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I See No More Ships (Posted on 2006-01-29) Difficulty: 4 of 5
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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Solution discussion | Comment 1 of 5

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 0.292625829
10000  6213.71 156058717  60254607 0.305287581
11000  6835.08 161786318  62466046 0.316492116
12000  7456.45 166890608  64436824 0.326477310
13000  8077.83 171468086  66204198 0.335431933
14000  8699.20 175596306  67798113 0.343507704
15000  9320.57 179338315  69242910 0.350827954
16000  9941.94 182745887  70558581 0.357493967
17000 10563.31 185861940  71761696 0.363589700
18000 11184.68 188722348  72866106 0.369185331
19000 11806.05 191357332  73883479 0.374339980
20000 12427.42 193792529  74823714 0.379103799
21000 13048.80 196049832  75695263 0.383519614
22000 13670.17 198148046  76505388 0.387624216
23000 14291.54 200103418  77260362 0.391449384
24000 14912.91 201930054  77965630 0.395022714
25000 15534.28 203640262  78625945 0.398368284
26000 16155.65 205244831  79245472 0.401507196
27000 16777.02 206753253  79827877 0.404458026
28000 17398.39 208173921  80376400 0.407237187
29000 18019.76 209514275  80893914 0.409859235
30000 18641.14 210780939  81382975 0.412337127
31000 19262.51 211979826  81845869 0.414682434
32000 19883.88 213116236  82284639 0.416905519
33000 20505.25 214194928  82701124 0.419015694
34000 21126.62 215220191  83096980 0.421021348
35000 21747.99 216195897  83473702 0.422930059
36000 22369.36 217125555  83832646 0.424748690
37000 22990.73 218012351  84175039 0.426483471
38000 23612.11 218859181  84502002 0.428140069
39000 24233.48 219668687  84814554 0.429723655
40000 24854.85 220443285  85113628 0.431238950
41000 25476.22 221185183  85400077 0.432690278
42000 26097.59 221896411  85674683 0.434081607
43000 26718.96 222578832  85938167 0.435416582
44000 27340.33 223234160  86191191 0.436698559
45000 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      797
178.9   0.29     0.18      964
178.8   0.35     0.22     1147
178.7   0.41     0.26     1347
178.6   0.48     0.30     1562
178.5   0.55     0.34     1793
178.4   0.62     0.39     2040
178.3   0.70     0.44     2303
178.2   0.79     0.49     2582
178.1   0.88     0.54     2877
178.0   0.97     0.60     3187
177.9   1.07     0.67     3514
177.8   1.18     0.73     3857
177.7   1.28     0.80     4216
177.6   1.40     0.87     4590
177.5   1.52     0.94     4981
177.4   1.64     1.02     5387
177.3   1.77     1.10     5810
177.2   1.90     1.18     6248
177.1   2.04     1.27     6703
177     2.19     1.36     7173
175     6.08     3.78    19935
173    11.92     7.41    39103
171    19.72    12.25    64705
169    29.50    18.33    96781
167    41.26    25.64   135380
165    55.04    34.20   180563
163    70.83    44.01   232398
161    88.69    55.11   290967
159   108.62    67.49   356362
157   130.66    81.19   428685
155   154.85    96.22   508054
153   181.23   112.61   594594
151   209.84   130.39   688446
149   240.72   149.58   789763
147   273.93   170.21   898714
145   309.52   192.33  1015481
143   347.55   215.96  1140262
141   388.09   241.15  1273271
139   431.21   267.94  1414739
137   476.99   296.39  1564917
135   525.50   326.53  1724073
133   576.83   358.43  1892498
131   631.09   392.14  2070504
129   688.37   427.73  2258426
127   748.78   465.27  2456626
125   812.44   504.83  2665492
123   879.48   546.49  2885442
121   950.04   590.33  3116924
119  1024.26   636.44  3360421
117  1102.30   684.93  3616454
115  1184.32   735.91  3885580
113  1270.53   789.47  4168403
111  1361.11   845.75  4465571
109  1456.27   904.88  4777786
107  1556.25   967.01  5105802
105  1661.29  1032.28  5450437
103  1771.67  1100.87  5812573
101  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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