Husam’ Solution of Yallop's Formula for Visibility Of Crescent of Luna With Examples

Yellop’s formula is a good formula for the prophecy of Crescent of Luna.

How ever it is difficult to solve.

We have modified this formula for some practical simplicity.

We may give an example for its solution.

Yallop’s Formula at a given place on Earth at Best Time within 60h of New Moon.

Yallop’s Formula in Husa:m’s Modified form: Q=[ARCV-(11.8371-6.322(‫)س‬+3^(‫)س‬0101-2^(‫)س‬0.731] [(10)^(-1)] Q=Q(ARCV,‫ )س‬A function of ARCV and ‫س‬

[ARCV inDegrees and ‫ س‬in Arc Min] ‫ س‬is called as Si:n=Seen. ‫= س‬ω(60), In Arc Min . ω= Sδ(1-Cos(ARCL)) In Degrees. [Topocentric Width Of Crescent] Sδ=SD(60)^(-2) In Degrees SD= [566204.92/RP][1+Sin((AltM)/RP)] In Arc Sec. [There are may formulae of SD, we HAVE CHOSEN THIS ONE]

RP= 61.52 [Generally] Alt M = 90-Z, Z = Zenith ARCV=

ArcCos [(Cos(ARCL))(Cos(DAZ))] in Degrees

ARCL= ArcCos((Cos(Lm-Ls))(Cos(Dm))) in Degrees Tb= Best Time for the visibility of Cresent Tb= Ts +(4/9)Lag or Tb= (4.3/9.3)Lag +Ts [A better Approximate Value] Lag= Tm-Ts Tm= Time for Lunar Set Ts= Time for Solar Set Other Variables required

Ls,Lm, Dm,Azs (Azimuth of Sun),Azm(Lunar Azimuth),Daz =|Azm-Azs| SP=Illumination of Lunar Disc RP={Geocentric Distence between Earth and Lunain Km}/{Radious of Earth in Km} May vary from time to time.

Tb,Ts,Tm may be in UTC or GMT. Q> 0.216 Visibility With Unaided Eye 0.216>Q>-0.014 Visibility with unaided eye under perfect conditions.

-0.16<Q<-0.14 Visibility With Optical Aid. Q<-0.16 Invisibility of Crescent IOf Luna Example 1 To find the visibility of Lunar Crescent at a place on Earth Where Tb = 14:48 UMT Ls=235.39 Lm=245.01 Dm=-3.76 Hs=Azs=250.63 Hm=Azm=241.97 All are in degrees Rp=61.52 We can find 0.248834=‫ س‬in Minutes Q=4.313 This is a visibility Example 2 Let the values of Variable at some place on Earth at best time are: Ls=235.65,Lm248.32,Dm=3.55,Azm=244.69,Azs=243.47,DAZ=1.22>0 All in degrees,AltM= 1.75 {Given, if given then we do not require Zenith}

Suppose that the best time is given and we do not require it. For sake of simplicity take the best time of the previous Example.

We can calculate ARCV and ‫س‬ ‫ س‬is calculated as 0.2464873 FOR ARCV Try yourself Try yourself as an exercise Q(ARCV,‫= )س‬-0.497 This shews invisibility We can use Shahid Quraishi’s formula as well, neglecting the difference between Topocentic Width and Width. Ψ= Q=[ARCV-(0.351964-2.2220(‫)س‬+3^(‫)س‬10.43418-2^(‫)س‬5.422] With Criteria Ψ> 0 Probebly Visible with out aid Ψ> 0.15 Visible Under Perfect Conditions 0.05<ψ< Probebly Visible under Perfect Conditions -0.06<ψ<0.05 May require Optical aid -0.16>ψ Invisible Try to solve by this method as well.

Leglect the differenceof Topocentic Width and Width