Transit and Culmination

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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)

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

⁽⁵⁾

β = ecliptic latitude
γ = obliquity of the ecliptic
L = ecliptic longitude

1. The Sun

Moon
Planets
ISS

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 in (6) can be approximated by:

(0.986 °/d<<1.019 °/d, error =±3.3 %)
(358.6 °/d<<361.8 °/d, error ≈±0.5 %)

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

(7)

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

At Φ = 50° we get H_C = 1,30·10^-3 rad = 0,0744° = 0.298 min = 17.9 s
The altitude of the Sun at culmination is 0.14'' higher than at transit, only 1/13000 of it's diameter, very difficult to measure !
Between transit and culmination the Sun moves 1/5.5 of it's diameter. The effect is of no visual consequence.

In polar regions there is a maximum:

At Φ = 85° we have H_C = 172 s, at Φ = 87,5° we have H_C = 345 s !

For summer and winter solstice (L=90°, L=270°) culmination occurs precisely at meridian passage:

H_C = 0 s.

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

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

Using my spreadsheet to compute ∆T by formula (6), but inserting the dayly value of ,
on 2018, Jan 1 at 40°N, 0°E I get the result:

∆T = 4.06 s

Evaluating the data of JPL HORIZONS Web Interface (apparent AZ and EL) results:

∆T = 4.07 s

Comparing JPL and spreadsheet results for azimuth and refracted elevation:

Date__(UT)__HR:MN:SC.fff     Azi_(r-apprnt)_Elev
********************************************************************
$$SOE
 2018-Jan-01 12:03:00.000 *   179.85467  27.05276

Spreadsheet latitude 0°:

Using formula (4) for ∆h=h_culm - h_trans and the approximation at 40°N:

The difference in elevation ∆h is less than 0.11 ''.

The results of my spreadsheet for 2018, Mar 1 are:

∆T=14.1 s and ∆h=0,106 ''.

Comparing with JPL HORIZONS Web Interface data:

∆T=14.1 s

mean dElev/dt=0.475 ''/min between transit and culmination:

∆h=0.112 ''

The difference in azimuth between culmination and transit is 0.08°.

*************************************

2. The Moon

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

a. Stellarium, a free planetarium, is quite comfortable to get the difference in time ∆T between transit and culmination, as well as the difference in elevation, and azimuth.

On 2024, Oct 15 there is a large value of ∆T. At 50°N, 0°E:

Transit (local time):

Culmimation (local time):

Result: ∆T=360 s, ∆h=-0.903 arcmin, ∆az=-1,89°

*****

b. My spreadsheet

is most comfortable: computing the elevation and azimuth of the Moon every 30 minutes of a day, and evaluating the data to get ∆T, ∆h, and ∆az by a few key strokes to insert the date.

It's made by LibreOffice 6.3.6.2 for MAC (.ods file). Saved as .xls file it can be opened by Excel.

Example: 2024, Oct 15, at 50°N, 0°E:

The numerical results by quadratic (elevation) and linear (azimuth) interpolation:

∆T = 360 s = 6 min 0 s, ∆h = -0.93 arcmin, ∆az = -1.87°

c. On 2024, Oct 15 at 50°N, 0°E (computed by MICA, Multiyear Interactive Computer Almanac by USNO):

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

d. The result from CalSKY is 6 min 03 s.

e. 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

f. The NavList formula, derived by Wilson, writes:

∆T = [10800°/(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 hour²/° < 10800°/[PI*(dH/dt)²]< 16.625 hour²/°
mean: 10800°/[PI*(dH/dt)²] = 16.365 hour²/°

On 2024, Oct 15 at 22:30 UT using MICA:
    vLat = 0 (observer is at rest)
    vDec = hourly declination change in arcminutes per hour = 18.19 arcmin/hour
=(18.19/60) °/hour
    δ = 0.025°
    dH/dt = 14.49°/hour
    10800°/[PI*(dH/dt)²] = 16.37 hour²/° = 16.37 hour·60 s/°

∆T = 355 s = 5 min 55 s

******************

Summary
2024, Oct 15 at 50°N, 0°E

	∆T / s	∆h / '	∆az / °
a. Stellarium	360	-0.90	-1.89
b. my spreadsheet	357	-0.91	-1.87
c. MICA	355
d. CalSKY	363
e. (3) & MICA	341
f. Wilson & MICA	355
g. JPL HORIZONS	355	-0.91	-1.86

******************

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

Urs Klaeger pointed out to me:
With increasing declinations AND at geo-latitudes north of the subsolar/sublunar point, Culmination is after Transit. Vice versa south of the Sun/Moon.

