If you stand a needle on the ground, why can its shadow tell you the time?
When the direction of the sun changes, the shadow of the needle tip moves along the disk. Test on a flat surface first, then mount the panel to the wall.
The sundial has no gears, but it can tell time by its shadow. But it’s not just a matter of just drawing a few random marks on any wall: where you are, which way the wall faces, and how much the disk tilts, all determine where your shadow falls.
Guess first
A wall faces away from the sun. Can the shadow on this wall tell the time now?
Show answer and explanation
cannot — It must be illuminated by the sun to the front of the disk.
Then connect experience and science
Recognize the needle and disk first
The needle is vertical to the disk surface. The origin in the picture is the pin, and the moving point is the position of the shadow of the needle tip on the disk.
Vertical needle length a; the needle is the origin of the coordinates
Let the sun go from morning to afternoon
The slider is true solar time: the sun crosses the local meridian at 12 o'clock. It is not the unified time zone time displayed on the phone.
H = 15° × (true solar time −12)
Change the season or turn the disk around
The shadow at the same time also changes with the sun's declination and disk orientation. The blue curve is a day in the same season, and the thin lines correspond to different moments.
The shadow is determined by the intersection of the direction of the light and the disk surface.
First check whether the disk can work
The sun must be above the horizon and must shine on the front of the disk. The polar needle may not have a finite center and should not be forced to make one.
h>0 and Q>0; when P=0, the polar needle center is undefined
First find H from true solar time, and then find the intersection of the sun's direction ray and the disk. a is the length of the vertical needle, and Q determines whether the sun shines on the disk; when h ≤ 0 or Q ≤ 0, the projected coordinates are not regarded as effective sun shadows.
give it a try
Compare a flat ground sundial to a wall sundial. At certain times, the sun is behind the disk. At this time, you cannot forcefully draw a shadow that can tell the time.
No need to memorize symbols
- D
- Normal azimuth angle, 0° south, 90° west, 180° north, 270° east
- z
- Normal zenith distance; 0 horizontally upward, 90 vertically
- a
- The needle is perpendicular to the disk and its length is meters
- x/y
- Shadow coordinates to the right and up of the disk, meters
- P/Q
- Polar relationship/illumination criterion
Algorithm, accuracy and scope of application
Realize the P/Q/Nx/Ny formula of any direction and tilted disk in Chapter 58, the polar needle center/length/angle, and the dual criteria of horizon and rising sun. Horizontal, vertical, and equatorial disks are special cases of the same general formula; mathematically extended intersections when the north-facing disk is not exposed to light are not shown as effective shadows. The original book 58.a/b is an independent sample, and a three-dimensional ray plane is used for intersection verification. True solar time has not been converted into civil standard time in a certain city.
my understanding check
Can 12 o'clock in this lesson be directly used as 12 o'clock on the mobile phone?
No, it's true solar time — Additional conversions such as mean time difference, longitude and time zone are required.
When the sun is on the horizon, does it necessarily illuminate the front of the sundial?
Not necessarily, it depends on the direction — The disk may be facing away from the Sun.
Does the vertical needle in this lesson always point to the North Celestial Pole?
uncertain — The needle is perpendicular to the disk surface; the polar axis needle is in another direction, and its fixed point and length are calculated separately in the book.
The shadow moves because the direction of the sun is changing; in order for it to tell time, the location, disk orientation, and time definition must be calculated correctly.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 58 chapter