Monday, July 02, 2012

Random orbital calculations for 9-graders. Just for the heck of it, I decided to avoid numeric value of Newton Gravitational Constant.

Earth radius R=6371 km. Free-fall acceleration is g=9.8e-3 km/sec^2
Circular orbital velocity at the surface:

V1=sqrt(gR)=sqrt(9.8e-3*6371)=7.90 km/sec

If the orbit is elliptical with perigee at the surface, and apogee at distance H from the center of the Earth, from laws of conservation of energy and angular momentum, denoting h=H/R, the velocity at apogee is:

Va=sqrt(2gR/h(h+1))

At Geostationary Transfer Orbit (GTO), H=42164 km [1], h=H/R=6.62, at apogee, velocity is:

Va=sqrt(2*9.8e-3*6371/6.62/(6.62+1))=1.57 km/sec

At Geostationary Earth Orbit (GEO) Vg=3.07 km/sec [1]. To raise perigee, delta-v is 3.07-1.57=1.50 km/sec.

If you launch not from equator, but from latitude 51.6 or 28.5 deg, you'd have to turn the orbit with delta-v

2*sin(51.6/2 deg)*1.57=1.37 km/sec (total: 1.37+1.50=2.87)
2*sin(28.5/2 deg)*1.57=0.77 km/sec (total: 0.77+1.50=2.27)

If you combine the two in the same maneuver, delta-v becomes:

sqrt(3.07^2+1.57^2-2*3.07*1.57*cos(51.6 deg))=2.43 km/sec (<2.87)
sqrt(3.07^2+1.57^2-2*3.07*1.57*cos(28.5 deg))=1.85 km/sec (<2.27)

By the time you reach apogee, compared to equatorial launch, delta-v penalty is

from 51.6 deg: 2.43-1.50=0.93 km/sec
from 28.5 deg: 1.85-1.50=0.35 km/sec

Or 56-to-28 deg penalty: 2.43-1.85=0.93-0.35=0.58 km/sec﻿

At GTO perigee velocity is

Vp=Va*h=1.57*6.62=10.40 km/sec.
delta-v from V1 to GTO is 10.40-7.90=2.50 km/sec

checking with alternative calculation (skipping intermediate numeric steps)

Vp = V1*sqrt(2h/(h+1))=7.90*1.32=10.41 km/sec (check)

[1] 1 sidereal day = 23 hours, 56 minutes, 4 sec = 86164 sec
H=(G*mass of earth*(86164s)^2/(4*pi^2) )^(1/3)=42164 km
Vg=42164m*2*pi/86164s=3.0746 km/sec or
Vg=(2*pi*G*mass of earth/(86164s))^(1/3)=3.0746 km/sec

Often quoted altitude 35,786 km is simply H minus Earth's equatorial radius of 6,378 km (35,786=42,164-6,378).

http://en.wikipedia.org/wiki/Geostationary_orbit

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