46 lines
8.6 KiB
Plaintext
46 lines
8.6 KiB
Plaintext
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GLOBULAR CELESTIAL NAVIGATION BY MEANS OF TRANSFORMATION OF GEOCENTRIC PLANAR MEASUREMENTS TO A SPHERICAL COORDINATE SYSTEM
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WHAT IS A TRANSFORMATION
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One common transformation equation is the 2D translation equation, which allows you to shift a point in a Cartesian coordinate system by a certain amount in the x and y directions. The equation is as follows:
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NewX = OldX + dx NewY = OldY + dy
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In this equation, (OldX, OldY) represents the original coordinates of the point, and (NewX, NewY) represents the transformed coordinates after the translation. dx and dy represent the amount of shift in the x and y directions, respectively.
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The purpose of a transformation equation is to convert a set of coordinates from one coordinate system to another. A coordinate system is a reference frame used to locate points in space. It consists of an origin (a fixed point) and a set of axes (lines) that define the directions and scales of measurement.
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Pg. 219
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Pg. 220
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WTF DID I JUST READ?
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Here’s the TL;DR: Using planar angle measurements of the sky, a 2D lat/long coordinate system can be made to fit a 3d spherical coordinate system using the provided transformations. Long version:
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1. Coordinate Systems: The fundamental reference systems in astronomy are based on the celestial equator and the ecliptic. Angular coordinates are measured from the ascending node of the ecliptic on the equator, known as the vernal equinox or the first point of Aries.
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2. Rectangular Coordinate Systems: Coordinate systems can be specified using either spherical coordinates (direction and distance) or rectangular coordinates (projections on three axes). The axes of the rectangular coordinate systems are right-handed, with the x-axis directed towards the equinox, the y-axis to a point 90° east, and the z-axis positive to the north.
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3. Origin of Coordinates: Coordinates can be based on different reference points: observer-topocentric, center of the Earth-geocentric, center of the Sun-heliocentric, or center of mass of the solar system-barycentric.
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4. Reference Planes and Directions: Spherical coordinates are determined by reference planes and directions, such as the horizon and local meridian (azimuth and altitude), the equator and local meridian (hour angle and declination), the equator and equinox (right ascension and declination), the ecliptic and equinox (ecliptic or celestial longitude and latitude), and the plane of an orbit and its node (orbital longitude and latitude).
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5. Reduction to Topocentric Coordinates: The conversion from geocentric to topocentric coordinates depends on the figure of the Earth and involves small differential corrections.
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6. Position Types: Positions in astronomy can be geometric (actual position at the time of observation), apparent (corrected for aberration and refraction), or astrometric (directly comparable with catalog positions of stars). 7. Coordinate Conversions: The passage provides formulas for converting between different coordinate systems, such as azimuth/altitude, hour angle/declination, right ascension/declination, and longitude/latitude.
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The geocentric rectangular coordinates (x, y, z) can be converted to heliocentric coordinates (xc, yc, zc) using the following equations: xc = x + X yc = y + Y zc = z + Z In these equations, (X, Y, Z) represents the geocentric coordinates of the Sun.
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The heliocentric coordinate system is derived from the geocentric coordinate system, specifically using the geocentric coordinates of the Sun. The geocentric coordinates serve as a reference point for determining the position of celestial objects relative to the Earth. From the geocentric coordinates, the heliocentric coordinates can be derived by adding the respective geocentric coordinates of the Sun. This conversion allows for a different perspective where the positions of celestial objects are described relative to the Sun instead of the Earth.
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EQUATION & VARIABLES DEFINED
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• 𝜙 represents the geographic or geodetic latitude, which is the angle measured from the equatorial plane to a point on the Earth's surface.
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• 𝜙' represents the geocentric latitude, which is the angle between the equatorial plane and the line joining the Earth's center to a point on the Earth's surface.
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• tan 𝜙' = (I - e^2) tan 𝜙 is an equation relating the geocentric latitude (𝜙') to the geographic latitude (𝜙) and the ellipticity (e) of the Earth's meridian. It states that the tangent of the geocentric latitude is equal to the product of the tangent of the geographic latitude and the factor (I - e^2), where I represents the spherical radius of the Earth.
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• 𝜙_1 represents the parametric latitude, which is another way of expressing the relationship between the geographic and geocentric latitudes.
