Spherical Astronomy
Lecture 01
Cristiano Guidorzi
Physics Department
University of Ferrara
Astrophysical Measurements
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Outline
1
Spherical Trigonometry
Basics
Spherical Triangle: angles and sides relations
2
The Earth
Coordinates
Geodetic/geocentric latitude
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Spherical Trigonometry
Basics
Outline
1
Spherical Trigonometry
Basics
Spherical Triangle: angles and sides relations
2
The Earth
Coordinates
Geodetic/geocentric latitude
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Spherical Trigonometry
Basics
Great Circles and Triangles
Small and Great Circles.
Spherical triangle: arcs of great circles
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Spherical Trigonometry
Basics
Spherical angles and sides
Angles & sides
Unique correspondence
a ↔ BC, b ↔ AC, c ↔ AB
Spherical Excess
E = A + B + C − 180◦
it’s not constant
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Spherical Trigonometry
Basics
Spherical Excess and triangle area
Spherical Excess
A (∆) : triangle ∆ area
S(A)/4 π r2 = A/π
A+B+C
π
4 π r2 = 4 π r2 + 4 A (∆)
A (∆) = (A + B + C − π ) r2 = E r2
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Spherical Trigonometry
Spherical Triangle: angles and sides relations
Outline
1
Spherical Trigonometry
Basics
Spherical Triangle: angles and sides relations
2
The Earth
Coordinates
Geodetic/geocentric latitude
Cristiano Guidorzi (Università di Ferrara)
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Spherical Trigonometry
Spherical Triangle: angles and sides relations
Spherical Triangle: angles and sides relations
Rotation around x axis by angle χ
x = cos ψ cos θ
y = sin ψ cos θ
z = sin θ
x′ = cos ψ ′ cos θ ′
y′ = sin ψ ′ cos θ ′
′
z = sin θ ′
x′ = x
y′ = y cos χ + z sin χ
z′ = −y sin χ + z cos χ
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Spherical Trigonometry
Spherical Triangle: angles and sides relations
Spherical Triangle: angles and sides relations
Replacing, we get:
cos ψ ′ cos θ ′ = cos ψ cos θ
sin ψ ′ cos θ ′ = sin ψ cos θ cos χ + sin θ sin χ
sin θ ′ = − sin ψ cos θ sin χ + sin θ cos χ
with spherical angles (P is now C)
ψ = A − 90◦ ,
ψ ′ = 90◦ − B,
Cristiano Guidorzi (Università di Ferrara)
θ = 90◦ − b
θ ′ = 90◦ − a,
χ =c
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Spherical Trigonometry
Spherical Triangle: angles and sides relations
Spherical Triangle: angles and sides relations
Replacing, we get:
cos (90◦ − B)cos (90◦ − a) = cos (A − 90◦ ) cos (90◦ − b)
sin (90◦ − B)cos (90◦ − a) =
sin (A − 90◦ ) cos (90◦ − b) cos c + sin(90◦ − b)sin c
sin (90◦ − a) = − sin (A − 90◦ ) cos (90◦ − b) sin c + sin(90◦ − b) cos c
or
sin B sin a = sin A sin b
cos B sina = − cos A sin b cos c + cosb sin c
cos a = cos A sinb sin c + cosb cos c
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Spherical Trigonometry
Spherical Triangle: angles and sides relations
Sine and Cosine Formulas
Sine Formula
sin b
sin c
sin a
=
=
sin A sin B sin C
Sine-Cosine Formula
sin c cos b = sin b cos c cos A + sina cos B
Cosine Formula
cos a = cos b cos c + sinb sin c cos A
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The Earth
Coordinates
Outline
1
Spherical Trigonometry
Basics
Spherical Triangle: angles and sides relations
2
The Earth
Coordinates
Geodetic/geocentric latitude
Cristiano Guidorzi (Università di Ferrara)
12/04/2010
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The Earth
Coordinates
Coordinates
Coordinates
Longitude: angle between the
meridian and that of Greenwich
Latitude
geodetic (geographical) φ (plumb
line)
geocentric φ ′
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12/04/2010
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The Earth
Geodetic/geocentric latitude
Outline
1
Spherical Trigonometry
Basics
Spherical Triangle: angles and sides relations
2
The Earth
Coordinates
Geodetic/geocentric latitude
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The Earth
Geodetic/geocentric latitude
Geodetic/geocentric latitude
Geodetic Reference System 1980 (GRS-80)
Ellipsoid:
equatorial radius: a = 6, 378 Km
polar radius: b = 6, 357 Km
flattening: f = (a − b)/b = 1/298
x2
a2
y2
+ b2 = 1
tan φ ′ =
y
x
tan φ = − dx
dy =
⇒ tan φ ′ =
b2
a2
a2 y
b2 x
tan φ = (1 − e2 ) tan φ
Max difference ∆φ = φ − φ ′ = 11.5′ (@ 45◦)
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