VELOCITY MATTERS
Escape velocity is important as I can show. The escape from
the sun example didn't help but I get inconclusive results
from published examples. The radius for light escaping increases
as mass increases, so in my mind that can't help.
The denser the object, the relativistic effect on light should
increase.
Now consider the rock at 600,000 miles, since earth is the closest
object, we want to know it will escape.
The calculation gives 8,190,000 ft/sec while the speed of light
is 900,000,000 ft/sec or it is going 100 times slower than light.
That is fast and I hope it's possible.
Here is some information gleaned from some of the sources on the
escape velocity page.
[Abell]
Escape velocity (Sec 5.4)
v * v = 2 * G * ( m1 + m2 ) / r
Black hole radius (Sec 31.10)
R = 2GM/(c*c)
This text book example has mass and radius directly
proportional.
Mass/Radius = (c * c)/2G
The gravity constant G:
G = .0667/1M dyne cm cm /gm gm [many places]
1M dyne = 2.247 lb , 14590 gm = 1slug , 1ft = 30.48 cm
G = 515,027/1M lb ft ft / slug slug
c = 900 M ft/sec , M = 1,000,000
Earth mass = 5,977 MMMM gm
= 409,600 MMM slugs
Earth radius = 6,378 km
= 4,368 miles
Sun mass = 1.985 10^33 gm
= 136 10^27 slugs
Sun radius = 69,510 M cm
= 476,000 miles
How far away is the rock, 600,000 miles.
Ratio radii of sun/earth = 109
Before going on to the forces on planetary objects, lets look
at the black hole radius.
M/R = 900 900 MMM / 2 515,027 slug slug/lb sec sec
= 786,367 MM slug / ft
Using the mass of the sun:
R = 136,000 MMMM slugs / 786,367 MM slugs/ft
= 173,013 M ft / 5280 ft/mi
= 33 M miles
So the mass of the sun must be concentrated within a radius
of 33 miles. The sun will develop or evolve into a white
dwarf but that's a discourse for another page.
Now for a table to demonstrate the distances between the planets.
[Fundamentals... Bate Mueller White]
Table of the G Forces on a satellite in 200nm orbit,
per M except Earth
Object Force
Earth .89
Sun 600
Mercury 260/M
Venus 19/k
Mars 71/k
Jupiter 32/k
Saturn 2.3/k
Uranus 80/M
Neptune 36/M
Pluto 1/M
Moon 3.3/M
Earth bulge k/M
So what do we say, the closest object wins.
So lets see what distances are involved.
Object Orbit Half the Half the M miles
in Axis Axis Axis Distance
Orbit Mkm M km M miles to Earth
Mercury 57.9 25 17 N/A
Venus 108.2 50 34 N/A
Earth 149.6 75 51 N/A
Mars 227.9 100 68 17
(Ceres) 414 200 136 85
Jupiter 778 400 273 222
Saturn 1427 750 513 462
Uranus 2870 1400 959 908
Neptune 4497 2200 1500 1450
Pluto 5912 3000 2054 2000
So at 600,000 miles distance from earth, earth has the biggest
influence. So half a million miles from earth is a lot closer
than the 17M miles to Mars.
So all we want to know is if anything big escapes from us.
Lets see how the orbit equation is derived. In physics we
take the energy approach but here is a nice vector approach.
All letters are vectors, a vector magnitude is donated as |a|
for acceleration, G M and m are scalars (not vectors).
The x denotes vector cross product.
The r is vector to object from the primary focus.
Take cross product of angular momentum, (r x v), with the opposing force
equation |F| = m|a| = GMm/(|r||r|) but in vector form.
a x r x v = ( GM (r x v ) x r) /( |r|*|r|*|r| )
Remove the time derivative
(d/dt)(v x ( r x v )) = GM (d/dt)(r/|r|)
Integrate ( or reverse differentiate ).
v x ( r x v ) = GM r/|r| + cev
Where cev is the constant (energy vector) of integration.
Take vector dot product with r,
r dot (v x r) = r dot GMr/|r| + r dot cev
gives
|rv||rv| = GM|r| + |r||cev|cos(angle between r & cev)
No more vectors and, letting h = rv the momentum, rearrange.
r = (h*h/GM)/[1 + (cev/GM)cos(angle)]
which looks like the conic equation
r = p/[1 + e cos(angle from foci axis to r)]
From analytical geometry, e determines the path. Back to vectors:
e = cev/GM
e = v x h (1/GM) - r/|r|
e = a vector from r to the lowest point of orbit
It represents the balance of kinetic and potential energy.
Get the balance just right and you have orbit, or too much
and you escape, or you fall back to earth.
Lets go back to calculate a mass at 600,000 miles from earth.
[Abell]
Escape velocity (Sec 5.4)
v * v = 2 * G * ( m1 + m2 ) / r
= 2 * 515,027/1M (lb ft ft / slug slug)
* ( Earth + Moon? + rock masses) slugs
/ 600,000 miles 5280 ft/mile
= (325 / MM) * ( 409,600 MMM slugs ) [Earth alone]
= 133 MM ft/sec ft/sec
v = 12 M ft/sec = 12,000,000 ft/sec
Now that's fast. I had 8M last time. That is one rock
we don't want to catch. This calculation must be incorrect.
A review is now underway. My error in the GM product.
GM = 1.407654 10^16 ft ft ft/ sec sec [for Earth]
R = 21 M ft Earth Radius
v * v = 2 GM / R
v = 36,000 ft/sec for earth orbit escape
v * v = 2 GM / 600,000 mi * 5280 ft/mi
= 888,670.46
v = 942.69319 ft/sec
BYE Bye the ARMAGEDDON ROCK, we hope.
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