Solution to Spherical Astronomy Practical Focus Matrix Problem

 Problem:

Apply the  matrix method for the same location in the example problem but for a sidereal time ST  = 12 h 28m.  Apply to the case of the planet Mars which is also visible at the same local time but at position: RA = 10h 28m, and d = +12 ^o 51’.

Solution:

The sidereal time (ST) is given as: 11 h 28 m

The new Right Ascension is 10 h 28 m

Next, we change the new Right Ascension to hour angle using:

h = ST - RA

So: h = 11 h 28 m – 10 h 28 m =  1h 00m

This is then converted into degrees, using the fact that there are 15 degrees/ hr.   So h = 1h 00m/15 degrees/ h   

So: 1 h 00 m  = 15 ^o.0

Perform the matrix operations in the specific order:

(x)
(y)
(z) A,a = R3(-180^o) R2(90 - lat.) (XYZ(h, d)) 

Where:

R3(180) =

(cos(180)..........sin(180)..........0)
(-sin (180)...... cos(180)...........0)
(0 ....................0.......................1)

Therefore: R3(-180) =

(-1       0      0)
(0       -1      0)
(0        0      1) 

And:  R2(90 ^o - lat.) =

(sin lat.       0         - cos lat.)
(0                 1                   0   )
(cos lat.      0.           sin lat.)

For which we have:

sin (lat.) = sin (25.^o 75)= 0.434 

cos (lat.) = cos (25.^o 75) = 0.900


Thence, R2(90 ^o - lat.) =

(0.434            0      -0.900 )
(0                   1               0   )
(0.900          0        0.434  )

Finally:

(x)
(y)
(z) h, d =

(cos d             cos h)
(cos d             sin h)
(sin d                 -    )

where: d = +12 ^o 51’ = 12. ^o  85

sin (d) = sin (12  ^o 85) =  0.28

cos (d) = cos (12. ^o 85) = 0.960

cos h = cos (15 ^o.0) = 0.966

sin h = sin (15^o.0)  =  0.259

Assembling the foregoing into the applicable matrix:

(0.96       0.966)
(0.96        0.259)
(0.28    .......... )     =

(0.927)
(0.249)
(0.28)

Whence:

R3(-180 ^o) R2(90 ^o - lat.) (XYZ(h, d)) =

=

(-0.150)
(-0.249 )
( 0.956 )

The last element in the column yields the altitude, so:

a = arc sin(0.956) and a = 72.^o 9

Meanwhile, the azimuth A =

arc tan (y/x) = arc tan (-0.249/ -0.150) = 1.66

Therefore: A = arc tan(1.66) = 58. ^o 96 

REM: You must convert radians to degrees (by multiplying by 57.3 deg/rad) in each case!