Day and Night
Porter Johnson

The earth moves in a nearly circular orbit about the sun, in which the motion is seen as counterclockwise (right-handed) from above; i.e., the direction of the earth's orbital angular momentum. The earth rotates counterclockwise about a fixed axis that is tilted at an angle a » 23^o with respect to this "above" direction. To represent the earth and the sun, let us define a Cartesian coordinate system in which the earth moves in the x-y plane (the ecliptic), and the above direction corresponds to the +z-direction. We may further specify that the axis of rotation of the earth lies in the x-z plane, and is given by the unit vector A:

Let us set up a geocentric coordinate system, so that the sun appears to rotate (also counterclockwise) about the earth in a roughly circular orbit. We may specify the position of the sun by giving the angle between its location, s, with respect to the  negative x-axis. Thus the vector going from the earth to the sun is given by the unit vector S:

Note: we have chosen s so that the sun is in the negative-x direction with respect to the earth at s = 0, and since the polar direction A has a positive x-component, the north pole is on the other side of the earth from the sun. Here is the relation between s and the seasons:

• Winter: 0 < s < p/2
• Spring: p/2 < s < p
• Summer p < s < 3 p/2
• Autumn: 3 p/2 < s < 2 p

Every point on earth at co-latitude q lies at this direction from the center of the earth once per day, and as the earth rotates about its axis A, the component parallel to A remains fixed, whereas the component perpendicular to A changes.

Let us introduce the angle j as the angle between direction of the unit vector Z from the center of the earth and the vector Z_o^. One may write Z as the following linear combination of Z_o||, Z_o^, and the vector Y_o = A ´ Z_o = - sinq (0, 1, 0):

We may express the unit vector Z in terms of its components:

As the earth rotates about its axis with 24 hour period, each point of co-latitude q (relative to the center of the earth) is at direction Z, where j passes through all values between 0 and 2 p.

For what fraction of the 24 hours is it daylight, and for what fraction is it night? One may answer that question by determining the angle between the unit vectors Z and S. If the angle between those vectors is less than 90^o then it is daylight [the sun is in our sky], whereas if that angle is greater than 90^o it is dark because the earth blocks the sun from us. We may state the daylight condition simply by requiring the scalar product of Z and S to be positive:

We may thus express the daylight condition as the following inequality:

We may write this condition as

• B₁ = coss sinq  cosa
• B₂ = sins sinq
• C =  coss cosq  sina

The parameters B₁, B₂, and C are defined in terms of the co-latitude location q , the time of year s, and the inclination a of the earth's axis.

Let us define the angle j_o by the relations

Let us define the angle b by the relation

and write

Note that, because coss = cos(2p - s),  is symmetric about s = p;  T(s, q) = T(2p - s, q). In other words, the daylight period is symmetric about the summer solstice, s = p.

Note also that, if s ® p - s , the period of daylight undergoes the transformation T® 24 - T. That is, the amount of daylight less 12 hours is anti-symmetric about the vernal equinox, s = p/2.

Let us consider several special cases:

• Vernal equinox: s = p/2; Autumnal equinox: s = 3 p/2.
  Note that the length of day is precisely 12 hours at all latitudes on these days.
• Equator: q= p/2.
  Note that 1 / tans = 0, so that the length of daylight is precisely 12 hours at all times of the year.
• North Pole: q = 0; South Pole: q = p.
  Note that there is 6 months of continuous sunlight, followed by 6 months of continuous darkness, at the poles.
• Arctic Circle: q = a
  T(s, q = a) = 24 / p cos^-1[ coss cosa /Ö(1 - sin²a cos²s) ]
  On the Arctic Circle it varies continuously between complete darkness at the winter solstice s = 0 and complete daylight at the summer solstice s = p.

Here is a table of the number of daylight hours as a function of the time of year for the first 26 weeks of the solar year, for several co-latitudes  in the Northern Hemisphere:

This angle may be written in the form W = q + b, where, as before,

Here is a graph of the Zenith angle at various co-latitudes in the Northern Hemisphere [note that when the angle is greater than  90^o the sun remains below the horizon] at various times of the year:

The vector Z is always vertical; since it is perpendicular to the surface of the earth at any time and place. It is convenient to define vectors that lie parallel to the surface of the earth, corresponding to the directions East: E and North: N. Note that the direction of the polar axis, A has a vertical component, as well as a North component, at any point, and the angle between A and Z is our co-latitude q. Thus, the vector produce A ´ Z has magnitude sin q and lies East. Furthermore the triple product Z ´ (A ´ Z) also has magnitude sinq and lies in the direction of N. That is

Here are the unit vectors Z, E, N expressed in component form:

We can determine the location of the sun  in the sky at any point on earth at any time by expressing the solar location vector S in terms of the unit vectors Z, E, N:

We have already calculated the vertical component, S · Z. The North component S_N is given by

whereas the East component S_E is 

In particular, at sunrise-sunset the vertical component S · Z is zero.

The direction of the rising or setting sun is dependent on of co-latitude, in that the sun does not always rise precisely in the East and does not always set precisely in the West.  In fact, there is substantial variation in the location of the rising sun.  The following table indicates the direction of the rising sun, as given by the number of degrees North [positive] or South [negative] of East for various co-latitudes in the Northern Hemisphere: