On a slightly flat earth model, does the "vertical foot" necessarily pass through the center of the earth?
First stand on a smooth reference ellipsoid and compare the direction of the vertical surface with the line connecting the center of the earth.
Sky coordinates and observations
If you were standing on a perfect sphere, a line vertical to the surface of your feet would pass through the center of the sphere. However, the earth model used in astronomical calculations is slightly bulging at the equator and slightly flat at the poles. At this time, there will be a little difference between "vertical model surface" and "connected to the center of the earth", so there are also two latitudes.
Guess first
On this slightly flattened reference ellipsoid, do lines perpendicular to the surface necessarily pass through the center?
Show answer and explanation
uncertain — The normals of a perfect sphere all pass through the center; an ellipsoid usually does not meet this requirement, as the equator and poles are special positions of symmetry.
Then connect experience and science
First describe the earth with an ellipsoid
Draw the difference curve between geographical latitude and geocentric latitude to avoid exaggerating the flatness of the earth and mistaking it for the true proportion.
Equatorial radius a = 6378.14 km
Distinguish between two latitudes
The geographical latitude corresponds to the normal of the reference ellipsoid; the geocentric latitude corresponds to the direction from the center of the earth to you. The two are not the same line.
φ: geographical latitude; φ′: geocentric latitude
Stand higher along the normal line
Altitude changes the observer's geocentric coordinates, and the geocentric latitude also changes slightly. Adjust the altitude from 0 to 5000 meters and observe the results.
Geocentric coordinates = ellipsoid surface position + height component
See where they are the same and where they are different
At the equator and the poles, the two latitudes are the same; the difference is more pronounced at mid-latitudes. Do not mistake the difference for instrument measurement error.
Sea level: tan φ′ = (b/a)² tan φ
Expand the calculation
b/a = 0.99664719 u = atan((b/a) tan φ) ρ sin φ′ = (b/a) sin u + (h/6378140) sin φ ρ cos φ′ = cos u + (h/6378140) cos φ φ′ = atan2(ρ sin φ′, ρ cos φ′)
h is in meters; ρ is the geocentric distance in units of the Earth's equatorial radius. The auxiliary angle u is not the geocentric latitude; first find the two components, and then use atan2 to restore φ′.
give it a try
First compare "equator" and "mid-latitude". The picture on the left retains the original flatness, so the two lines almost overlap; the picture on the right magnifies the slight angle difference to help you see the difference in direction.
No need to memorize symbols
- φ / φ′
- Geographical latitude/geocentric latitude
- a / b
- Equatorial radius/polar radius of the reference ellipsoid
- ρ
- Distance from the center of the Earth in units of equatorial radius
- H
- Height above the reference ellipsoid in meters
Algorithm, accuracy and scope of application
According to Chapter 11 and local AA+ constants a=6378.14 km, b/a=0.99664719. Height increases along the normal line of the reference ellipsoid; local plumb line deviations and geoid fluctuations are not added. The model is not a reconstruction of real terrain, nor does it purport to replace modern geodesic reference systems. Use AA+ RhoSinThetaPrime and RhoCosThetaPrime for grid comparison. Scenario description: Use the reference ellipsoid constants from the original book and Astro.globe. The meridional section on the left uses the original major and minor axis ratio; the right side only magnifies the directional angle difference by 120 times. The reference ellipsoid normal is not equal to the normal to the real terrain, nor is it equal to the local plumb line determined by gravity; gravity anomalies, terrain, and the geoid are not calculated in this lesson.
my understanding check
What does the blue line represent in this experiment?
Surface normal of the reference ellipsoid — It comes from the geometric relationship of the smooth reference ellipsoid; the true plumb line is also related to the local gravity field.
At which locations would the two latitudes be the same?
Equator and poles — The symmetry of these positions causes the surface normal to coincide with the line connecting the center of the earth.
The two lines in the picture on the right are widely separated. Does it mean that the real earth is so flat?
No, the picture on the right magnifies the angle difference. — The left picture only retains the proportions of the original model; the right picture only magnifies the direction angle difference by 120 times, and the actual angle difference must be read in numbers.
The geographical latitude looks at the normal of the reference ellipsoid, and the geocentric latitude looks at the line connecting the center of the earth; the same position can be described by two different direction angles.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 11 chapter