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• tan 𝜙_1 = (I - ƒ) tan 𝜙 is an equation relating the parametric latitude (𝜙_1) to the geographic latitude (𝜙) and the flattening (ƒ) of the Earth. It states that the tangent of the parametric latitude is equal to the product of the tangent of the geographic latitude and the factor (I - ƒ), where I represents the spherical radius of the Earth.
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• e represents the ellipticity or eccentricity of the Earth's meridian. It is a measure of the departure of the Earth's shape from a perfect sphere.
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• ƒ represents the flattening factor, which quantifies the oblateness or departure from a perfect sphere. It is defined as I - ƒ = (I e^2)^1/2, where I represents the spherical radius of the Earth.
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• p represents the geocentric distance, which is the distance from the center of the Earth to a point on the Earth's surface, measured in units of the Earth's equatorial radius.
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• S and C are auxiliary functions used in calculations. They are defined such that p sin 𝜙' = S sin 𝜙 and p cos 𝜙' = C sin 𝜙 = cos p sin 𝜙' = S sin 𝜙_1. These functions are related to the trigonometric values of the latitudes and the geocentric distance.
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TRANSFORMING A GEOCENTRIC RECTANGULAR COORDINATE SYSTEM INTO A SPHERICAL COORDINATE SYSTEM
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(x) ξ = cos 𝜙₁ sin ϴ This equation represents the x-coordinate in the spherical system, where 𝜙₁ is the parametric latitude and ϴ is the longitude. (y) 𝜂 = 𝜂₁ sin 𝜙₁ cos d₁ - cos 𝜙₁ sin d₁ cos ϴ This equation represents the y-coordinate in the spherical system. It involves the parametric latitude (𝜙₁), the latitude (d₁), and the longitude (ϴ). (z) ζ = ζ₁ sin 𝜙₁ sin d₁ + cos 𝜙₁ cos d₁ cos ϴ This equation represents the z-coordinate in the spherical system. It also involves the parametric latitude (𝜙₁), the latitude (d₁), and the longitude (ϴ). The parametric latitude (𝜙₁) is derived from the geocentric latitude (𝜙') and is used to calculate the coordinates in the spherical system. Additionally, the relationship between ζ and 𝜂₁, ζ₁, and the derived quantities (p₁, p₂, sin(d₁ - d₂), cos(d₁ d₂)) is expressed by: ζ = p₂ (ζ₁ cos (d₁ - d₂) - 𝜂₁ sin (d₁ - d₂)). This equation provides an alternative way to calculate ζ using the derived quantities and 𝜂₁, ζ₁. In summary, these equations allow for the transformation of geocentric rectangular coordinates (ξ, 𝜂, ζ) to spherical coordinates (𝜙₁, ϴ, ζ)
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APPLYING THE TRANSFORMATIONS
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The given equations facilitate the transformation from geocentric rectangular coordinates (ξ, 𝜂, ζ) in the XYZ system to spherical coordinates (𝜙₁, ϴ, ζ) in a 3D space, representing latitude and longitude. To summarize the process:
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1. Start with geocentric rectangular latitude and longitude coordinates (ξ, 𝜂, ζ)
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2. Calculate the parametric latitude (𝜙₁) using the geocentric latitude (𝜙’).
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3. Apply the flattening correction to account for the Earth's spheroid shape: • Compute derived quantities p₁, p₂, sin(d₁ - d₂), and cos(d₁ - d₂) using the geocentric latitude (𝜙') and Earth's ellipticity (e). • Use these derived quantities and 𝜙₁ to calculate the coordinates (ξ, 𝜂, ζ) in the spherical system, as described in the equations mentioned previously.
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4. The resulting spherical coordinates (𝜙₁, ϴ, ζ) represent the latitude and longitude in the 3D spherical coordinate system, considering the Earth’s alleged spheroidal shape.
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The flattening correction adjusts for the Earth's flattened shape, ensuring that the 2D latitude and longitude coordinates match up with the 3D spherical coordinates. It is incorporated into the calculation of the derived quantities p₁, p₂, sin(d₁ - d₂), cos(d₁ - d₂), which are then used to determine the spherical coordinates. By considering the flattening correction, the transformation accurately maps the geocentric rectangular latitude and longitude coordinate system to a 3D spherical latitude and longitude coordinate system.
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