Transit and Culmination

The culmination of a celestial body means that the body is at its greatest altitude, whereas the transit is the passage of its center through the meridian.

Only the fixed stars culminate really in the meridian. The Sun, Moon, and the planets culminate out of the meridian.
At mid-latitudes (50°) the difference may be up to 18 seconds for the Sun, and more than 6 minutes for the Moon.

To compute the difference in time between transit and culmination we start by the well known equation

⁽¹⁾

    h = altitude
    δ = declination
    Φ = latitude
    H = hour angle = Local Sidereal Time - Right Ascension

Differentiating the above equation with respect to time:

⁽²⁾

At the instant of culmination we have . If (constant declination) the culmination is at H=0°.

For non constant declination the hour angle of culmination H_C is very small

and we get as an approximation of (2):

⁽³⁾

The altitude at culmination is by ∆h greater than the altitude on the meridian:

(4)

The Sun

In (3) the derivative may be replaced by differentiating the classical equation

⁽⁵⁾

β = ecliptic latitude
γ = obliquity of the ecliptic
L = ecliptic longitude
The ecliptic latitude β of the Sun is very small: | β |<0.0002°, , cos β=1, and therefore from (5):

Equation (3) then becomes:

⁽⁶⁾

L = heliocentric ecliptic longitude
H = hour angle = Local Sidereal Time - Right Ascension
γ = obliquity of the ecliptic = 23.44°
δ = declination
Φ = latitude

For the Sun the rates of change in time (derivatives) of L and H can be approximated by:

For equinoxes (δ=0°, L=0°, L=180°) H_K is an extremum:

(7)

At Φ = 40° we get H_K = 9,14·10^-4 rad = 0,0524° = 0.210 min = 12.6 s

At Φ = 50° we get H_K = 1,30·10^-3 rad = 0,0744° = 0.298 min = 17.9 s
The altitude at culmination is only 0.14'' more than at transit.

For solstices (L=90°, L=270°) H_C = 0.

Between winter solstice and summer solstice culmination occurs later then transit, between summer solstice and winter solstice culmination occurs earlier then transit.

horizontal axis: day of the year
vertical axis: time difference in seconds between culmination and transit
blue: latitude Φ = 40°, red: latitude Φ = 50°.

In NavList the following formula for the difference can be found (as a first order approximation):

∆T = (48/PI)*(tanΦ-tanδ)*(vLat-vDec) (8)

∆T = difference is seconds of time between culmination and meridian transit

On 2012, Mar 20 at 05:14 UT (spring equinox):
    vLat = hourly latitude speed is arcminutes per hour = 0 (observer at rest)
    vDec = hourly declination change is arcminutes per hour = -0,988 arcmin/hour
    δ = 0.0°
    at Φ = 50°:

∆T = 18.0 s

This result agrees with (7).

***

The Moon

The orbital plane of the Moon is inclined (| β |< 5.1°) aganst the ecliptic, and formula (6) is not valid.

On 2024, Oct 15 at 50°N, 0°E (computed by MICA, Multiyear Interactive Computer Almanac by USNO) there is an extreme value of difference:

    culmination    22:31:14.5 UT
    transit           22:25:19.3 UT
    difference      5 min 55 s

The result from calsky is 6 min 03 s.

On 2024, Oct 15 at 22:30 UT we apply the formula (3) for the Moon using MICA:
    = dδ/dt = 0.303°/hour
    dH/dt = 14.56°/hour
    δ = 0.00°
    H_C = 0.0248 rad = 1.42° = 5.68 min

H_C = 5 min 41 s

The NavList formula, derived by Wilson, writes:

∆T = [10.800°/(PI*(dH/dt)²)]*(tanΦ-tanδ)*(vLat-vDec) (9)

    ∆T = difference is seconds of time between culmination and meridian transit
    ∆H/dt = hourly change of hour angle, H in degrees
   vLat = hourly latitude speed of the observer is arcminutes per hour

For the Moon:

14.38°/hour < dH/dt < 14.61°/hour
mean: 14.495°/hour = 360°/24h 50min

16.105 hour·s/° < 10.800°/[PI*(dH/dt)²]< 16.625 hour·s/°
mean: 10.800°/[PI*(dH/dt)²] = 16.365 hour·s/°

On 2024, Oct 15 at 22:30 UT using MICA:
    vLat = 0 (observer is at rest)
    vDec = hourly declination change is arcminutes per hour = 18.19 arcmin/hour
    δ = 0.025°
    dH/dt = 14.49°/hour
    10.800°/[PI*(dH/dt)²] = 16.37 hours·s/°

∆T = 355 s = 5 min 55 s

The culmination occurs earlier than the transit if the ecliptic latitude B of the Moon is negative, and vice versa. The difference seems to be great near the perigee of the Moon (dL/dt greater than near apogee).

