[geocentrism] Re: Geosynchronous satellites paper
- From: "Robert Bennett" <robert.bennett@xxxxxxx>
- To: <geocentrism@xxxxxxxxxxxxx>
- Date: Sat, 28 Jul 2007 14:06:06 -0400
Robert B
I find maths tedious if I already know the answer
The writer reveals an interesting philosophy. Why let quantitative facts get in
the way? If the belief is known from an MS source, then would their maths
also be tiresome?
To reject all boring maths means mindlessly accepting MS theoretical proofs
based only on their presumed competence è the logical fallacy of appealing to
authority. è putting the fox in charge of the hen house.
so I have not seriously waded through your post From Robert Bennett Tue Jul 17
03:31:26 2007,
If only I were disposed to respond reciprocally…
but thanks for rejoining the issue anyway. Instead, I Googled 'Holmann transfer
orbit' (you did offer it as a viable alternative)
I don’t recall this offer. The author should either give us the corroborating
quote, or stop misquoting my posts.
and got 40000+ hits. Top of the list -- wikipedia! (If you don't like this
one, I'm sure the next one will give the same answer, and the next, and the
next ... ). The site gives all the equations necessary to understand the
problem, several of which you have already quoted to me, but they come to a
decision opposite from yours and in line with mine.
Wait a sec…. My maths were tedious. But now the writer throws up all the
equations necessary for my understanding from Wiki. What about his
understanding of the maths?
The Wiki diagram below shows the Hohmann transfer to reach the parking orbit,
but raising Artemis to the geostat orbit is done by a low continuous burn of
the Rita thrusters. This produces a slow spiral ascent for almost a year – a
quasi-circular trajectory – not a Hohmann transfer
According to the ESA site Rita thrust was aimed to accelerate slowly upward,
while its angular and linear velocity slowed down. This could also have been
done by firing Rita retro to the direction of motion(antapex), which would slow
down Artemis and cause it to rise. A reversal of cause and effect.
Firing Rita continuously to increase the linear velocity WOULD CAUSE ARTEMIS TO
SPIRAL DOWN, NOT UP.
So why does the Hohmann Xfer work? The strong impulsive thrust forward forces
the satellite to accelerate linearly for the duration of the burn (usually less
than a minute). At the end of the burn gravity alone is acting, so the
satellite finds itself moving too fast for its orbit, so it immediately
attempts to plunge to a lower level, changing direction towards the earth. The
change in direction means it must now move in an ellipse of higher energy – not
in a circle. Momentum causes it to overshoot its new lower stable circular
orbit. It is now too low for a stable orbit, so it reverses descending and
starts ascending (just the opposite condition to its descent), until it
achieves its greatest distance from the Earth(apogee).
The first diagram from Wiki is at best misleading, at worst duplicitous. The
first burn at delta V shows the yellow path continuing on the circle orbit
(dashed), as though the thrust had a delayed effect. But Newton’s laws require
an immediate orbit response, shown in the second diagram, in black. The correct
graphic of the Hohmann effect is an ellipse inside the boundary of the original
orbit, showing that the satellite DESCENDS, NOT RISES, after the first burn.
NOT ONLY IS THE WIKI DIAGRAM DECEPTIVE, BUT ALL OTHERS ON NET SHOW THIS
INCORRECTLY. THE CORRECT DIAGRAM IS THE SECOND ONE!
http://en.wikipedia.org/wiki/Hohmann_transfer_orbit
Explanation - with needed corrections
The Hohmann transfer orbit is one half of an elliptic orbit that touches both
the orbit that one wishes to leave (labeled 1 on diagram) and the orbit that
one wishes to reach (3 on diagram). The transfer (2 on diagram) is initiated by
firing the spacecraft's engine in order to accelerate it
momentarily with an impulse
so that it will follow the elliptical orbit;
dropping immediately in altitude for ¼ period, rising to cross the original
circle orbit to reach the new apogee, where it has the lowest speed of its new
orbit
this adds energy to the spacecraft's orbit. When the spacecraft has reached its
destination orbit
distance
, its orbital speed (i.e., orbital energy) must be increased again
since it is now moving too slow for a stable circular orbit at its present
distance.
in order to make its new orbit circular; the engine is fired again to
accelerate it to the required velocity.
But this acceleration does not make it rise; it doesn’t even change its height
!!
E=\frac{1}{2}m v^2 - \frac{GM m}{r} = \frac{-G M m}{2 a} \,
Solving this equation for velocity results in the Vis-viva
<http://en.wikipedia.org/wiki/Vis-viva_equation> equation,
v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)
Therefore the delta-v <http://en.wikipedia.org/wiki/Delta-v> required for the
Hohmann transfer can be computed as follows:
\Delta v_P = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1+r_2}} - 1
\right), Delta-v required at periapsis <http://en.wikipedia.org/wiki/Periapsis>
.
Two questions for the writer:
What is the derivation of this equation? It was offered in support of the
Hohmann transfer, even though it’s tedious math.
Why does the analysis ignore the mass loss after both rocket firings? Artemis
is ~ 1/3 payload and 2/3 fuel.
Example
For the geostationary transfer orbit we have r2 = 42,164 km and e.g. r1 = 6,678
km (altitude 300 km).
In the smaller circular orbit the speed is 7.73 km/s, in the larger one 3.07
km/s.
Could it possibly be clearer, folks, that speed decreases with height?
In the elliptical orbit in between the speed varies from 10.15 km/s at the
perigee to 1.61 km/s at the apogee.
The average is < 6 km/s, meaning the new elliptical orbit has less average
speed and kinetic energy than the original orbit.
………
Low-thrust transfer => The Artemis final orbit model
It can be derived that going from one circular orbit to another by gradually
changing the radius costs a delta-v of simply the absolute value of the
difference between the two speeds. Thus for the geostationary transfer orbit
7.73 - 3.07 = 4.66 km/s, the same as, in the absence of gravity, the
deceleration would cost. In fact, an acceleration impulse is applied to
compensate for half of the deceleration due to moving outward.
Could it possibly be clearer, folks, that speed decreases with height?
Therefore the acceleration impulse due to thrust is equal to the linear speed
deceleration due to the combined effect of thrust and gravity.
Since deceleration means to slow down, could it possibly be clearer, folks,
that speed decreases with height?
Summary: The Hohmann orbit maneuver does not apply to the Artemis spiral
transfer, nor does it imply that satellite speed increases with altitude.
The diagram commonly used for illustrating the Hohmann transfer is seriously
(and deliberately?) flawed.
Robert B
………
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