Latitude
Gemini ai on the Copilot equation:
https://share.gemini.google/5qZ0lWWl60ru
* * *
* * *
Washington daylight hours from the Microsoft Weather App:
With respect of Montreal and Edmonton longitude calculations...
Copilot:
Does this formula assume the Earth is a sphere?
Yes. The standard formula I showed assumes the Earth is a perfect sphere.
In reality, the Earth is slightly flattened at the poles and bulges at the equator, making it an oblate spheroid. The equatorial radius is about 21 km larger than the polar radius.
For sunrise and sunset calculations, however, the spherical assumption works remarkably well because:
The Earth's radius is about 6,371 km.
The flattening is only about 0.3%.
The resulting error in sunrise or sunset times is typically much smaller than other effects, such as atmospheric refraction.
In fact, the largest source of uncertainty is often the atmosphere. Variations in temperature and pressure can shift the apparent sunrise or sunset by a minute or more because they cange how strongly light bends near the horizon.
Modern astronomical software therefore includes corrections for:
Earth's actual shape,
atmospheric refraction,
observer elevation,
the Sun's finite size,
and the precise Earth-Sun geometry on a given date.
An interesting consequence is that if you stand on a hill or in a tall building, you can see the Sun for longer than someone at sea level because your horizon is farther away. From the top of a 100 m hill, sunrise occurs about a minute earlier and sunset about a minute later than at sea level.
For a place like St-Jean-sur-Richelieu (about 45°N), treating the Earth as a sphere gives sunrise and sunset times that are extremely close to the values published by weather services. The atmosphere usually matters more than the Earth's slight flattening.
In a sense, the spherical model is to astronomers what a straight line is to engineers: not perfectly true, but often accurate enough to get very useful answers.
* * *
Quick review of Latitude and Longitude:
The roofers are here, with very large and impressive equipment.
They are starting at my end of the building, and I am on the top floor.
The old roof is being torn off and it is dramatic. All this to say I will share the
main reference points for the math I was planning to blog about today but
leave the discussion for tomorrow. No puttering around the place for me today!
Yesterday I looked at Longitude, the East/West position of Montreal and Edmonton,
Alberta.
360 degrees, divided over 24 hours, gives 15 degrees. So knowing the Longitude
of both cities allows us to see they are 2 hours and 40 minutes apart.
But to discuss patterns of enlightenment, we need Latitude, the North/South component.
And that is more complicated because the great sphere that represents Earth is itself
bent by 23 degrees and its position with respect to the Sun changes with time of the
year and time of day.
Below, the formula proposed by Copilot:
Gemini's explanation:
https://share.gemini.google/Hr9g5bHDbZXD
* * *
Gemini also gave me a Python script which allowed me to generate daily enlightenment
for a city from its position numbers. The daylight hours are blue curve minus the orange
for any given time.
The code:
import matplotlib.pyplot as plt
import numpy as np
# --- 1. CONFIGURATION & LOCATION PARAMETERS ---
latitude_deg = 45.31 # Saint-Jean-sur-Richelieu latitude
longitude_deg = -73.25 # Saint-Jean-sur-Richelieu longitude (West is negative)
lat_rad = np.radians(latitude_deg)
tz_offset = -4.0 # Forced EDT (UTC-4) year-round
# --- 2. ANNUAL SOLAR CALCULATIONS (365 Days) ---
n = np.arange(1, 366) # Day of year (1 to 365)
gamma = (2 * np.pi / 365) * (n - 1) # Fractional year angle
# Solar Declination (radians) - Spencer (1971) approximation
delta = (
0.006918
- 0.399912 * np.cos(gamma)
+ 0.070257 * np.sin(gamma)
- 0.006758 * np.cos(2 * gamma)
+ 0.000907 * np.sin(2 * gamma)
- 0.002697 * np.cos(3 * gamma)
+ 0.00148 * np.sin(3 * gamma)
)
# Equation of Time (minutes)
eot = 229.18 * (
0.000075
+ 0.001868 * np.cos(gamma)
- 0.032077 * np.sin(gamma)
- 0.014615 * np.cos(2 * gamma)
- 0.040849 * np.sin(2 * gamma)
)
# Solar Noon (UTC)
# Corrected: West longitude adds time to UTC because Montreal is behind Greenwich
solar_noon_utc = 12.0 + (-longitude_deg / 15.0) - (eot / 60.0)
# Hour Angle ($\omega_s$) accounting for refraction (-0.833 degrees)
alt_rad = np.radians(-0.833)
cos_omega = (np.sin(alt_rad) - np.sin(lat_rad) * np.sin(delta)) / (
np.cos(lat_rad) * np.cos(delta)
)
cos_omega = np.clip(cos_omega, -1.0, 1.0)
omega_s = np.degrees(np.arccos(cos_omega))
# --- 3. TIME CONVERSION ---
sunrise_utc = solar_noon_utc - (omega_s / 15.0)
sunset_utc = solar_noon_utc + (omega_s / 15.0)
# Convert UTC to uniform EDT (UTC-4)
sunrise_local = sunrise_utc + tz_offset
sunset_local = sunset_utc + tz_offset
