M-750 航線經我天空的軌跡 air route trail across my sky

Here in central Taiwan, the M-750 air route from Hong Kong to Japan passes near me. From a pilot's point of view, I am 3.5 nautical miles (6.5 kilometers) southeast of its track. Since I know the exact path, and 我的位置的座標 my coordinates, let's see all the things I can compute as a simple ground observer.

[2013 update: I used programs/3d to make m750_3d.kml. See also ../routes/programs.]

Sound

Angle between where the plane is vs. sounds like it is

The eye and ear have different impressions of where the plane is. (Here the speed of light is much greater than sound, so we ignore it.)

$ make see_hear_angle

 knots   km/h    m/s    mph   mach ang-20C
   150    278     77    173   0.23     14
   200    370    103    230   0.31     19
   250    463    129    288   0.39     23
   300    556    154    345   0.47     28
   350    648    180    403   0.54     33
   400    741    206    460   0.62     38
   450    833    231    518   0.70     43
   500    926    257    576   0.78     48
   550   1019    283    633   0.85     53
   600   1111    309    691   0.93     58
   650   1204    334    748   1.01     63

True when the position of the sound and plane form an isosceles triangle. Distance is not a factor.

Noise vs. time

Given a airplane at speed 600 knots, whose track at its closest point passes by one at 15 kilometers away (elevation angle not important,) what are the formulas that would describe the shape of the decibel readings over time? (which are something like 25 25 40 40 40 38 36 34 30 25 25, with 25 dbA being the background.) Discussion in sci.physics.acoustics, answered in askthephysicist.com with graph too.

d horizontal distance of closest approach
v speed
t=0 time of closest approach

The dB level relative to the maximum is given by 10*log10(d2/(d2+v2t2)).

It turns out the speed of the plane has nothing to do with the imbalance over time, but instead it is the fact that standing in front of a jet plane with its motors running even still stationary on the ground, is less noisy than standing behind it...

Distance and vision

Location of point closest to me on route

Using makefile, section: point_closest_to_dan, we find it is 6.5 km away at bearing 314 degrees.

Elevation angle at nearest point

That gives

100FT METER DEG
FL410 12496 61
FL390 11887 60
FL370 11277 58
FL350 10668 57
FL330 10058 55
FL310  9448 53
FL290  8839 51
FL270  8229 49

Computing the speed of the aircraft

Taking the number of seconds between when a plane passes directly to my west, and then north, we can compute the speed, as we know the length traveled. Using the makefile, section length_of_points_90_degrees_to_dan, we find the length is 13.0 km. We then divide this by the time elapsed (makefile section "speed").

On our first observation, it seems the plane took about a minute to fly the 13 km. So we get 780 km/h; 421 knots; 484 mi/h; 0.65 mach; 2013: the web flight tracker sites show that they often are close to 600 knots!

Arc length of aircraft

Let's compute the arc length of say a Boeing 747-400, 70 meters long, against the sky. (makefile, section arc_length.) We use the point on the trail nearest to us.

height(ft, km), length(arc minutes)
25000  7.6 24.0
30000  9.1 21.5
35000 10.7 19.3
40000 12.2 17.4

For comparison, the full moon is on average "just over half a degree" wide.

Actually, the heights that mostly matter according to Mar. 2004 Taiwan government aeronautical publications are flight levels 410, 380, 370, 340, 330, 300, 290, where 1 flight level is 100 ft., also depending on atmospheric pressure...

Anbu to Xigang

有時韓國下來的飛機很多改由 APU → TNN。 紅則 geodesic 線。 其半路點為中橫公路的天冷。

Tracks for astronomy or satellite programs

I should enter route M-750's path across the heavens in star charting software.

A satellite TLE file to describe the air route?

I want to make a satellite TLE file for the air route, if possible, so I can say plug it into various software. It seems I must fake one of the parameters, say rotations per day, to get the altitude down to where I want, at the price of making times unreal ... but my focus is on the path across the heavens anyway. I got as far as Kepler's third law.

Try just plotting it as an "horizon"

A simpler option would be for me to use simple trigonometry to compute an elevation angle for the air route at each degree of azimuth from me, and plug this into star charting software as a "horizon" line. See target "horizon" in the makefile.

Supposedly I will notice a plane crossing very close to some star one day, and plugging in different heights' horizons, see which height matches.

Distance from azimuth

As a side effect of our horizon calculation, we get the horizontal distance from us as a function of azimuth, which is independent of altitude. Looking at a map, we notice that M750 leaves the island of Taiwan 143 km from us NEward, and 106 km SWward. The azimuth of the latter is 227.5 deg...


積丹尼 Dan Jidanni Jacobson

Last modified: 2013-12-22 20:56:00 +0800