Jean Meeus writes:
If culmination occurs south of the zenith and ∆δ is positive, the highest altitude is reached after the meridian passage; if the culmination occurs between the pole and the zenith, the situation is reversed.
(„More Mathematical Astronomy Morsels“, page 320)

Moon at 50°N, May-Jun 2012: daily change of declination (∆δ),
and ∆T = t_Trans - t_Culm (minutes)

e.g. on May 10:
∆δ>0, and ∆T<0: culmination is after transit.

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
StarryNight SN7 CSAP
    transit:             00h 40m 34.2s UT
    culmination:      00h 46.8m UT
     ∆T = 6 min 15 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).

******************************************

3. The Planets

As an example transit and culmination of Mars at 50°N, 0°E:

Mars, 2012 May 25 (computed by MICA):

    Transit            18:41:50 UT, 18:41:51.2 UT
    Culmination     18:41:43 UT, 18:41:45 UT
     ∆T = 7 s

Mars, 2012 May 25 (computed by StarryNight):

    Transit            18:41:50 UT, 18:41:51.2 UT
    Culmination     18:41:43 UT, 18:41:44.5 UT
     ∆T = 6,7 s

Mars, 2012 Aug 01 (computed by MICA):
    Transit            16:17:38 UT
    Culmination     16:17:26 UT
      ∆T = 12 s

Mars, 2012 Aug 01 (computed by StarryNight):
    Transit            16:17:38 UT, 16:17:40.7 UT
    Culmination     16:17:26 UT, 16:17:28.5 UT
      ∆T = 12.2 s

Mars, 2013 Apr 01 (computed by MICA):
    Transit            12:18:19.5 UT
    Culmination     12:18:32 UT
      ∆T = 12.5 s, ∆h = 0.07 ''

Mars, 2013 Apr 01 (computed by StarryNight):
    Transit            12:18:20 UT, 12:18:22.9 UT
    Culmination     12:18:34 UT, 12:18:36 UT
      ∆T = 13.1 s

Mars, 2020 Oct 13 (computed by Stellarium):
    Transit            23:50:25 UT
    Culmination     23:50:22 UT
      ∆T ≈ 3 s

Body	dH/dt in °/hour	10800°/(dH/dt)² in hour²/°
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 instruments.

*********************************

4. International Space Station

At CalSKY I found for my location

Lon: +7d58m00.00s Lat: +51d37m00.00s Alt: 122m

Tuesday, 30 June 2020

ISS

Appears       4h53m36s -1.1mag az:211.1° SSW h:2.8°
at Meridian   4h56m36s -2.3mag az:180.0° S    h:17.6°
Culmination   4h57m57s -2.3mag az:145.7° SE   h:22.5°
distance: 953.4km height above Earth: 421.4km elevation of Sun: -2° angular velocity: 0.47°/s
Disappears    5h03m01s   2.0mag az: 77.5° ENE horizon
Time uncertainty of about 2 minutes

For the difference between transit (meridian) and culmination:

∆T = 1m 21s, ∆h = 4.9°, ∆az = 34.3°

***

The same passage event at HEAVENS ABOVE :

Evaluating the table results:

∆T = ∼ 2m, ∆h = ∼ 7°, ∆az = ∼ 45°

Links

D. A. Pio: Longitude from the Moon Culminations, Monthly Notices of the Royal Astronomical Society, from November 1998 to November 1999, Volume LIX, London 1899.

[NavList 9938] Re: Time of meridian passage accuracy

James N. Wilson: Position from Observation of a Single Body (Appendix I)

Books

Jörg Meyer: Die Sonnenuhr und ihre Theorie, Verlag Harry Deutsch, Frankfurt 2008, ISBN 978-3817118243

Jean Meeus: More Mathematical Astronomy Morsels, Willmann-Bell, 2002,
ISBN 978-0943396743

Software

MICA Multiyear Interactive Computer Almanac 1800-2050 by U.S. Naval Observatory (Willmann-Bell)

StarryNight 7 CSAP

Last update: 2023, Oct 07