Equation_of_Time

 Equation of Time:

Help me make sense of this: I have read that, because the Earth's rotation around the Sun is somwhat oval, the sun sets a bit longer before the winter Solciste!?

(source:Meteo Media)

Quick Answer: Yes, you’re right! The earliest sunset of the year actually happens a few days before the winter solstice. This isn’t just because Earth’s orbit is slightly oval (elliptical), but also because of how our clocks interact with the Sun’s apparent motion — a phenomenon called the equation of time.

🌍 Why the Earliest Sunset Isn’t on the Solstice

Winter Solstice basics: Around December 21, the Northern Hemisphere has its shortest day of the year. The Sun follows its shortest arc across the sky, giving the least daylight.

But… not the earliest sunset: The earliest sunset usually happens about 1–2 weeks before the solstice (for example, around December 8–10 in many mid-latitude cities).

Latest sunrise mismatch: Similarly, the latest sunrise happens after the solstice, in early January.

🕰 The Equation of Time

This is the key to understanding the mismatch:

Earth’s orbit is elliptical and Earth’s axis is tilted.

Because of this, the Sun doesn’t cross the sky at a perfectly uniform pace.

Our civil clocks run on mean solar time (averaged out), but the Sun’s apparent motion follows true solar time.

The difference between these two is the equation of time, which shifts sunrise and sunset times relative to the solstice.

📅 What This Means in Practice

In the Northern Hemisphere:

Earliest sunset: ~Dec 4–12 (varies by latitude).

Winter solstice (shortest day): ~Dec 21–22.

Latest sunrise: ~Jan 2–8.

So, even though the solstice is the shortest day, the timing of sunrise and sunset doesn’t line up neatly with it.

✨ How to Visualize It

Think of it like this:

The solstice is about day length (shortest daylight).

The earliest sunset is about clock time drift (when the Sun dips below the horizon earliest).

These don’t coincide because Earth’s orbit and tilt make solar time “run fast” or “slow” compared to our clocks.

In short: The Sun sets earlier before the solstice because of the combined effects of Earth’s elliptical orbit and axial tilt, which cause a mismatch between solar time and clock time. That’s why the darkest day (solstice) isn’t the same as the earliest sunset.

Would you like me to show you a visual diagram of how the equation of time shifts sunrise and sunset around the solstice? It makes this much easier to grasp.

But the total number of light hours in a day is as expected...

Exactly — you’ve nailed the subtlety here 🌞.

The total daylight duration (the number of hours between sunrise and sunset) is governed purely by the Sun’s declination (its position north or south of the celestial equator) and your latitude. That’s why the shortest day of the year — the winter solstice — really does have the least daylight, no matter what quirks the orbit introduces.

What shifts is the clock time of sunrise and sunset, not the total daylight span. Here’s why:

⏳ Two Different Effects at Play

Day length (light hours): Determined by Earth’s tilt and the Sun’s declination. This is what makes the solstice the shortest day.

Clock time of sunrise/sunset: Influenced by the equation of time (the mismatch between solar time and our civil clocks). Because Earth’s orbit is elliptical and tilted, the Sun doesn’t move evenly across the sky relative to our clocks.

📅 The Result

Around early December, the Sun is still setting earlier each day, even though the daylight span is shrinking toward its minimum.

By the solstice, the daylight span is shortest, but the sunset has already started shifting later again.

Sunrise compensates by continuing to get later into January, so the total daylight hours line up exactly as expected.

🌍 A Simple Analogy

Think of it like this:

The solstice is about the length of the rope (daylight hours).

The equation of time is about how the rope is shifted left or right on the clock.

So yes — the total daylight hours behave exactly as expected, but the distribution of those hours (when sunrise and sunset happen) gets nudged around by orbital mechanics.

Would you like me to show you a table of actual sunrise/sunset times for your location (St-Jean-sur-Richelieu, QC) around December, so you can see the earliest sunset vs. the solstice vs. the latest sunrise? It makes the pattern crystal clear.

# Generating equation of time plot for the year 2025 with monthly x-axis labels

                                                                                       

I’ve generated the visualization above. It shows how many minutes the Sun runs “ahead” or “behind” clock time through 2025. Positive values mean the Sun is ahead of mean solar time; negative means it’s behind.

What you’re seeing

Amplitude: The swing is about ±16 minutes across the year.

Major extrema (approximate):

Early November: peak around +16 minutes (Sun “early”).

Mid–late February: trough around −14 minutes (Sun “late”).

Late July: smaller trough around −6 minutes.

Mid–late April: smaller peak around +4 minutes.

Shape: The curve isn’t symmetric; it’s the sum of two effects that don’t align perfectly in phase.

Why this happens

Axial tilt (obliquity): The projection of Earth’s tilted axis onto the ecliptic makes the apparent solar day vary seasonally.

Orbital eccentricity: Earth’s speed in its elliptical orbit isn’t constant; solar time slips relative to a uniform clock.

Together, they create the familiar lopsided figure‑eight (analemma) and this year-long timing offset.

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import numpy as np

import matplotlib.pyplot as plt

from datetime import datetime, timedelta

# Use matplotlib style for better aesthetics

plt.style.use('seaborn-v0_8')

# Generate dates for each day of 2025

start_date = datetime(2025, 1, 1)

end_date = datetime(2025, 12, 31)

days = (end_date - start_date).days + 1

dates = [start_date + timedelta(days=i) for i in range(days)]

day_of_year = np.array([date.timetuple().tm_yday for date in dates])

# Calculate the Equation of Time (EoT) using an approximation formula

# EoT in minutes = 7.5 * sin(B) - 9.87 * sin(2B + 24.99°)

B = 2 * np.pi * (day_of_year - 81) / 364

eot = 9.87 * np.sin(2 * B) - 7.53 * np.cos(B) - 1.5 * np.sin(B)

# Plotting

fig, ax = plt.subplots(figsize=(10, 6))

ax.plot(dates, eot, color='darkblue', linewidth=2)

ax.axhline(0, color='gray', linestyle='--', linewidth=1)

# Formatting the x-axis to show months

months = [datetime(2025, m, 15) for m in range(1, 13)]

month_labels = [date.strftime('%b') for date in months]

ax.set_xticks(months)

ax.set_xticklabels(month_labels)

# Labels and title

ax.set_title('Equation of Time for 2025', fontsize=16)

ax.set_xlabel('Month', fontsize=12)

ax.set_ylabel('Time Difference (minutes)', fontsize=12)

ax.grid(True)

# Save the plot

output_path = "/mnt/data/equation_of_time_2025.png"

plt.savefig(output_path)

plt.show()

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Asked Copilot to take me through the math: