Subject: (whorl) Voyage of the Whorl: Calculations of Thrust, Distance and Peak Velocity

From: Talarican <exultnttalarican at mindspring.com>
Date: Tue, 04 Sep 2001 07:32:55 +0000

Assuming a "Bussard ramjet-style" 1/2 continuous acceleration, 1/2
continuous deceleration flight profile, to calculate the necessary
acceleration/deceleration and actual distance covered, the elapsed times
were divided in half, as the acceleration and deceleration phases are (ah,
so Wolfean!) symmetrical.

Bruce Bowden, a.k.a. "Dr. Matrix", has been kind enough to post various
forms of the Lorentz equations for relativistic motion on his "Starships of
the Mind" website. (scientium.com/essays/startrip/startrip.html)

Assume a value for {a'}, the subjective acceleration, and set half the
elapsed Urth time {t} to 550 yr. Calculate elapsed subjective time {t'} for
either the accelerating or decelerating leg:

t' = c / a' * arcsinh(a' t / c)              (*1)

Keep iteratively adjusting {a'} until close enough to that value for which
{t'} = 150 yr, half the subjective voyage time. (A process of "relaxation",
if you will)

Having approximated {a'}, calculate x, 1/2 the distance to B&G, for either
the accelerating or decelerating leg
x = c^2 /a' (cosh(a't'/c) - 1)

Then peak velocity (at turnover) can be calculated
v[peak] = c * sqrt(1- (1/(1 + a'x/c^2)^2)

It is of course convenient to perform all the actual calculations in SI
units. The following constant values and conversion factors were used.

c = 299792458 m/s     (*2)
1 year = 31556926 s      (*2)(*4)
1 parsec = 3.08573 * 10^16m     (*3)
1 light-year = 9.46055 * 10^15m     (*3)
1 std. gee = 9.80665 m/s^2     (*2)

Using that most wondrous of software inventions, the spreadsheet    (*5), a
value of {a'} was found, by trial and error, which fulfilled the two elapsed
time conditions {t} and {t'}. Then {d} and {v[peak]} could be directly
calculated.

{a'} was found to be equal to about 0.02026 gee, which gets you within about
a fortnight of exact. And if _that's_ not close enough, there's no more
help, hope, or comfort _I_ can offer you. (especially since {t} and {t'}
aren't known to the nearest year anyway).

Therefore, {d}= 2*{x} was 9.54 * 10^18 meters, or 309 parsecs, or 1008
light-years (as we'd expect);
and {v[peak]} at turnover was 2.99 * 10^8 m/s, or 0.996 c.

by way of comparison, using 2*{t} = 1000yr,
{a'} ~ .0194 gee, {d} = 277 psc, {v[peak]} = 0.995c.

The slight acceleration required, about one fiftieth gee, should hardly be
noticeable to ordinary Cargo except possibly a few physicists, and it could
be that the interior of the Whorl is slightly sloped forward in such a
fashion as to compensate. Perhaps the "mountains" which divide segments of
_Whorl's_ interior longitudinally (_Exodus_ ch 15) are actually escarpments
at the edges of sections tilted slightly forward ("eastward")?

Of course, presumably "turnaround" took place some 180 years prior to the
events in BLS; were any strange perturbations in _Whorl's_ perceived gravity
noticed by the Cargo at that time, and recorded in history, as the
starcrosser executed its maneuver? Unfortunately, there are no indications
in the text that I could find.

The Bussard Ramjet, unfortunately, has one glaring limitation, provided you
grant that they and their vast "ramscoops" are even possible. A Bussard
ramjet's terminal velocity is said to limited by the resistance of the
interstellar gas against the ramscoop to about 0.90 c. This unfortunately
limits the maximum time dilation to about 40%, and the effective actual time
dilation, given that the Whorl had to gradually accelerate and then
decelerate, to quite a bit less; i.e. subjective trip time had to be well
over 400 years minimum. That just isn't going to fit well with the textual
evidence.

Perhaps the starcrosser reached 0.90 c using Bussard ramjets, and then
accelerated without the ramscoop using matter-antimatter annihilation.

Notes:
(*1) Note my "spreadsheet style" notation: ^ is of course an exponent
operator. Since ASCII includes no radical sign, the FOTRAN-like
pseudo-function "sqrt" is used to indicate a square root.
(*2) source: CRC Handbook of Chemistry, 63rd ed.
(*3) source: CRC Mathematical Tables, 26th ed.
(*4) standard 20th century Earth year used in satellite orbit calculations.
Possibly the second weakest assumption in the whole analysis, after the
value of {t}.
(*5) "Excel", ported with enormous trouble from your "Very Small-Pliable
Portals" environment to that of the great thinking engine that allows me to
communicate down the Corridors of Time with your Internet. The thinking
engine now reports an overwhelming urge to display an endless blue expanse
then lapse into a comatose state. Furthermore, it has generated a most
annoying aquastor that appears to be a piece of bent wire with eyes.

The Mad Exultant with red eyes.

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