139   431.21   267.94  1414739137   476.99   296.39  1564917135   525.50   326.53  1724073133   576.83   358.43  1892498131   631.09   392.14  2070504129   688.37   427.73  2258426127   748.78   465.27  2456626125   812.44   504.83  2665492123   879.48   546.49  2885442121   950.04   590.33  3116924119  1024.26   636.44  3360421117  1102.30   684.93  3616454115  1184.32   735.91  3885580113  1270.53   789.47  4168403111  1361.11   845.75  4465571109  1456.27   904.88  4777786107  1556.25   967.01  5105802105  1661.29  1032.28  5450437103  1771.67  1100.87  5812573101  1887.68  1172.95  6193168 99  2009.62  1248.72  6593257 97  2137.86  1328.40  7013967 95  2272.75  1412.22  7456520 93  2414.70  1500.43  7922248 91  2564.16  1593.30  8412600 89  2721.61  1691.13  8929162 (Alpha is 180 degrees minus half the subtending arc.) `

Note that the use of 180 degrees makes the calculation of the angle of the two intersecting great circle arcs, tangent to the ends of the visible portion of the horizon, easier to calculate than if some other separation were used.  That would involve some spherical trig; the 180-degree case involves only comparing arc lengths to the ends of the 180-degree long line (great circle).

Edited on January 29, 2006, 3:21 pm
 Posted by Charlie on 2006-01-29 15:18:13

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