----------
> De : reto ambrosini
<[EMAIL PROTECTED]>
> A : [email protected]
> Objet : cos(T)=-tan(lat)*tan(decl)
> Date : vendredi 20 mars 1998 20:01
>
> Hello All !
>
> I tried to calculate the sunrise time with the
formula
>
> cos(T)=-tan(lat)*tan(decl)
>
> I took the datas from the astronomical Almanac,
here is an
> example for the 3. of september 1998 at
latitude +46
>
> Sun declination : 7 42 13.5 = 7.704 deg
> Rise time : 5h23m
> Set time : 18h35m
> Day length : 13h12m
>
> T=arccos(-tan(46)*tan(7.704))=98.05 deg =6.537
hours
>
> Calculated rise time : 12-6.537 = 5.463h =
5h27m47s
> Calculated set time : 12+6.537= 18.54h =
18h32m13s
> Calculated day length : 6.537*2 = 13.07 h =
13h04m25s
>
> As you see there is a difference between the
calculated datas
> and the datas of the nautical almanac, the
difference in the day
> length is about 8 minutes (2 degrees!) that's
quite a lot.
>
> Why do I have such a difference?
> Did I do some wrong calculation or is this
formula only
> approximative?
>
> Thanks
>
> Reto Ambrosini
>
> [EMAIL PROTECTED]
>
>
>
>
Hello Reto
Your calculation is right, but not complete.
Because of two points: astronomical refraction
and change in declination.
Indeed sunrise and sunset times are relative to
the apparent rise or set of the upper limb of the
sun. At these instants the altitude of the sun
center is not zero but -50' : -34' due to the
astronomical refraction and -16' due to the sun
semi-diameter. So the formula to be used is the
usual one for sun's altitude :
sin(alt) = sin(decl) * sin(lat) +
cos(decl)*cos(lat) *cos(H) , with alt= -50' or
-0;01454 rad, whence :
cos(H) = (-0;01454 - sin(decl) * sin(lat) ) /
cos(decl)*cos(lat)
Twice the hour angle computed give you the day
length. That hour angle added or substracted from
12 gives you the rising and setting times in
solar time. To get UT times add the Equation of
time and the longitude. Let's take 0 for it. And
happily Equation of time is almost zero for that
day : -0m 26s (in the french way...). So the
times computed are almost UT times.
The second point is the change in the sun
declination during the day. By 3 of september it
is pretty fast : 22' per day. If you take the
average declination for that day (7°31'13") you
will find the the rising and setting times given
by the A.A..
Best regards
J-P CORNEC
48°44'24"N 3°27'26"W
