From which we easily find, e= ✓599 = 0.08158. 300 75. The Geocentric Zenith of a place is the point in which a straight line from the earth's centre passing through the place, meets the celestial sphere. The Geocentric Latitude of a place, sometimes called the reduced latitude, is the arc of the meridian intercepted between the equator and the geocentric zenith of the place. The difference between the latitude and the geocentric latitude is called the reduction of latitude. 76. The tangent of the latitude of a place is to the tangent of the geocentric latitude as the square of the equatorial radius of the earth is to the square of the polar radius. Let Z, Fig. 4, be the zenith of the place A, and z the geocentric zenith. Then ZGQ is the latitude of A, and 2CQ is its geocentric latitude. Let AD be drawn perpendicular to eq. Put = ZGQ ZGQ the latitude, and ' = zCQ = the geocentric latitude. = = Then in the right angled triangle ACD, we have AD = CD tang ', and in the right angled triangle AGD, we have AD GD tang 4. Hence CD tang ?' GD tang ; or, CD: GD: tang tang '. But by conic sections, CD : GD: qC2 pC2. Consequently tang tang': : qC2 : PC'. Cor. If qC1, and e = earth's eccentricity, we have (74), pC' —1—e'. Hence, tang : tang ' :: 1 : 1 —e2 ; or, tang '(1-e2) tang ? (A) CHAPTER V. ASTRONOMICAL REFRACTION. 77. Earth's atmosphere. From the science of pneumatics we learn that the earth's atmosphere is an elastic medium, the density of which continually decreases as the distance from the general surface of the earth increases. The law of decrease is such that as the height increases in arithmetical progression, the density decreases nearly in geometrical progression. The actual decrease is such that at the height of 3 miles, the density is only about one half as great as at the earth's surface; at the height of 7 miles, about one fourth as great; at the height of 10 miles, about one eighth as great; and thus on. whole height of the atmosphere is not known. But from the preceding law, it follows that, at the height of 40 or 50 miles its density must be extremely small, so as to be nearly or quite insensible. The density at any given place varies with the pressure, as indicated by a barometer, and with the temperature, as indicated by a thermometer; but this variation is not great. The If the density of the whole mass of the atmosphere was uniform throughout, and the same that it is at the earth's surface when the barometer stands at 30 inches, and Fahranheit's thermometer at 50°, which is regarded as being nearly the mean density at the earth, the height would then be 5.13 miles. 78. Refraction. The science of optics teaches us, that when a ray of light passes obliquely from one medium into another, of different density, it becomes bent, or refracted; the ray in the second medium, called the refracted ray, taking a different direction from that in the first, which is called the incident ray. Both rays lie in the same plane with a perpendicular to the common surface of the two mediums at the point of passage from one to the other. When the ray passes from a rarer to a denser medium, the refracted ray is bent towards the perpendicular to the common surface, making with it a less angle than that made with it by the incident ray. Thus an incident ray SA, Fig. 5, entering obliquely a second medium of greater density at A, takes a direction AB, making the angle BAD, which is called the angle of refraction, less than the angle SAE, which is called the angle of incidence. The angle BAC, which expresses the difference between the directions SA and AB, of the incident and refracted rays, is called the refraction. For the same two mediums, the amount of refraction changes with a change in the angle of incidence. The law of this change is such that the sine of the angle of incidence is to the sine of the angle of refraction in a constant ratio. Thus if I be the angle of incidence, R the angle of refraction, and m a constant quantity, the value of which for different mediums is determined by experiment, we have sin I: sin R::m: 1; or, sin I = m sin R. For the passage of a ray of light from a vacuum into air of a mean density, or that which it has when the barometer stands at 30 inches and the thermometer at 50°, the value of m is 1.000284. When a ray passes through a medium composed of strata of different densities, bounded by parallel planes, the whole refraction is the same, as if the incident ray had at once entered the last stratum with its first angle of incidence; the direction of the ray in the last stratum being the same in either case. Thus, if a ray SA, Fig. 6, in passing through such a medium, takes the directions AB, BC, a ray S'A' entering the last stratum at the same angle of incidence with SA, will take a direction A'C', parallel to BC. When the strata are indefinitely thin and their number indefinitely great, or which amounts to the same, when the density continually varies from A to C, the broken line ABC becomes a curve. The whole refraction is however still the same, provided the density at the surface C remains unchanged; that is, the whole refraction for a given angle of incidence, depends entirely on the density at the second surface. 79. Astronomical Refraction. As the density of the earth's atmosphere continually increases from its upper surface to the earth (77), it follows from the last article that when a ray of light from any of the heavenly bodies, enters the atmosphere obliquely, it becomes bent into a curve, concave towards the earth. The density in the upper parts of the atmosphere being very small, the curve at first deviates very little from a straight line, but the deviation becomes greater as it approaches the earth. Both the straight and curved parts of the ray must necessarily lie in the same vertical plane; for as the corresponding parts of the atmosphere on each side of a vertical plane may be regarded as of equal density, there is no cause for a deviation to either side. The whole change produced in the direction of the ray in traversing the atmosphere is called the astronomical refraction. 80. Astronomical refraction increases the altitude of a heavenly body, but does not affect the azimuth. Let SaA, Fig. 7, be a ray which proceeding from a body S, enters the atmosphere at a, and being bent by refraction, meets the earth's surface at A; and let AS' be a tangent to the curve Aa at A. Then will the ray enter the eye of an observer at A, in the direction S'A, and consequently the body S will appear to be in the more elevated position S'. As the tangent AS' must be in the same vertical plane with the ray AaS, the azimuth of the body is not affected by refraction. It follows that the altitude of a heavenly body is obtained by subtracting the refraction from the observed altitude, and the zenith distance, by adding the refraction to the observed zenith distance. 81. At the zenith, the refraction is nothing. For in consequence of the corresponding density of the atmosphere on every side of a vertical line, there is no cause for a ray entering it in that direction, to deviate from its rectilineal course. 82. To obtain approximate formula for computing the refraction due to any altitude or zenith distance. As the upper and under surface of that portion of the atmosphere, through which a ray from any of the heavenly bodies passes in its course to a place on the earth's surface, do not differ much from parallel planes, we may obtain approximate formulæ for the refraction, by assuming the density to be uniform throughout, and the same that it is at the earth's surface (78). Let bd, Fig. 8, be a part of the boundary of the atmosphere on this supposition, Sa a ray from a body S, which being refracted at a, meets the earth'ssurface at A; and let C be the centre of the earth, and Z the zenith of the place A. Then to an observer at A, the body will appear in the direction AS', and the angle SaS' will be the refraction corresponding to the apparent zenith distance ZAS'. Put, = CA = radius of the earth, assumed to be a sphere, h = ea = height of a uniform atmosphere, or, (App.* 13), sin R cos r + cos R sin r = m sin R, or, dividing by cos R, we have, tang R cos r + sin r = m tang R But since m differs but little from a unit (78), it is evident from equat. (A), that R + r must differ but little from R, and consequently r must be a small angle. Taking Appendix to part 1. |