Caption: Illustrated are 3 types of horizon: true horizon, visible horizon, and celestial horizon (AKA astronomical)---which we usually just call horizon for short.

Features:

1. As illustrated, the true horizon is circle defined by the line of sight from an observation point that is a tangent line to a sphere.

2. The visible horizon is the actual local line dividing Earth features and sky. The visible horizon is affected by surface features: e.g., trees, mountains, water waves, weather, atmospheric refraction, etc.

3. The celestial horizon is the great circle on the celestial sphere cut by the tangent plane to the astronomical object (considered as an exact sphere) the observer is on at the location of the observer.

   The celestial horizon is the limit of the true horizon when the observing point is actually on the sphere for which it is specified.

4. Which meaning of horizon is meant is usually known by context.

   The celestial horizon is usually what we always mean in astronomy---except when we don't.

5. We can derive a simple approximate formula for the true horizon:

   Let the radius of a exactly spherical astronomical object be R, the height of the observation point be h, and the line-of-sight distance to the true horizon be d.

   The angle at the horizon between a radius and the line of sight is a right angle since the line of sight is tangent to the spherical surface of the astronomical object at the true horizon by definition.

   From the Pythagorean theorem, we find

       d=sqrt( (R+h)**2-R**2 ) = sqrt( 2Rh+h**2 ) = sqrt(2Rh)*sqrt(1+(h/(2R))

           ≅ sqrt(2Rh)*(1+h/(4R)) ≅ sqrt(2Rh) ,

   where we have assumed h/R << 1 for the approximate formulae: the first one is a 1st order approximation and the second, a zeroth order approximation which can be used whenever h/(4R) << 1. Thus, to zeroth order

       d ≅ sqrt(2Rh)= (3.5696 km)*sqrt(R/R_⊕)*sqrt(h/1 m) ,

   where for the fiducial-value formula we have used Earth mean radius R_⊕ = 6,378.1 km and divided h by 1 meter.

   Using the approximate formula we find the following reference results given in Table: True Horizon Reference Distances below for a fiducial tall human of observing from h = 2 meters.

   
   Table:  True Horizon Reference Distances
   
   Astronomical   Mean Radius   True Horizon Distance   True Horizon Distant
   Object                        Zeroth Order Formula      Exact Formula
                                    with h = 2 m           with h = 2 m
                     (km)              (km)                  (km)
   
   General            R           5.0482*sqrt(R/R_⊕)   5.0482*sqrt(R/R_⊕)*sqrt(1+(0.002 km)/(2R))
   Earth            6371.0            5.0482                5.0482
   Mars             3389.5            3.6821                3.6281
   Moon             1737.1            2.6360                2.6360
   Ceres             473              1.38                  1.38
   1950 DA             1.1            0.066332              0.066363
   

   Notes:

   1. The linked names for the astronomical objects: Earth, Mars, Moon, Ceres (asteroid, dwarf planet), 1950 DA (asteroid).
   2. We have quoted 4 digits beyond significant figures for 1950 DA to show how small the 1st order correction term is even for astronomical objects of order 1 kilometer in size scale.

   ---

6. From the above Table: True Horizon Reference Distances we can see that the on Earth, the true horizon is fairly far away to a typical observer: ∼ 5 kilometers.

   In fact, it seems almost impossible to notice the curvature of the Earth just from looking at the visible horizon even over calm bodies of water where the visible horizon approximates the true horizon.

   However, a person with good eye sight under clear, calm weather conditions can see ships, etc., rise or sink below the visible horizon over bodies of water. This sinking-below-the-horizon effect was recorded first by Strabo (63? BCE--24? CE) in Classical antiquity (see Wikipedia: Spherical Earth: Roman Empire). He thought effect had been well know for centuries by sailors and other well informed people. It has probably been known well into prehistory by many people who never bothered to interpret it as implying a spherical Earth.

   In the many years of looking over Lake Erie, yours truly---and yours truly was born right on the shoreline---staring at boats (and that country yours truly used to refer to when very young as the United Steaks) yours truly has completely failed to notice the sinking-below-the-horizon effect.

7. On the Moon, the true horizon and approximately the visible horizon are about 2.5 km away---about half the distances as on Earth.

   Looking at images taken on the lunar surface, it seems as if the visible horizon is noticeably closer than what we see on Earth (see Wikimedia commons: Category: Astronaut photography on lunar surface).

   This may be partially due to the shorter distance to the true horizon, but it may also be partially due to the sharpness of images because of lack of an atmosphere and the stark black of the lunar sky. See the discussion at Wikipedia: Examination of Apollo Moon photographs: Apparently identical backgrounds in Apollo 15 photographs taken at different locations, but it doesn't seem definitive to me.