When are the days getting longer the fastest?

We're way past the winter solstice, and approaching the equinox. The sun is noticeably staying up later and later every day, which raises an obvious question: when are the days getting longer the fastest? Intuitively I want to say it should happen at the equinox. But does it happen exactly at the equinox? I could read up on all the gory details of this, or I could just make some plots. I wrote this:

#!/usr/bin/python3

import sys
import datetime
import astral.sun

lat  = 34.
year = 2025

city = astral.LocationInfo(latitude=lat, longitude=0)

date0 = datetime.datetime(year, 1, 1)

print("# date sunrise sunset length_min")

for i in range(365):
    date = date0 + datetime.timedelta(days=i)

    s = astral.sun.sun(city.observer, date=date)

    date_sunrise = s['sunrise']
    date_sunset  = s['sunset']

    date_string    = date.strftime('%Y-%m-%d')
    sunrise_string = date_sunrise.strftime('%H:%M')
    sunset_string  = date_sunset.strftime ('%H:%M')

    print(f"{date_string} {sunrise_string} {sunset_string} {(date_sunset-date_sunrise).total_seconds()/60}")

This computes the sunrise and sunset time every day of 2025 at a latitude of 34degrees (i.e. Los Angeles), and writes out a log file (using the vnlog format).

Let's plot it:

< sunrise-sunset.vnl                   \
  vnl-filter -p date,l='length_min/60' \
| feedgnuplot                          \
  --set 'format x "%b %d"'             \
  --domain                             \
  --timefmt '%Y-%m-%d'                 \
  --lines                              \
  --ylabel 'Day length (hours)'        \
  --hardcopy day-length.svg

Well that makes sense. When are the days the longest/shortest?

$ < sunrise-sunset.vnl vnl-sort -grk length_min | head -n2 | vnl-align

#  date    sunrise sunset     length_min   
2025-06-21 04:49   19:14  864.8543702000001


$ < sunrise-sunset.vnl vnl-sort -gk length_min | head -n2 | vnl-align

#  date    sunrise sunset     length_min   
2025-12-21 07:01   16:54  592.8354265166668

Those are the solstices, as expected. Now let's look at the time gained/lost each day:

$ < sunrise-sunset.vnl                                  \
  vnl-filter -p date,d='diff(length_min)'               \
| vnl-filter --has d                                    \
| feedgnuplot                                           \
  --set 'format x "%b %d"'                              \
  --domain                                              \
  --timefmt '%Y-%m-%d'                                  \
  --lines                                               \
  --ylabel 'Daytime gained from the previous day (min)' \
  --hardcopy gain.svg

Looks vaguely sinusoidal, like the last plot. And looks like we gain/lost as most ~2 minutes each day. When does the gain peak?

$ < sunrise-sunset.vnl vnl-filter -p date,d='diff(length_min)' | vnl-filter --has d | vnl-sort -grk d | head -n2 | vnl-align

#  date       d   
2025-03-19 2.13167


$ < sunrise-sunset.vnl vnl-filter -p date,d='diff(length_min)' | vnl-filter --has d | vnl-sort -gk d | head -n2 | vnl-align

#  date        d   
2025-09-25 -2.09886

Not at the equinoxes! The fastest gain is a few days before the equinox and the fastest loss a few days after.