Week 3

This is the third lab week for the year 2 project and our progress is quite slow. This week we were mainly focused on the calculations. All members of the group where trying to understand the complex algorithms that needed to be implemented into the code which was easier said than done, as it seemed each equation led to another unknown!

 A complete calculation was finally achieved by Dominyka! Her entire working is shown below. This demonstrates how complex the algorithm is as all of this working merely shows the position at one specific time on a certain date. 

Calculating the position of the Moon for 05/03/2017 at 11:32. (first quarter) [1]

1)      Conversion of time and date to Julian days

Julian day = 864000s

Julian year = 365.25d

Julian century = 36525d

JD=365.25*year’+30.6001*(month’+1)-15+1720996.5+day+(hour+minute/60+second/3600)/24

If month<=2, then month’=month+1, year’=year-1,

if month>2, then month’=month, year’=year.

Number of Julien centuries: T=(JD-JD₀)/36525

JD₀ is the Julian day for 01/01/2000 at 12:00UT

JD₀=365.25*1999+30.6001*14-15+1720996.5+1+12/24=2451546.151

JD is the Julian day for 05/03/2017 at 11:32UT

JD=365.25*2017+30.60001*4-15+1720996.5+5+(11+32/60)/24=2457818.631

T=6272.479596/36525=0.171731132

2)      Astronomical algorithms

·         Ecliptic latitude B and longitude L of the Moon [2]

B=-5.171º

L=75.062º

·         Convert B and L to right ascension RA and declination delta

eps=23+26/60+21.448/3600-(46.8150*T+0.00059*T²-0.001813*T³)/3600

X=cosB*cosL

Y=cos(eps)*cosB*sinL-sin(eps)*sinB

Z=sin(eps)*cosB*sinL+cos(eps)*sinB

R=
delta=180/π*arctg(Z/R)                                                                //π=180º

RA=180/π*arctg(sinL*cos(eps)-(tgB*sin(eps))/cosL)

eps=23+26/60+21.448/3600-(46.8150*0.171731132+0.171731132²*0.00059-0.001813*0.171731132³)/3600=23.43705789º≈23.437º

X=cos(-5.171º)*cos(75.062º)=0.2567

Y=cos(23.437º)*cos(-5.171º)*sin(75.062º)-sin(23.437º)*sin(-5.171º)=0.9187

Z=sin(23.437º)*cos(-5.171º)*sin(75.062º)-sin(23.437º)*sin(-5.171º)=0.3

R=0.953926

delta=1*arctg(0.3/0.953926)=17.4578º

RA=arctg(sin(75.062º)*cos(23.437º)-

(tg(-5.171º)*sin(23.437º))/cos(75.062º))=45.73877694º≈45.74º

·         Sidereal time at Greenwich [3]

theta₀=280.46061837+360.98564736629*(JD-JD₀)

Local time: theta=theta₀+longitude(E⁺,W^-)

Hour angle: tau=theta-RA

theta₀=280.46061837+360.98564736629*6272.479596=2264555.568

2264555.568/360=6290.432134

6290.432134-6290=0.4321338119

theta₀=0.4321338119*360=155.5681723º

theta=155.5681723º-2.963827º=152.6043453º

tau=152.6043453º-45.74º=406.8643453º

  

3)      Final results

·         Convert tau and delta to horizon coordinates h(altitude) and az(azimuth) of the observer (53.406773º, -2.965723º)

sinh=sin(beta)*sin(delta)+cos(beta)*cos(delta)*cos(tau) //beta-latitude

tg(az)=-sin(tau)/(cos(beta)*tg(delta)-sin(beta)*cos(tau))

h=sin^-1(sin(53.406773º)*sin(17.4578º)+

cos(53.406773º)*cos(17.4578º)*cos(106.8643453º))=4.353º

az=arctg(-sin(106.8643453º)/(cos(53.406773º)*tg(17.4578º)-sin(53.406773º)*cos(106.8643453º)))=66.285º

·         Compute the parallax in altitude

horParal=r/a (rad)                                    //r-radius of the Earth(6378km)

                                                                  //a-distance to the Moon(384400km)

Converting to degrees: horParal=horParal*57.2957795

altParal=sin^-1(cosh*sin(horParal))

Apparent altitude: H=h-altParal

horParal=0.0165921rad=0.9506568201º

altParal= sin^-1(cos(4.353º)*sin(0.9506568201º))=0.9479142571º

H=4.353º-0.9479142571º=3.41º

Notes for week 4:

•  Finish implementing the code and begin testing. 
• Design the poster and discuss what should be included. 

[1] – http://www.geoastro.de/elevazmoon/basics/index.htm

[2] - http://www.satellite-calculations.com/Satellite/suncalc.htm

[3] - http://www2.arnes.si/~gljsentvid10/sidereal.htm