Visible Horizon

Assume:

  • field of view ==> FOV = 90 degrees
  • Atmosphere Altitude Aa ==> 100 km
  • Earth Radius Re ==> 6371 km
  • Theta ==> 1/2 of Angle FOV make with Center of earth at top of Atmosphere

Theta:

Theta  = arcsin( Aa * tan(FOV/2) / Re)

Cap Area:

Area of Spherical Cap: Acap=2*pi*r*h   
h = height of cap above plane slicing cap
h = r * ( 1 - cos (theta))
A = 2 * pi * r ^ 2 * (1 - cos(theta))

Fraction of Sphere Covered by Cap:

F = Acap/Asphere
F = 2 * pi * r ^ 2 * (1 - cos(theta)) / (4 * pi * r ^2)
F = (1 - cos(theta)) / 2
In [8]:
import math
FOV = math.pi * 90 / 180.0
Aa = 100 #km
Re = 6371 #km
theta = math.asin( Aa *  math.tan(FOV/2) / Re)
print "theta = ", theta, " Radians"
F = (1.0-math.cos(theta)) / 2.0 
print "Sphere Fraction covered by FOV: F = ", F

theta =  0.0156967676335  Radians
Sphere Fraction covered by FOV: F =  6.15958638103e-05

Meteor Calcs

Rates

Worldwide Mass Rate Rm = 5 tons/day
Average Size of visible meteor Mm = 1.5 g / meteor

Meteor Count/day : Rc 
Meteor Count/hr : Rh

Meteor in FOV

Streaks in FOV per hour: Rfov = F * Rh

Meteor Data from AMS

Norminal Observed Rates:
    2-4 Streaks/hr in Evening twilight
    4-8 Streaks/hr in morning twilight

Nominal Mass of visible object before entry: <1-2 grams

Nominal Entry Speed:
    11 - 72 km/sec

Meteor Density Info

Iron Meteors: 7-8 g/cm^3
chondrites: 3.0 to 3.7 g/cm^3

Reference:

In [9]:
Rm = 5e3  #kg/day
Mm = 1.5e-3  #kg/meteor
Rc = Rm / Mm
Rh = Rc / 24.0
Dm = 1.5 # g/cm^3
print "Meteors/day = ", Rc
print "Meteors/hr = ", Rh
print
Rfov = F * Rh
print "Rfov = ", Rfov , " streaks/hr"
print "Average chrondrite Meteor Diamter: %3f cm" % (2.0 * math.pow((Mm*1000/Dm)*3.0/4.0/math.pi,1.0/3.0))

Meteors/day =  3333333.33333
Meteors/hr =  138888.888889

Rfov =  8.55498108477  streaks/hr
Average chrondrite Meteor Diamter: 1.240701 cm

Earth Constant Info

Me = 5.97219e24   # Earth Mass in kilograms
RE = 149.59787e6  # Radius of Earth's Orbit (kilometers)
Ty = 365.256      # Year in days

Other Constants & Formulas

G = 6.674e−11        # Gavitational Constant  (N*m^2/kg^2)
Vo = sqrt(G*M / r)   # Orbital Velocity for circular orbit. M=mass of larger body, r=radius

Reference

In [10]:
Me = 5.97219e24   #Earth Mass in kilograms
RE = 149.59787e6  #Radius of Earth's Orbit
Ty = 365.256      #Days in Year
G = 6.674e-11     #Gavitational Constant  (N*m^2/kg^2)

print "Earth Orbit Velocity", 2.0*math.pi*RE/(Ty*24*3600), " km/sec"
print "Earth weight gain rate per 1000000 years: ",1000000*Ty*Rm, " kg"

Earth Orbit Velocity 29.7847649488  km/sec
Earth weight gain rate per 1000000 years:  1.82628e+12  kg
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