During June 21-22, determine the length of daylight at the following latitudes. Any help is greatly appreciated.

latitudes
---------------
70 degrees north
40 degrees south
40 degrees north
90 degrees south
0 degrees

http://en.wikipedia.org/wiki/Sunrise_equation

To determine the length of daylight at different latitudes during June 21-22, we need to consider the summer solstice, which is the longest day of the year in terms of daylight hours.

On the summer solstice (June 21 or 22), the Earth's axial tilt causes the North Pole to be tilted towards the Sun, while the South Pole is tilted away from the Sun. As a result, latitudes closer to the North Pole experience long daylight hours, while latitudes closer to the South Pole have shorter daylight hours.

To calculate the length of daylight for each latitude, we can use various sources, such as online calculators or astronomical software. However, I'll provide a method using a simple formula that approximates the daylight hours for each latitude during the summer solstice.

The formula to estimate the length of daylight at a given latitude on the summer solstice is:

Daylight hours = 2 * (cos^(-1))(−(tan L) * tan δ))

Where:
- L is the latitude in radians.
- δ is the solar declination (angular distance of the sun above or below the celestial equator).

Let's calculate the length of daylight at each latitude:

1. 70 degrees north:
- Convert 70 degrees to radians: 70 * (π/180) = 1.22 radians.
- The solar declination on the summer solstice is approximately +23.5 degrees.
- Substitute the values into the formula: Daylight hours = 2 * (cos^(-1))(−(tan 1.22) * tan 23.5)

2. 40 degrees south:
- Convert 40 degrees to radians: 40 * (π/180) = 0.70 radians.
- The solar declination on the summer solstice is approximately -23.5 degrees (negative since it's south of the equator).
- Substitute the values into the formula: Daylight hours = 2 * (cos^(-1))(−(tan 0.70) * tan -23.5)

3. 40 degrees north:
- Convert 40 degrees to radians: 40 * (π/180) = 0.70 radians.
- The solar declination on the summer solstice is approximately +23.5 degrees.
- Substitute the values into the formula: Daylight hours = 2 * (cos^(-1))(−(tan 0.70) * tan 23.5)

4. 90 degrees south:
- 90 degrees south refers to the South Pole, where it experiences 24 hours of daylight during the summer solstice.

5. 0 degrees:
- 0 degrees is the equator, which generally experiences roughly 12 hours of daylight throughout the year due to its proximity to the celestial equator.

Note that these calculations provide an approximation, and actual daylight hours may vary slightly due to factors such as atmospheric conditions and local geography.

If you need precise calculations for a specific date and time, I recommend using online calculators or astronomical software that takes into account various factors affecting daylight hours.