# --- 4. PLOTTING ---
fig, ax = plt.subplots(figsize=(10, 6), dpi=100)
ax.plot(
n, sunrise_local, color="#d97706", linewidth=2, label="Sunrise (EDT Year-Round)"
)
ax.plot(
n, sunset_local, color="#4f46e5", linewidth=2, label="Sunset (EDT Year-Round)"
)
ax.fill_between(
n,
sunrise_local,
sunset_local,
color="#fef3c7",
alpha=0.5,
label="Daylight Span",
)
ax.set_title(
f"Annual Sunrise & Sunset Curves — EDT Year-Round ({latitude_deg}°N)",
fontsize=12,
pad=15,
)
ax.set_xlabel("Day of Year", fontsize=10)
ax.set_ylabel("Local Time (Hours)", fontsize=10)
month_days = [1, 32, 60, 91, 121, 152, 182, 213, 244, 274, 305, 335]
month_labels = [
"Jan",
"Feb",
"Mar",
"Apr",
"May",
"Jun",
"Jul",
"Aug",
"Sep",
"Oct",
"Nov",
"Dec",
]
ax.set_xticks(month_days)
ax.set_xticklabels(month_labels)
ax.set_xlim(1, 365)
ax.set_ylim(4, 23)
ax.set_yticks(np.arange(4, 24, 2))
ax.grid(True, linestyle="--", alpha=0.5)
ax.legend(loc="upper center", frameon=True)
plt.tight_layout()
plt.show()
For St-Jean-sur-Richelieu:
* * *
And because a bit of fun is always welcome as one trudges through Math, the Wikipedia
page on 'Terminator', a concept in this scientific field:
Terminator (solar) - Wikipedia
Just changed all the clocks in the appartment, including my watch. The
new Quebec government has said we are doing this for the last time. They are going
to let a panel of experts advise on what time system we should be going to.
If health experts decide, I would expect standard time. If it is a popular choice,
it might be Daylight Savings. Ontario would like to align with Manitoba to go
with Daylight, but is waiting for Quebec and New York State...
Copilot:
Has New York State indicated what it would like on this matter?
Yes. The general position of New York State has been quite similar to Ontario's: many state legislators have expressed support for permanent daylight saving time (keeping the "summer" clock year-round and eliminating the twice-yearly clock changes). However, New York has not actually made the change and continues to observe the normal daylight-saving schedule.
In fact, New York has considered legislation that would allow permanent daylight saving time if federal law permits it. One such bill, Senate Bill 3380, was still pending as of 2026
The complication is that U.S. federal law plays a role. States can opt out of daylight saving time entirely and stay on standard time year-round (as Arizona and Hawaii largely do), but a state generally cannot switch unilaterally to permanent daylight saving time without a change in federal law.
This is why Ontario has been watching developments south of the border. If New York were able to move to permanent daylight saving time, Ontario would be much more likely to activate its own legislation so that Toronto, Ottawa, Montreal, and New York City remained aligned.
An interesting wrinkle is that not everyone agrees on which permanent time is best:
Permanent daylight saving time means more evening light in winter.
Permanent standard time means earlier sunrises in winter.
For places as far north as southern Ontario, Quebec, and New York, permanent daylight saving time would push some winter sunrises quite late. In New York City, for example, the latest winter sunrise would be about 8:20 a.m. under permanent daylight saving time.
So, to answer your question succinctly: New York's political discussion has generally leaned toward permanent daylight saving time rather than permanent standard time, but neither New York nor the U.S. federal government has yet completed the legal changes needed to make that happen.
Didn't the original move to Daylight originatein the US? Are the countries elsewhere than North America that use it?
No, the original idea did not originate in the United States.
The history is a bit surprising:
In 1784, Benjamin Franklin (while in Paris) wrote a humorous essay suggesting people could save candles by getting up earlier. He did not propose changing the clocks.
In 1895, George Hudson, a New Zealand entomologist, made one of the first serious proposals for a seasonal clock change.
In 1907, British builder William Willett became a prominent advocate of the idea in the United Kingdom.
The first country to adopt daylight saving time nationally was Germany (with Austria-Hungary) during the First World War in 1916, largely to conserve fuel and electricity. Britain followed shortly afterward, and the United States adopted it in 1918.
As for the rest of the world, yes, daylight saving time is used well beyond North America, though not nearly as widely as many people think.
Today it is mainly concentrated in:
Most of Europe (including France, Germany, Spain, Italy, the UK, etc.)
The United States and most of Canada
Parts of Australia and New Zealand
A few countries in the Middle East, such as Israel and Lebanon.
Chile and some other jurisdictions in South America.
