Calculation of Streamer Motion

Here's how we determine the motion of the streamer relative to the earth. We start with some spherical trigonometry:

Definitions:

N: North celestial pole

Z: Zenith

O: direction of earth's orbital vector

R: radiant of Leonid shower

The two points O and R define an arc.

W is the point where that arc intersects the horizon

P is the point on that arc closest to the zenith

l is the latitude of the observer

g is the zenith distance (90 - the altitude)

e is the distance along the arc O - R, starting from W

a is the right ascension

d is declination

w is azimuth

m n k y q are the various angles shown

So here we go. Rather than present the formulae in standard mathematical format, which is a major pain to edit, I will simply paste in my C++ code. You may need to widen your window to see this clearly:

GammaR = acosf(sinf(Latitude) * sinf(DeltaR) + cosf(Latitude) * cosf(DeltaR) * cosf(AlphaR - SidTime));

OmegaR = acosf((sinf(DeltaR) - sinf(Latitude) * cosf(GammaR)) / (cosf(Latitude) * sinf(GammaR)));

GammaO = acosf(sinf(Latitude) * sinf(DeltaO) + cosf(Latitude) * cosf(DeltaO) * cosf(SidTime - AlphaO));

OmegaO = acosf((sinf(DeltaO) - sinf(Latitude) * cosf(GammaO)) / (cosf(Latitude) * sinf(GammaO)));

Theta = asinf(cosf(Latitude) * sinf(AlphaR - SidTime) / sinf(GammaR));

EpsO_EpsR = acosf(sinf(DeltaR) * sinf(DeltaO) + cosf(DeltaR) * cosf(DeltaO) * cosf(AlphaR - AlphaO));

Kappa = asinf(sinf(GammaO) * sinf(OmegaO - OmegaR) / sinf(EpsO_EpsR));

Mu = Pi - Kappa - Theta;

Nu = asinf(sinf(GammaR) * sinf(OmegaO - OmegaR) / sinf(EpsO_EpsR));

Psi = Pi - Nu;

GammaP = asinf(sinf(Psi) * sinf(GammaO));

EpsP_EpsO = acosf(cosf(GammaO)/cosf(GammaP));

EpsP = Pi / 2.0;

EpsO = EpsP - EpsP_EpsO;

EpsR = EpsO - EpsO_EpsR;

OmegaP = OmegaO - asinf(sinf(EpsP_EpsO) * sinf(Psi) / sinf(GammaP));

Now we shift to an elevation view, tilted by GammaP degrees, of the ground, atmosphere, and meteor:

I: projected impact point of meteor

M: termination point of meteor

h: termination height

ve: heliocentric velocity of earth

S: point in atmosphere where second meteor will terminate

Beta = EpsR_EpsO;

d = Ve * t * sinf(Beta) / sinf(EpsR);

And now we return to a plan view:

OmegaL = 90 + OmegaP;

Phi = OmegaM - OmegaL;

n = h * tanf(GammaM);

m = powf(n * n + d * d - 2.0 * n * d * cosf(Phi), 0.5);

GammaS = atanf(m / h);

if (OmegaM > OmegaL)

OmegaS = OmegaM + asinf(d * sinf(Phi) / m);

else

OmegaS = OmegaM - asinf(d * sinf(Phi) / m);

This gives us GammaS and OmegaS, the position in the sky where the second meteor should appear.