The sundial said twelve o'clock, why hasn't the clock arrived yet?
The sun is not a uniform clock. First compare the two sets of solar times at the same instant.
The sundial follows the real sun, while mean solar time follows a uniform "mean sun". The Earth's orbit is not a perfect circle, and the Sun also moves along a tilted ecliptic, so the two readings will move faster and slower throughout the year.
Guess first
If the sundial differs by more than ten minutes from the mean solar time, does that mean the sundial must be broken?
Show answer and explanation
Not necessarily, this may be caused by the average time difference. — The sun's apparent motion is not uniform, and the sundial and uniform mean solar time may be different.
Then connect experience and science
First let the two clocks mark the same moment
To see the difference clearly, it is recommended to set mean solar time to 12:00; add the sundial reading to the average time difference of the day.
Apparent solar time = mean solar time + E
The motion of the sun is not completely uniform
Orbital eccentricity changes the speed of the Earth's revolution; the tilt of the ecliptic relative to the equator also changes the speed of the sun's right ascension.
The hour angle of the real sun is different from the mean sun
The plus or minus sign determines who is in front
E>0 in this book means that the sundial reading is leading and the real sun crosses the meridian earlier than the mean sun. The reverse is true for negative signs.
E = L₀ − 0.0057183° − α + Δψ cos ε
Beware of units vs. another notation convention
The formula first gives the angle, then multiplies it by 4 to get the time in minutes. Some books use opposite signs, so you can't just look at the positive and negative numbers.
1° = 4 minutes
Expand the calculation
ESpend = L₀ − 0.0057183° − α + (Δψarc seconds/3600) cos ε Eminute = 4 × ESpend Apparent solar time = mean solar time + Eminute
L₀ and α cost; if α is an hour, multiply it by 15 first. The angular difference is taken as the branch close to zero according to the course agreement. Close to 360° cannot be misinterpreted as about 24 hours.
give it a try
Let’s look at February first, then November. The two readings belong to the same instant; they are not two observations, nor are they in different time zones.
No need to memorize symbols
- E
- Apparent solar time − mean solar time, in minutes
- L₀
- mean solar longitude
- α
- Solar apparent right ascension
- Δψ cos ε
- The nutation term that reduces the apparent equinox point to the equinox point
Algorithm, accuracy and scope of application
Realize formula 28.1/28.2: The solar mean ecliptic longitude is reduced by apparent right ascension and nutation calculation is performed. The angle is first reduced to −180° to 180° and then multiplied by 4 to convert to time minutes; five low-precision formulas of 28.3 are also provided. The apparent right ascension and nutation directly reuse the truncated periodic term pipeline in Appendix III of Chapter 25 solarHigh, without the need for the first page approximation model. The symbol is taken from the book "apparent solar time minus mean solar time", which is contrary to some literature. The figure sets mean solar time to 12:00, and only demonstrates the time scale difference; E is calculated based on 0 o'clock on the selected TD date, and does not claim to find the precise solar midday at a certain longitude, nor does it convert UTC, time zones or daylight saving time.
my understanding check
According to the definition of E as positive according to this book, which reading is ahead?
Sundial apparent solar time — E = apparent solar time − mean solar time.
The formula yields a time angle difference of 1°. How long does it correspond to?
4 minutes — 360° corresponds to 24 hours, so 1° corresponds to 4 minutes.
By adding this result directly to Beijing time, can we get the sundial time at any location?
No, it still lacks the relationship between longitude and time zone. — This lesson only compares the local solar time scale and does not perform civil time zone conversion.
The sundial follows the real sun, and the mean solar time follows the uniform ruler; first explain the symbol convention, and then talk about who is faster and who is slower.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 28 chapter