Navigation and Time on the Sphere, Etc.

A taut string between two points in the plane gives the shortest path between those points, and that path is a straight line.

Navigation on the surface of a sphere is different from navigation in the plane. A taut string on the surface of sphere is curved -- it is not straight line. But a short taut string gives the shortest path between two nearby points in either case.

• Rule Assumption 1: Extension of the taut string results in a great circle through both points.
• Rule Assumption 2: Following a great circle path in one direction or another provides the shortest taut string path between any two points on the surface of the sphere.
• Rule Assumption 3: Specifying a point and direction through it determines a great circle. (Directions can be given with respect to the great semicircles (lines of longitude) passing through a point, but starting at the North end at the South Pole. Observe the angle between the lines or great semicircles or lines of longitude and another great circle changes as one follows the latter. The great circle between the North Pole and Greenwich England gives the line of zero magnitude. (Altitude is given by circles parallel to the equator).
  

Airline routes around the globe try to follow great circles -- the shortest distance between two points on the globe. As an exercise, locate the great circle routs between the capitals of various countries with the help of a taut string held against a globe.

Spherical Triangles and the Sum of their Angles. Three nearby points not on a great circle can be used to form a spherical triangle by joining each them, pairwise, by taut strings held against the sphere or globe. Now measure and add together the sum of the (interior) angles by sides of the spherical triangle. The sum is greater than 180 degrees. But if you make the triangle smaller, the sum of the angles will approach 180 degrees.

Determining Line of Longitude

. An old-fashion (relatively low tech) way of determining your line of longitude is to know what time it is in Greenwich, England, the international reference point, when it is noon at your present location according to a sunclock -- the sun is highest at noon. For instance if you are in North or South Atlantic ocean, three hours behind of Greenwich time, then difference in longitude then you are (3/24) x 360 degrees = 45 degrees west of Greenwich -- the 0 degree line of longitude.
Ship navigators in principle can determine their longitude if they know Greenwich (solar) time and can observe locally when the sun is highest in the sky. The British Admiralty offered a prize for a mechanical clock, a chronometer, which could travel with a ship but keep Greenwich time. The prize was offered and collected. The invention of a ship chronometer aided in sea and ocean navigation and map (sea chart) creation. Questions: When was the prize offered, who collected it and when?

Altitude Determination

Using the North Star

The North-South axis of the earth's revolution is aligned with the North Star (Polaris).

    

               

             .       .    rays from North Star (Polaris)

             .       .    are // to earth's axis of revolution

             .       .   

             .       . 

             .       .   /   Ray OA is perpendicular

             North   .e /    to earth surface at A.

            + +      . /  f

             |  .____./_________________________________ 

             |     . /|                         

             |     A/ |

             |     /   .           

             |    /     .                            

             |   /       .

             |c /         .

             | /  b       .

           O +------------------------------------------------.

             |           Equator

             |

             axis of planetary

             revolution

  

Ray OA goes from the center of the earth to your location A. The ray OA is perpendicular to the earth's surface at A. It points in the upward direction. Focusing a telescope on the North Star gives an angle d between the vertical and the direction of the North Star.

Now angle d+f=90 degrees. Moreover, angles f and b are equal. Therefore d+b=90 degrees. measurement of d gives the altitude b = 90 degrees - e and the polar angle c = d

Using the Sun -- Approach 1 (correction required)

        North

            + + .  

             |      .

             |         ._________________________________________ 

             |           .a/                             To Sun:

             |           /  .

             |         /     .                         = Rays from Sun

             |       /        .                            

             |     /           .

             |   /             .

             | /  b            .

             +------------------------------------------------.

             |           Equatorial plane

             |

             axis of planetary

             revolution

  

This diagram falsely assumes a planet orbits in an plane about a distance sun and that planet also rotates on a North-South axis perpendicular to that plane. In this situation, the shadow angle a that the sun's rays make with a vertical pole at the surface at noon equals the angular of altitude b.

Using the Sun -- Corrected Approach

In the case of the earth and the sun, the North-South axis of rotation of the earth makes an angle q with the orbital plane of the sun. The equatorial plane of the earth is tilted and not in the plane of the earth's orbit around the sun. (By observation, all the sun's planet except for one, orbit the sun in a single plane.)

one can measure the shadow angle a at noon (on a cloud-free day) and then add a correction factor q to obtain the altitude.

    

                                                               

                                                              o

                                                        o

            North                                 o        ray from sun

            Pole                            o    

            + + .                     o

             |      .          o     

             |          o    X                                    o

             |    o  |    .X  A                            o  ray from sun   

             |       a   X  .                       o    in orbital plane

             |        \X     .              o             

             |       X        .      o                                

             |     X  \a       o      \

             |   X     |o       .      angle q == angle of ascension

             | X   o            .      |

             +------------------------------------------------.

             |               Equatorial plane

             |

             |

             |   

             |

             |

             |

             |

           South

           Pole  

  

In the above diagram, at high noon, the sun rays make angle of ascension q with the equatorial plane of the earth. This angle q depends on the time of year (Problem: Find where it is tabulated.) Now the altitude angle b of the vertical pole equals the shadow angle a + the angle of ascension q.

Measurement of Angle of Ascension

the angle if the altitude b is known, for instance, from measurement with respect to the North Pole, the tilt q of the earth equatorial plane from the plane of the earth's orbit (the rays of the sun) can be computed from a+ q = b or q = b - a.

The ascension angle decreases from 23+(26/60) degrees at the summer solstice (June 22) to -[23+(26/60)] degrees at the winter solstice (December 22) and then increases from -[23+(26/60)] degrees at the winter solstice (December 22) to [23+(26/60)] degrees at the summer solstice (December 22). On June 22 and December 22 the axis of revolution of the earth, and rays from the sun lie in plane perpendicular to the orbital plane of the earth.

The tropics of cancer and Capricorn are meridian circles at altitudes 23+(26/60) degrees above or below the equator. Between these circles, people may see the sun directly overhead once or twice during the year. Outside these circles, the sun is always in the southern or northern portion of the sky, and never directly overheard. The axis of revolution of the earth is tilted 23+(26/60) degrees away from the perpendicular to the orbital plane of the earth and all but one planet around the sun.

Direction of the Earth's Revolution

. Each day the Sun raises in the East and sets in the West. From a fixed point on the earth's surface the sun apparently moves from east to west across the sky. But the same motion would be observed if the Sun was drawn in a fixed position and the earth rotated so that the Sun rays appeared over the eastern horizon in the morning and disappeared over western horizon in the evening.

To illustrate this further, draw a large circle, stand at the center without moving. Now ask a friend to walk around you a few times in one direction, say clockwise. You will see the friend appear out of the corner of your left eye (friend-rise) and then disappear out of the corner of your right eye (friend-set). Next ask the same friend to stay in one position on the circle, but turn around slowly in an anti-clockwise direction. You will see again the friend appear out of the corner of your left eye (friend-rise) and then disappear out of the corner of your right eye (friend-set). The effect of friend-rise and friend-set can thus be seen in two situations. One of these situations requires less motion than the other.

Solar-Based Clocks -- Common Time

The speed at which the hands of a clock travel can be calibrated (set), so that 24 hours by the clock is on average, the time between noon one day and noon the next day. The clocks we use each day are based on solar time.

Star-Based Clocks --- Sidereal Time.

The earth rotates on axis which points at the North Star. During one sidereal, the earth rotates once on its axis. In the North hemisphere the night ski star apparently rotates 360 degrees (one revolution) around the North Star Polaris. Star-based (sidereal) clocks can be calibrated (set) so that 24 hours corresponds to one of these revolutions -- one star-based day.

The earth travel around the sun in 366.2422 revolutions about it axis of revolution == a line through the North Star Polaris. This implies the earth travels (1/366.22) of its orbit every 24 star-based clock hours. Because the sun rays spread out radially, the direction of the sun rays changes by about (360/366) degrees (almost one degree) per day. This affects the star-based time of sunrise and sunset. There is a delay representing the extra star-based time needed for the sun rays to appear or disappear over the horizon. Between each sunrise the earth has to rotate, not 360 degrees, but almost 361 degrees. Rotating that extra degree requires 24 star-based hours divided by 366. (But 24 hours = 24 x 60 minutes and 360 = 6 x 60. So rotating that extra degree requires about 4 star-based minutes. There is a difference, but very small between one star-based and one-solar based minute).

On average, each solar based day is longer than one star-based day by the time needed to for the earth to rotate (360/366) degrees. So there is one fewer solar based days in one year (= one earth revolution around the sun) than there are star-based days.

One solar based day is about 4 minutes longer than a star-based day. The position of the stars in the night sky changes by one degree, every 4 minutes of times. Every 24 solar-based hours, the Northern hemisphere astronomer finds that the night sky appears to rotate nearly (360/366) degrees about the North Star. This explains the apparent movement of the constellation through the night sky.

If the earth rotated in the opposite direction about its axis, there would be 359 degrees between each sunrise, and the solar-based day would be 4 minutes shorter than the star-based day.

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