Mean longitude

[salvomic]

hi,
Given that I believe that Skyfield always provides the correct and most accurate results,
I'm preparing a script for calculating the equation of time with Skyfield, using the mean longitude of the Sun.

I notice that, from the comparison with the formula provided by Jean Meeus, Astronomical Algorithms, second edition, there is a slight but significant difference with the value of "mean longitude", that for past dates taken as an example comes to a few arcseconds and for today's date reach 19' 57.7", bringing the equation of time calculated with Skyfield to an excess of 01m 19.85s (for today's date many sources report -8.08 seconds, approximating the Meeus formula, while with my routine I get -9m27s, that is more than 1 minute over.
I'm using something like this code:

...
planets = load('de440.bsp')
ts = load.timescale()
t = ts.now()
dt = t.delta_t 
jd  = t.tdb
jde = jd - dt/86400

...
sun, earth = planets['sun'], planets['earth']
barycentric = earth.at(t)
astrometric = barycentric.observe(sun)
apparent = astrometric.apparent()
ra, dec, distance  = apparent.radec('date')
position = (sun - earth).at(t)
elements = osculating_elements_of(position)
i  = elements.inclination.degrees
L0 = elements.mean_longitude

for Meeus formula I input:
taw = (jde - 2451545)/ 365250 Lm = 280.4664567 + 360007.6982779 * taw + 0.03032028 * (taw**2) + (taw**3) / 49931 - (taw**4) / 15300 - (taw**5) / 2000000

where taw are Julian Millennials;
then both approaches use the Equation of time formula:

Et = Ld - 0.0057183 - rah*15 + dpsid * math.cos(math.degrees(i))

where

observer = planets['Earth'] + mylocation
psi, eps = nutation_module.iau2000a(jd, 0.0) 
dpsi = (60*60*psi/tau*360) 
deps = (60*60*eps/tau*360)

Definitely I get for today (2025, 1, 11, 20, 15) here (19:15 utc):

Mean longitude of Sun (Skyfield):  291deg 13' 11.2" (291.2198)
Mean longitude of Sun (Meeus):     291deg 33' 09.2" (291.5526)
Δ Lon. (Skyfield - Meeus):         -00deg 19' 58.0"
Equation of time (Sf):             -09m 27.65s
Equation of time (Meeus):          -08m 07.78s
Δ EoT Skytime - Meeus:             -00h 01m 19.87s

So, where am I wrong? Or what I'm missing? or Meeus is approximated or wrong?
Thank you so much in advance!

::
EDIT2: aligned and corrected some typos

[brandon-rhodes]

@salvomic —

EDIT: sorry if the code is not correctly aligned.

Here is the GitHub guide to quoting code:

https://docs.github.com/en/get-started/writing-on-github/getting-started-with-writing-and-formatting-on-github/basic-writing-and-formatting-syntax#quoting-code

As you write a new question, or "Edit" an existing question like this one, you can click "Preview" to see ahead of time if GitHub understands which sections of your text should be formatted as code. Maybe you can click "Edit" and adjust your original text, to make the code format correctly?

[salvomic]

Good, thank you. Yes, I was really trying to fix the code. I will follow the guide you kindly suggested.

[salvomic]

I did other tests and compared the data with Astronomical Almanac, which coincides with the calculations of Meeus (delta only 1s).
I would like only to understand why in the calculation of the mean longitude of the Sun there is a difference (slightly variable) of the above amount:

– now (2025 January) about 21' corresponding to about 1m20s 
– 1992  it was only about 5' = 20s
– 2030 it will be     about 55' = 3m.

For example, today the equation of time according to AA (and Meeus) is -9m 17.3s, while from the calculations with Skyfield (using the obsculatory mean longitude) results in -10m 46.6s.

Mean longitude Skyfield:  294deg 15' 12.8" (294.2536)
Mean longitude Meeus:     294deg 36' 56.9" (294.6158)

I am grateful if you can give me some advice.