Sunrise
This fairly mindless thought experiment came out of one of those Middle East to HKG through the night when somebody needs to know the time of sunrise and can't just look out the window and wait for the sky to brighten. Unfortunately Josh had to suffer me trying to figure this en-route so my apologies to him. This is a corrected version of what we came up with.
You can do this as a visual graph which works pretty well using the back of a NOTAM sheet, the edge of the checklist and the distance scale off the edge of the Jepp map. First draw yourself to vertical axis at the left and right edge of the page with an X axis between the 2. The left hand Y axis is your departure airport, the right hand Y axis is your destination airport and the X axis is your flight time (FT) between the 2.
Label your 2 Y axis with Z time running up such that your take-off time (TOT) can appear on the left axis and the time of sunrise on the right axis; this you can get from the TIME section of the Flight Info Supp or PDA/PC software. Mark both these times on their respective axis.
Now comes the more tricky bit. First we need a bit of maths; numbers to find are your flight time and longitude of your departure and destination airport. From this figure the change of longitude, ΔLONG. Now calculate δ = ΔLONG/(15 x FT); this is the slope of a line you need to draw.
Now draw 2 lines. The first starts on the left axis at the departure time from the departure airport. This line has a slope of one - in other words it goes up one hour for every hour down route. The second line on the right axis from the sunrise time at the destination airport; it has a slope of δ. This means that it rises by δ hours for every hour across the X axis.
These 2 slopes will intercept. From the interception point, move left/right to the Y axis and read of the sunrise time.
But first we need to correct it for a couple of errors. Since you are sitting around 35kft, the effective sunrise will be earlier than at ground level. If you are keen, you can figure the accurate value yourselves.) I've changed my thinking in this area since I started on this experiment but I'm a little uncomfortable about this simplification; I think that the value is dependent both on latitude and time of year too.)
At higher latitudes the effective radius of the Earth reduces so the horizon distance decreases. All other things being equal, sunrise in the Tropics will be earlier at altitude than further north/south. A simple piece of trig using the cosine law suggests that the furthest you will be from the horizon is approx 3.3°. Obviously in Winter in the Arctic Circle, the sun won't rise. With the Earth spinning at 15°, 3.3° gives about 13 mins of gain. I cannot see an easy way to accurately approximate this. Ideas?
Finally you need to be aware that the sunrise is affected by latitude; for a given day, the sunrise in winter is later, the further north you are. The converse is true. Now it's a bit hard to give a rule of thumb but 2 mins per degree near the summer/winter solstice and zero mins near the spring/autumn equinox will do us. For the rest of the year, feel free to interpolate. So you need to correct your estimate for the latitude difference between your destination airport and your estimated position at sunrise.
Thus heading south from HKG to SYD in January, you calculate an altitude corrected time of (say) 1955Z. Review the CFP to get an approx latitude of 5° south at that time, then correct sunrise by the latitude difference (34-5=) 29 x 2 = 58 mins to delay estimated sunrise until 2053Z.
Now you have your answer. A rough and ready graphic looks like this:
For those preferring a straight mathematical solution, the elapsed time after take-off to the point of sunrise is (SR - TOT + δxFT) / (1+δ) where SR is the sunrise at destination and TOT is the take-off time. Just add the answer to TOT. Correct for altitude and latitude. Easy, eh? QED
Created over Christmas 2006