Astronomy, Trigonometry, and the Fajr Call to Prayer

Well VC was just (I say "just" even though I began writing this post more than 90 minutes ago) awakened by one of the only calls to prayer he's heard since coming to Kabul -- there is none of the ruckus and overlap that characterizes the Cairo prayer calls, and that has caused to much controversy in Egypt (see this BBC article for more info).

Anyway, it was surprising to hear Allahu Akbar not once, but THREE times, and not at the usual 4:30AM-ish time (depending on the time of year) but at 3:20, 3:30, and 3:50!

Which of the muzzeins was correct? What is the proper call to prayer time for this time of year in this part of the world?

Place your bets, as we insomniac chunks go through a rigorous calculation of the REAL call to prayer time!

Because this website warns of the use of online programs to automatically give you the call to prayer time that might ignore certain geospatial positioning issues, we are going to calculate the morning call to prayer time for today, 10 August 2005, for Kabul ourselves! This call to prayer is known as “Fajr” and will be calculated according to the following equations borrowed from this website:

Fajr = Z - V

Z = 12 + (R-L)/15 + T/60

V = arcos[(-sinG – sinDsinB)/cosDcosB]/15

Where:

B= latitude of place
L= longitude of place
R= reference longitude (i.e. TIME BAND x 15)
D= declination angle of sun from celestial equator (-ve in southern hemisphere)
T= equation of time
G= twilight angle

For Kabul (see this website for coordinates):

B = 34.28 (degrees N)
L = 69.11 (degrees E)
R = 15 X 3.5 = 52.5

Now we need D and T. These can be approximated using visual charts, but why should we settle for approximations when we can DO the trig ourselves?

According to this website, the declination for a particular day can be represented by the formula:

23.45 * sin[( j_day_value+284) * 360/365]

Where:

j_day_value = Julian day value of the day

and

the Julian Days can be calculated thus:
month_index = month number -1
date_index = date - 1
function calcJDay(month_index,date_index) {
var j_day=0
var date_val=date_index+1
if (month_index==0)
j_day=0+date_val
else if (month_index==1)
j_day=31+date_val
else if (month_index==2)
j_day=59+date_val
else if (month_index==3)
j_day=90+date_val
else if (month_index==4)
j_day=120+date_val
else if (month_index==5)
j_day=151+date_val
else if (month_index==6)
j_day=181+date_val
else if (month_index==7)
j_day=212+date_val
else if (month_index==8)
j_day=243+date_val
else if (month_index==9)
j_day=273+date_val
else if (month_index==10)
j_day=304+date_val
else if (month_index==11)
j_day=334+date_val
return j_day
}

So calculating the Julian day value for Kabul for today, 10 August 2005 (we can TOTALLY read computer script!):

month index = 8 - 1 = 7
date index = 10 - 1 = 9
which means that the j_day = 212 + date_val = 212 + 10 = 222 (incidentally, this is just a complicated way of saying that August 10th is the 222nd day of the year!, which I did not figure out until I manually counted the days on the calendar to get a N value for the Equation of Time calculation, below...grrrrrrr).

Using a Julian day value of 222 in the declination formula:

23.45 * sin(( j_day_value+284) * 360/365)
= 23.45 * sin((222+284)*260/265))
= 23.45 * sin (496.45)
= 23.45 * 0.69
= 16.18

D = 16.18, which is a feasible value looking at the approximate charts.

As for T, the Equation of Time, this website tells us the following:

The equation of time is the sum of two offset sine curves, with periods of one year and six months respectively. It can be approximated by:

E = 9.87sin(2B) – 7.53cos(B) – 1.5sin(B)

Where:

B = 360(N - 81)/364 if sin and cos operate on degrees.

For today’s date, N = 222, so B = 360(222-81)/364 = 139.45.

Therefore:

E = 9.87 sin(2*139.45) – 7.53cos(139.45) – 1.5sin(139.45)
= 9.87(-0.99) – 7.53(-0.76) – 1.5(0.65)
= -9.77 + 5.72 – 0.98
= -5.03 = T

T = -5.03, which also looks feasible according to the visual charts (thank God!).

We now how our values for B, L, R, D, and T, and all we say that G = 18 in Afghanistan, according to regional custom.

Let’s crunch our numbers (I sound like a trigonometric aerobics instructor!) à

Z = 12 + (R-L)/15 + T/60
= 12 + (52.5 – 69.11)/15 – 5.03/60
= 12 – 1.11 – 0.08
= 10.81

V = arcos{[-sin(18) – sin(16.18)sin(34.28)]/cos(16.18)cos(34.28)}/15
= arcos{[-0.31 – (0.28)(0.56)]/(0.96)(0.83)}/15
= arcos(-0.59)/15
= 2.20rad/15
Unfortunately, our online inverse trig calculator works in radians, and we are using degrees. Converted (recall pi rad = 180 degrees and use your unit circle!):
V = 126.16/15 = 8.41

Now, for our Call to Prayer Calculation!

*drum roll*

Fajr = Z – V = 10.81-8.41 = 2.4

Hmmmm – 2.4 seems like an odd number. If it means “2:24AM” then VC either had NOTHING to complain about when the Muzzein let him sleep in for an hour and waited until almost 3:30 for the morning call – OR, what VC woke up to was not the Fajr Call, but something else. In any case, I’ve been working on this for more than an hour (it’s now 5AM), and I still have to edit this text and create links to make this calculation sheet bloggable. Meaning: I’ll find my error later. I will note, though, that sunrise in Kabul, according to this website, is supposed to be at 5:10AM, today, so even the latest call to prayer I heard (3:50AM approx.) was way early for Fajr.

Mystère et boulle de gomme!

If you are in desperate need to do trig and inverse trig online (like I was), then you can use these websites for your calculations. This website is also interesting because of the “services” it offers with regard to sun positioning calculations (“forensic analysis”!?!?).

Hopefully at some point today I can figure out this mystery!! Having spent 2 hours and consumed nearly 20 (ok, exactly 20) of these AMAZING Iranian Werther's Original-like (with peanut butter in the middle!) candies, I am determined to not have my efforts (and calories!) go to waste. By the way, here is the website for Aidin, the Iranian candy company sent to us from above (it's bilingual and has GREAT music!!).

VC