Vertical axis: Difference ∆T = t_Trans - t_Culm (minutes) at 50°N 0°E
horizontal axis: day in May-Jun 2012
computed by my Planet applet

moon ecliptic
latitude

Vertical axis: ecliptic latitude B of the Moon (degrees)
horizontal axis: day in May-Jun 2012

Moon
transit culmination difference time 2012

horizontal axis: day of the year 2012
vertical axis: time difference in seconds between culmination and transit
latitude Φ = 50°
computed by my Planet applet

Transit Culmination 2015 lunar
standstill

Transit Culmination 2015 lunar
standstill

Vertical axis: Difference t_Trans - t_Culm (minutes) at 50°N 0°E
horizontal axis: day in Sep-Oct 2015
computed by my Planet applet

lunar standstill 2024

Vertical axis: Difference t_Trans - t_Culm (minutes) at 50°N 0°E
horizontal axis: day in Sep-Oct 2024
computed by my Planet Applet

It seems that the variation of ∆T is dominated by the 18.6-year cycle of lunar standstills:
small values of ∆T (up to 4 minutes) are occuring in years of minor lunar standstills (1996, 2015), and large values (more than 6 minutes) in years of major lunar standstills (2006, 2024/2025).

Max. declination in 2024 on Sep 24 at 17 UT:    28.70°
Min. declination in 2024 on Oct 09 at 12 UT: -28.70°
Change in declination: -14.22 arcmin/h < < 18.21 arcmin/h

Max. declination in 2015 on Jan 03 at 18 UT:    18.65°
Min. declination in 2015 on Jan 18 at 06 UT:    -18.58°
Change in declination: -9.42 arcmin/h < < 11.64 arcmin/h

2006 (major lunar standstill) Aug 10, 50°N, 0°E

MICA:

    transit            00:40:34 UT
    culmination     00:46:48 UT
       ∆T = 6 min 14 s
Planet Applet:
    transit            00:40:34 UT
    culmination     00:46:49 UT
       ∆T = 6 min 15 s
calsky:

transit: 00h 40m 34.2s UT

    culmination:      00h 46.9m UT
      ∆T = 6 min 20 s
The altitude at culmination is only 0.81' more than at transit.

by formula (3) and MICA:
    = dδ/dt = -0,2585°/hour
    dH/dt = 14.45°/hour
    δ = -15.55°
     ∆T = 6 min 02 s

Vertical axis: hourly change of Declination (degrees)
horizontal axis: day in Sep-Oct 2015
minor lunar standstill

Vertical axis: hourly change of Declination (degrees)
horizontal axis: day in Sep-Oct 2024
major lunar standstill

The result by formula (3) or formula (8) will be large for northern latitudes, if δ<0,
and if ||is large. dH/dt is nearly constant (14.38°/hour to 14.61°/hour).

The Planets

As an example transit and culmination of Mars (computed by MICA) at 50°N, 0°E:

Mars, 2012 May 25:

    Transit            18:41:50 UT
    Culmination     18:41:43 UT
    ∆T = 7 s

Mars, 2012 Aug 01 25:
    Transit            16:17:38 UT
    Culmination     16:17:26 UT
      ∆T = 12 s

Mars, 2013 Apr 01 25:
    Transit            12:18:20 UT
    Culmination     16:18:34 UT
      ∆T = 14 s

Body	dH/dt in °/hour	10.800°/(dH/dt)² in hour·s/°
Sun	15	48
Planets	15 +	47.89
Stars	15.04	47.73
Moon	14.32 +	50.74 - 52.07

Table from Wilson

factor

Using the diagram above to get ∆T for the Sun, look up the value of F (depending on latitude and declination), and multiply F by the hourly declination change in arcminutes per hour:
Φ = 50°, δ = 0°, F = 18, = 1 arcmin/h, ∆T = 18 s

***

A paper by Pio (1899) describes the determination of the longitude (at known latitude) from the culmination of the Moon: The instants of two equal altitudes are measured by a chronometer and a sextant, and the time of culmination is "reduced to the meridian". The practical advantage of the method is that it does not require any transit intruments.