What is perhaps more interesting is that most of the world does not use daylight saving time at all. Most of Asia, most of Africa, and many countries in South America either never adopted it or abandoned it. Countries near the equator have little reason to use it because daylight hours change very little over the year.
In fact, the long-term trend has been away from clock changes. Countries such as Russia (2014), Brazil (2019), and much of Mexico (2022) have abolished daylight saving time.
One irony is that while many Canadians think of the spring and autumn clock changes as a universal practice, they're actually a feature mainly of the mid-latitude industrialized countries of Europe and North America. Most people on Earth never change their clocks at all.
But both Britain and France switch during the year?
Yes. Both Britain and France still switch their clocks twice a year.
In the United Kingdom, clocks move forward by one hour in spring (to British Summer Time, BST) and back in autumn (to Greenwich Mean Time, GMT).
In France, clocks move forward from Central European Time (CET) to Central European Summer Time (CEST) in spring and back in autumn.
An interesting point is that the UK and France make the change on the same dates, so France remains one hour ahead of Britain throughout the year.
The European Union discussed abolishing the clock changes several years ago. The European Parliament voted in favour of ending them, but the proposal stalled because countries could not agree on whether to keep permanent summer time or permanent standard time. As a result, the twice-yearly clock changes remain in place across most of Europe.
For someone in Québec, the surprising thing is that France changes its clocks on almost the same schedule as we do, despite being an ocean away. When Québec "falls back" in late autumn, France and Britain do too, although not always on exactly the same weekend.
In fact, if you've ever travelled to France in winter, you've experienced something slightly odd: Paris is much farther north than Montréal, yet because of France's time zone and daylight-saving arrangements, winter sunsets can occur surprisingly late by European standards. That is one reason the debate over permanent summer time versus permanent standard time remains lively in both France and Britain.
So here we are: Alberta and Quebec are three hours apart in physical terms (2 hours and 40
minutes), but currently clocking in at two in the summer and one in the winter:
Time zones in Canada are getting complicated. These maps lay them out | CBC News* * *
On a personnal note, the body has its own way of adapting to things.
In my older years, I have set fixed rules for myself, and one has to do with
breakfast time. I can have hot drinks when I get up, but breakfast is no sooner
than 10 AM. So that during Daylight Savings Time periods, I can eat at 9 AM Standard.
But guess what, in the Fall I am not hungry at 10 AM, or am too busy to notice.
Lately, I have been eating fruit and dry toast around 11 AM Daylight. Which is 10 AM
Standard. So my body has been gradually keeping up with my 10 AM rule and I am
marginally hungrier into the winter months!!
Yes. The general position of New York State has been quite similar to Ontario's: many state legislators have expressed support for permanent daylight saving time (keeping the "summer" clock year-round and eliminating the twice-yearly clock changes). However, New York has not actually made the change and continues to observe the normal daylight-saving schedule.
In fact, New York has considered legislation that would allow permanent daylight saving time if federal law permits it. One such bill, Senate Bill 3380, was still pending as of 2026.
The complication is that U.S. federal law plays a role. States can opt out of daylight saving time entirely and stay on standard time year-round (as Arizona and Hawaii largely do), but a state generally cannot switch unilaterally to permanent daylight saving time without a change in federal law.
This is why Ontario has been watching developments south of the border. If New York were able to move to permanent daylight saving time, Ontario would be much more likely to activate its own legislation so that Toronto, Ottawa, Montreal, and New York City remained aligned.
An interesting wrinkle is that not everyone agrees on which permanent time is best:
For places as far north as southern Ontario, Quebec, and New York, permanent daylight saving time would push some winter sunrises quite late. In New York City, for example, the latest winter sunrise would be about 8:20 a.m. under permanent daylight saving time.
So, to answer your question succinctly: New York's political discussion has generally leaned toward permanent daylight saving time rather than permanent standard time, but neither New York nor the U.S. federal government has yet completed the legal changes needed to make that happen.
Made a cake for Halloween. something I haven't done in a while. It
looks a tad weird and it is: I had no instant coffee so put in actual coffee
in the chocolate layer, which was then more liquid than the other... see
the result below. But it tastes great, isn't too caloric and it is festive.
I also didn't have chocolate chips for the topping, but have been eating it
with fruit and yogurt and chomping on chocolate candy now and again.
Even Copilot approves!!
So is Canada the sleazy one, signing an accord with the EU just as
things migt be picking up again on a deal with the US. Because one cannot
really have both going at the same time. If one did, one could just buy things in
one market, and sell them in another: for example, I buy up pumpkins from the US
and sell maple syrup with pumpkin in Paris, and all the while the US pumkin
farmer can only sell with tariffs attached in that market. Hooooo!
Looking at our other commitments was just one thing Team Canada woudn't let
the US do. Or maybe this wasn't a totally innocent request. In any event, we do
appear to be making a choice for the EU...
