The star looks higher. Is it really lifted by the air?
Neither the people nor the stars have been moved away, what the air changes is the direction in which the light reaches the eyes.
Sky coordinates and observations
When you look at a star close to the horizon, you think you are looking toward its true position. But as starlight passes through the atmosphere, its path gradually bends. The eyes can only receive the last section of incoming light. Looking back in this direction will "place" the stars in a slightly higher position.
Guess first
The atmosphere makes stars appear higher. What is the main change?
Show answer and explanation
The direction of light reaching the eye — The light path is bent by the atmosphere, and the eyes push back the light source in the direction it finally arrived. The stars themselves were not moved.
Then connect experience and science
Distinguish between real and seen directions
The blue dots indicate the geometric direction, and the bright dots indicate the viewing direction. The vertical scale in the figure is adjusted with height to make the difference clearer.
True height h ≠ apparent height
air bending light path
Under the standard atmospheric conditions used in this lesson, the refraction correction is positive and the apparent position is higher than the true position.
Apparent height = h + R/60
The lower the value, the stronger the impact
Drag the slider toward the horizon and the refraction increases rapidly. The impact at high altitude is much smaller.
R ≈ 1.02 / tan(h + 10.3/(h + 5.11))
Remember this is an approximation
The angle in the formula is expressed in degrees, and the output R is expressed in angle minutes. Here the air pressure is fixed at 1010 hPa and the temperature is 10°C.
1° = 60′
Expand the calculation
Rarc minutes = 1.02 / tan(h + 10.3/(h + 5.11)) h视 = h + Rarc minutes/60
All angles in brackets are measured in degrees, and the innermost fraction is calculated first. R is output in arc minutes and must be divided by 60 to be added to the height in degrees. See the description of this lesson for the scope of application.
give it a try
First compare "close to the horizon" and "high in the sky." See the reason on the left, and use the same set of angle scales on the right to see how much it is actually raised.
No need to memorize symbols
- h
- Geometric height unaffected by refraction, unit: degrees
- R
- Refraction correction, units: arc minutes
- h + R/60
- Observed apparent height, unit: degrees
Algorithm, accuracy and scope of application
Demonstrates the Meeus equation (16.4), Sæmundsson approximation, restricted to 0°–80°. The AARefraction.cpp of local AA+ has switched to the NOAA segmented model in 2025, so the two sets of results are not required to be completely consistent, nor does it claim that this formula has been compared with the AA+ algorithm. The diagram only explains the difference in position and does not simulate the path of real light through the layered atmosphere. Scenario description: The picture on the left is a qualitative light path diagram, and the bends and position differences are intentionally exaggerated; the picture on the right is partially enlarged with a real angle scale. The calculation follows Meeus (16.4), true altitude 0°–80°, 1010 hPa, 10°C; no layered atmospheric light is required, and no actual sunrise is predicted. Local AA+ has switched to the NOAA model, and the two cannot be said to be verified by the same algorithm.
my understanding check
What is the golden dotted line in the picture?
The position where the eyes are pushed back along the direction of eye entry — The dotted line is the reverse extension of the observation direction, not the actual curved optical path.
Where in the same standard atmospheric model do larger refraction corrections usually need to be made?
close to the horizon — Refraction is usually greater near the horizon. Real weather also affects the amount of correction.
Can the precise correction be measured directly using the exaggerated light path diagram on the left?
No, you have to look at the calculation scale on the right — The picture on the left only explains cause and effect, and the bends are exaggerated. The picture on the right and the numbers use the same real angle unit and the approximate formula in the original book.
The air bends the starlight, and the eyes push back in the direction in which it enters, so the stars are seen at a slightly higher position.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 16 chapter