地球科学概论 第4讲 地球形状及自转

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2 Sphere
l Is the Earth a sphere? l No ! l Then, which kind of figure? l Ellipsoid [first approxiamtion!]
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3 Ellipsoid
l The earth is rotational ellipsoid (flattened slightly at the poles and bulges somewhat at the equator)由于地球自转，地球必定是两极 扁平、赤道隆起的（旋转）椭球。 l Theoretical inference that the Earth is an ellipsoid（牛顿） l olive ellipsoid（卡西尼） l 地球自转证明（后面讲）
10
2 Sphere
l Eratosthenes’s method:
Cited from Baki iz 2010
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练习:
2 Sphere
l 进一步思考:
p Notice that the sun source is a distance from the Earth (not parallel light ray), calculate the radius of the Earth p Based on the Moon light ray, please precisely measure the spherical Earth’s radius p Suppose the Earth is a rotational ellipsoid, how to measure the semi-major and minor axes, based on the sun ray ?【练习】
3 Ellipsoid
l Relationship between Meridian arc length and geocentric latitude
p 随着地心纬度的增加而减小
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3 Ellipsoid
l Relationship between Meridian arc length and geodetic latitude【地理纬度关系如何？】
x = r cos φ y = r sin φ
r= a 1− e
2 2 2
x y + = 1 2 2 a b
2
2
1 − e cos φ
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e = a −b / a
2
2
3 Ellipsoid
l Relationship between Meridian arc length and geocentric latitude
4
1 History
l China: orbicular sky and rectangular earth （天圆地方）
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1 History
l Han:张衡(78-139 A.D, Astro, Math, Inventor, Geogra, Map, Liter)《浑天仪注》“天如鸡卵 ，地如卵黄”
15
3 Ellipsoid
l The problems lies in the two kind of latitude:
p geocentric latitude (地心纬度) p geodetic latitude (大地纬度)
φ
geocentric latitude geodetic latitude reduced latitude
9
2 Sphere
l Eratosthenes’s method:
p On summer solstice, he observed the midday sun shone to the bottom of a well in the town of Syene p At the same time, he observed the sun was not directly overhead at Alexandria p It casts a shadow with the vertical equal to 1/50th of a circle (7° 12') p The distance from Alexandria to Syene is 5000 stadia (1 stadium = 157.5 m) p The radius of the Earth is 6267km (Error only 2%)
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1 History
l 思考:
p What does the Earth look like for the ancient people? p If you live in 500B.C, could you find any clues indicating that the earth is spherical? p Can you prove your viewpoint？
p The negative value of co-tangent φ can be expressed by the two coordinates of corresponding point on ellipsoid:
dy ! = tan(90 + ϕ ) = − cot ϕ dx
第 4 讲
地球形状及自转
申文斌，邓洪涛，徐新禹，罗佳 2014.5.4
1
内容提要
l 1 历史 l 2 球形 l 3 椭球形 l 4 大地水准面 l 5 地球自转 l 6 地球自转参数观测 l 7 地球自转应用举例
7
2 Sphere
l 思考:
p Assume the earth is spherical, how to calculate it’s radius?
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2 Sphere
l Euclidean geometry (Euclid, 325-265 BC, Greek Math)
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2 2
did
l Relationship between Meridian arc length and geocentric latitude
p Then
a 1 − e 2 [1 − 2e 2 cos 2 φ + e 4 cos 2 φ ]1/2 dl = dφ 2 2 3/2 (1 − e cos φ )
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3 Ellipsoid
l Arguments in history
p Is the earth oblate spheroid or olive ellipsoid？ p Newton: according to mechanics, the earth should be oblate spheroid. Thus the 1° meridian arc length should increases with the increase of the latitude p Cassini: the earth is olive ellipsoid, supported by measurement results —— the 1° meridian arc length decreases with the increase of the latitude p Who made mistakes? What mistakes?
3
1 History
l Greek
p Plato (427-347BC): Earth’s circumference is 400,000 stadia (by speculation)（62,800 km - 74,000km), and Archimedes (287-212BC): 30,000 stadia [Euclid: 325-265BC] p Eratosthenes (276-194BC, Math, Georagr, Histo, Peot, Astro): earth’s radius is 6267km[39,377stadia] (by calculation, 240B.C)
2
1 History
l Greek(Homer, Thales, Pythagoras, Plato, Eratosthenes et al.)
p Homer (800-1000 BC ? poet): the earth resembling a flat disc p Thales (624-546BC,Think, Sci, Phi): the earth is spherical (600B.C) p Pythagoras (572-497 BC, Math, Phi) and Aristotels (384-322BC, Phi, Sci, Educ): sphere
p Then we have
x=
y= a 1 − e 2 cos φ 1 − e 2 cos 2 φ a 1 − e 2 sin φ 1 − e 2 cos 2 φ
p according to differential equation
dx a 1 − e 2 sin φ =− dφ (1 − e 2 cos 2 φ )3/2 dy a 1 − e (1 − e )cos φ = dφ (1 − e 2 cos 2 φ )3/2
dl a 1 − e [1 − 2e cos φ + e cos φ ] f (φ ) = = 2 2 3/2 dφ (1 − e cos φ )
2 2 2 4 2 1/2
p substitute parameter