**21 Dec is the shortest day of the year and 4 Jan is the longest.**
We all know that the day of the winter solstice is considered by many the shortest day of the year, but did you know that we can also consider the day of perihelion (4 Jan) as the longest day? It all comes down to how you define the length of a day.
**How long is a day?**
Ask
someone how long a day is and you will probably get the answer, "24
hours". However, ask someone at the US Naval Observatory and you will
get "1420x10^{6}x60x60x24 vibrations of a hydrogen maser". Ask a
farmer and you will get "From sunup to sunset". We have any different
definitions. For this post, I am going to use the synodic definition,
the time it takes the Sun to go from due south (local solar noon) to the
next time it is due south.
**It starts with Kepler.**
** **Thanks
to Johannes Kepler, we know that the Earth orbits the Sun in an
eliptical orbit, and that the Earth is moving fastest when it is closest
to the Sun (perihelion) and slowest when it when it is farthest (aphelion).
This year,

perihelion occurs on 4 Jan, which will be the longest synodic day of the year.

**Let's figure it out.**
Since
on the above diagram, the Earth is also spinning counter-clockwise, the
diagram below shows (with the added lines) one complete rotation of the
Earth for both the day of aphelion (3-4) and the day of perihelion
(1-2). Note that the diagrams are exaggerated for effect; the Earth does not really move this far in it orbit in one day.

Notice
that it takes a little more than one rotation of the Earth to have the
Sun directly due south again. Also notice that the Earth has to rotate a
little more at perihelion than at aphelion to be facing the Sun 1
synodic day later. So a synodic day is longest at perihelion and shortest at aphelion.

**But how much longer is it?**
** **
While
the above argument is enough to show that 4 Jan 2014 is the longest
(synodic) day of the year, it does not say by how much. Is it minutes
or milliseconds? To answer this, we will need some numbers and the more
precise form of Kepler's Law used above. Kepler figured out that
orbiting bodies sweep out equal areas in equal time. And by looking at
the diagram above, you can see that the angle swept out by the Earth in
one

sidereal day is equal to the extra angle the Earth has to rotate to get the Sun due south.

Since the area of a circular sector is

this gives the relationship between for angles swept in equal time.

The Earth-Sun distance is 152,098,232 kilometres (or
1.01671388 AU) at aphelion and is 147,098,290 kilometres (or 0.98329134 AU) at perihelion. This gives

**Microseconds or Hours?**
So, the Earth rotates
around the Sun just a little bit less on 5 July than on 22 December.
Since the Earth rotates about 1 degree (actually 360/364 degrees), it
rotates about 0.033 degrees less on 5 July for 1 synodic day than on 4
January. How long does it take the Earth to rotate that angle? Since it
takes 24 hours to rotate 360 degrees, it takes about 8 seconds to rotate
that 0.033 degrees. So the synodic day on 4 January is about 8 seconds longer than the synodic day on 5 July.

**Classroom use.**
Unless
you are teaching summer school, this is not much use in the classroom.
However, if you flip the seasons, you can show that the day of
perihelion is the longest day of the year (and it is close the the
"shortest" day of the year, 22 Dec). Since the first day back from the
Christmas break is close to 4-5 Jan, I use this for a gradual
welcome-back class. In a regular physics class, we just do the
qualitative approach with students playing the parts of the Sun and the
Earth.

The discussion starts with what causes the
seasons. It helps to have a globe handy at this point. We then discuss
Kepler's laws. I get one student to play the part of the Sun and one the
Earth. Depending on time, I might have the class discuss how the Earth
should rotate (the Sun rises in the east). If not, I will just get the
Earth rotating in the correct way. The class decides when the Earth has
rotated one complete rotation relative to the room and then notices
that it is the Earth (the student) is not facing the Sun. That 5 minute
activity sets up the above discussion.

A related post (

http://canisiusphysics.blogspot.com/2013/03/if-spring-starts-tuesday-why-does.html)