What Do the Twilight Angles Behind Fajr and Isha Times Mean?

Short answer: Fajr and Isha times are tied to twilight — the period when the sun is below the horizon but the sky is still lit. The U.S. Naval Observatory's rise, set, and twilight definitions set out three standard thresholds: civil twilight at 6° below the horizon, nautical twilight at 12°, and astronomical twilight at 18°. Different timetables adopt different depression angles, which is the main reason two tables for the same city can differ by several minutes. This page explains the terminology; it is not a timetable and not religious guidance.

Sunrise and sunset are defined events, not impressions

Before twilight, the two anchor events have precise definitions. The Naval Observatory's definitions page describes sunrise and sunset in terms of the sun's upper edge at the horizon, with an allowance for atmospheric refraction bending the light. Two consequences:

The sun's center is already slightly below the geometric horizon at the listed sunset moment.
Published times assume a flat, unobstructed horizon — hills, buildings, and altitude change what you actually see.

In short: sunrise and sunset are computed to a convention, not observed locally.

The three twilights, in plain terms

Using the Naval Observatory's thresholds:

Civil twilight — sun 6° below the horizon. The brightest stage; outdoor activity is still possible without artificial light.
Nautical twilight — 12°. The horizon at sea becomes hard to distinguish.
Astronomical twilight — 18°. Effectively the boundary of full darkness for observing faint objects.

These are measured as solar depression angles, which is why timetable documentation talks about "15°", "17°", "18°", or "19.5°" rather than clock offsets.

In short: a twilight angle is a measure of how far the sun has sunk, not how many minutes have passed.

Why two tables for the same city disagree

Five structural reasons, none of which means either table is broken:

Different depression angles. A convention using 18° for dawn will list an earlier Fajr than one using 15°.
Exact coordinates. A city is not a point; a few miles east or west shifts the result.
Elevation and horizon. Height above sea level moves the apparent horizon.
Rounding and safety margins. Some tables round to the minute in a fixed direction.
Time zone and daylight handling. The conversion from solar time to clock time is where many discrepancies actually originate.

Our overview of how the five daily prayer times are set in Saudi Arabia covers the institutional side of this; for the official Saudi calendar and prayer-time reference, see Umm al-Qura.

In short: the differences come from conventions and coordinates, not from one table being wrong.

Solar noon, and the one time that isn't about twilight

Dhuhr is anchored to solar noon — the moment the sun crosses the local meridian and reaches its daily high point. Solar noon rarely lands at 12:00 on the clock, because clock time is set by broad time zones while solar noon depends on longitude and on the equation of time. The NOAA Solar Calculator computes sunrise, sunset, solar noon, and solar position for any coordinates, so you can see that offset for a specific city. (NOAA notes the tool remains online but is no longer actively maintained.)

In short: solar noon is a local astronomical event; the clock is an administrative convenience.

How to sanity-check the astronomy yourself

Enter the coordinates in the NOAA Solar Calculator and note sunrise, solar noon, and sunset.
Compare those against the sunrise/sunset figures on your timetable — they should match closely.
Expect Fajr and Isha to vary more between sources, because they depend on a chosen twilight angle.
For anything you plan to act on, use the timetable published locally or by the official Saudi reference rather than a derived calculation.

If you are timing a trip rather than a day, Makkah weather by season and Makkah vs Madinah for pilgrims are the more useful pages.

Bottom line: twilight angles are the astronomical vocabulary behind dawn and night times — 6°, 12°, and 18° are the standard definitions, and the angle a table chooses explains most of the minutes separating one printed schedule from another.