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 The Voyage of the Mako (Posted on 2011-04-05)

The space freighter ‘Mako’ trades out of Antares. It travels at constant speed through hyperspace and its route is as follows:

ITINERARY
Antares-Alniyat ...........7 days
Alniyat- β Normae .......4 days
β Normae-1 Scorpii .....8 days
1 Scorpii-Antares ........5 days
Antares- π Scorpii .......7 days
π Scorpii-1 Scorpii.......3 days
1 Scorpii-Alniyat .........5 days
Alniyat-Dschubba ........3 days
Dschubba-Akrab .........6 days
Akrab-π Scorpii ..........4 days
π Scorpii- β Normae .....6 days
(Parts of days ignored throughout)

How many days would it take the ‘Mako’ to travel back home from β Normae to Antares?

 See The Solution Submitted by broll No Rating

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 an attempt at a solution | Comment 4 of 14 |

The distances from pi to Akrab, Akrab to Dschubba and Dschubba to Alniyat seem superflouous, as they add up to 13, which is much looser than the constraint imposed on the Alniyat-pi distance imposed by Alniyat-1 Sco-pi total of 5+3=8.

As seen from 1 Scorpii, the angular separations in degrees in the sky that matter, as determined by plane trigonometry, are:

Aln to Beta: 24.14684799650239
Beta to pi: 39.57121945723089
pi to Antares: 120
Antares to Aln: 88.8540080016114

These form a quadrilateral on 1 Scorpii's celestial sphere, which allows some leeway in its angles.

We are interested in the number of decrees separating Antares from Beta Normae in 1Sco's sky, from which, knowing their respective distances of 5 and 8 respectively, using plane trig, we can figure their mutual distance.

Let's try varying the Antares-Alniyat-Beta angle of the quadrilateral to see what its effects are.

DEFDBL A-Z
pi = ATN(1) * 4
dr = pi / 180
DEF fnac (x) = pi * ABS(x < 0) + ATN(SQR(1 - x * x) / x)

antAln = 88.8540080016114# * dr
alnBeta = 24.14684799650239# * dr
betaPi = 39.57121945723089# * dr
piAnt = 120 * dr

FOR angle1 = 91 TO 179 STEP 2
cAntBeta = COS(antAln) * COS(alnBeta) + SIN(antAln) * SIN(alnBeta) * COS(angle1 * dr)
distAntBeta = SQR(25 + 64 - 5 * 8 * cAntBeta)
cosPi = (cAntBeta - COS(piAnt) * COS(betaPi)) / (SIN(piAnt) * SIN(betaPi))
PRINT USING "### ###.#### ###.####"; angle1; distAntBeta; cosPi
NEXT

The program uses spherical trig to find the angular distance from Antares to Beta Norma in 1Sco's sky, and then uses plane trig to find the linear distance from one to the other. As a check, the cosine of the angle in the 1Sco's sky quadrilateral at pi Scorpii to see if it's a reasonable cosine. In all cases it does seem to be that:

`varied    sought   checkangle    distance  cosine 91       9.4104   0.7188 93       9.4407   0.6929 95       9.4708   0.6671 97       9.5007   0.6413 99       9.5304   0.6157101       9.5599   0.5902103       9.5891   0.5649105       9.6179   0.5398107       9.6464   0.5149109       9.6745   0.4903111       9.7022   0.4660113       9.7295   0.4420115       9.7562   0.4184117       9.7825   0.3951119       9.8082   0.3723121       9.8334   0.3499123       9.8580   0.3279125       9.8820   0.3065127       9.9053   0.2855129       9.9280   0.2651131       9.9500   0.2453133       9.9713   0.2261135       9.9919   0.2075137      10.0117   0.1895139      10.0308   0.1722141      10.0491   0.1556143      10.0666   0.1396145      10.0832   0.1244147      10.0990   0.1099149      10.1140   0.0962151      10.1281   0.0833153      10.1413   0.0711155      10.1537   0.0598157      10.1651   0.0493159      10.1756   0.0396161      10.1852   0.0307163      10.1939   0.0227165      10.2016   0.0156167      10.2084   0.0093169      10.2142   0.0040171      10.2190  -0.0005173      10.2229  -0.0041175      10.2258  -0.0068177      10.2278  -0.0086179      10.2287  -0.0095`

So it would seem that 9 or 10 would work as an answer, though there might be other constraints that I overlooked. And of course, I'm here treating the numbers as exact, even though they are rounded down.

 Posted by Charlie on 2011-04-06 14:50:41

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