Sunday, December 30, 2012

MASS TO MOTION

Converting mass to motion
is a simple fact of life.

Our bodies do it
when we walk around the block.

  We do it with planes,
trains and automobiles.

  Our thought experiment's spaceship does it
when it accelerates to neighboring planets.








Robert Hutchings Goddard (October 5, 1882-August 10, 1945). Throughout his extremely productive life, Goddard realized the potential of missiles for space flight and greatly contributed to practical realization.

As a teenager, Robert Goddard, read a serialized version of H. G. Wells's War of the Worlds and began filling notebooks with ideas for interplanetary travel. In 1907, Goddard first gained public notice from a powder rocket which produced a huge cloud of smoke in the physics building of Worcester Polytechnic Institute.
In 1914, Goddard received his first two U.S. patents.
  • Liquid fueled rocket.
  • Multi-stage rocket using solid fuel.

Altogether, Robert Goddard developed 214 patents; many were awarded posthumously.



In Goddard's" autobiographical essay: "on the afternoon of October 19, 1899, I climbed a tall cherry tree and ….. looked towards the fields at the east, I imagined how wonderful it would be to make some device which had even the possibility of ascending to Mars ...."

He died on August 10, 1945, four days after the first atomic bomb was dropped on Japan.
Expanded Text


Most people routinely buy fuel for their automobiles. We all know the purpose of gasoline is to enable vehicles to travel; thus, the fuel (mass) converts to energy which powers the engine which moves the car down the road (motion). Stop refueling, and the gas gauge eventually drops to zero; thus, we intuitively know that fuel consumption is another fact of life.

%TOGW = Fuel/MShip  
Of course, aircraft flight also depends on fuel; some aircraft designers use the term, Percent Take Off Gross Weight (%TOGW), to indicate portion of aircraft's initial gross weight needed for fuel at beginning of flight.
EXAMPLE: At beginning of trip, let a ship's total weight be 100 tons with 60 tons of fuel, then
 %TOGW = 60 tons/100 tons = 60%
for that particular trip. With a %TOGW of 60%, ship must limit all other mass (infrastructure, payload) to 40% of ship's mass. Since payload is the purpose of any trip, we want to minimize %TOGW.
Basic Assumptions:
In free space (no air resistance), an object will travel at a constant velocity until a force acts upon it. Let our thought experiment's spaceship eject particles for an entire second; the particle momentum (collective mass times their velocity) will force the spaceship to react accordingly by changing it's velocity.
MShip×VShip=mfuel× vExh
Thought experiment assumes that collective exhaust particle momentum throughout a second can be described by one vector (direction and length); thus, thought experiment further assumes an equal and opposite vector describes spaceship's momentum for same duration (one second).

MShip × VShip

1 sec
=mfuel × vExh

1 sec


Thought experiment's main premise: spaceship travels at g-force throughout entire voyage; thus, the vessel's velocity must increase by 10 m/sec for every second. Of course, this value approximates acceleration rate of free falling objects near Earth's surface. According to Einstein's famous thought experiment, occupants will feel same weight sensation as if they were static on Earth's surface. Thus, we're constrained to use the value 10 m/sec for the term, VShip.  Since we specify that value for every second of flight, we can substitute g (=10 m/sec2) for VShip/sec.

MShip × g=mfuel × vExh

1 sec
Progressive Elaboration
Following series of tables will be modified in constructive ways.
Independent Variable (IV). Thought experiment (TE) presumes that humanity will learn to design and construct particle accelerators onboard interplanetary space vessels for the purpose of expelling continuous stream of high speed exhaust particles.  This particle stream will be of sufficient quantity to propel the space vessel at continuous g-force.  TE further presumes continual improvement of these onboard particle accelerators such that the particle exhaust velocity (VExh) will continually increase as time goes on. Thus, first table will independently vary VExh in a range of values. Subsequent tables will show VExh in higher ranges.
Dependent Variable (DV). Previous work shows that MShip depends on VExh. Thus, TE further presumes that as VExh increases, that initial results will see a corresponding increase in ship size, MShip, capability. Consequently, this will decrease the %TOGW value.
MShip= mfuel

1 sec
×

vExh

g
Fuel. For each second of space flight, a collective mass of fuel is consumed; then, transformed into ions and accelerated into near light speed exhaust particles.  Since this is a Thought Experiment, we can choose a quantity which best helps us draw useful conclusions.  Thus, we initially choose a value of one kilogram of mass (mfuel) for every second of powered flight. Furthermore,let us express this mass per second (mfuel/sec) as fuel flow per second (ffsec).
MShip= ffSec×

vExh

g
TE assumes for every second of flight, one kilogram of fuel can be superheated and ionized into plasma particles which can be collected and accelerated by well placed electromagnetic fields to very high exhaust speeds.  Since there are 86,400 seconds per day; fuel flow per day can be described:
ffDay = 86,400× ffsec = 86,400 kg/day == 86.4 mT/day
With trip time, t, in days, determine total fuel, F, for trip:
F = ffDay × t
Finally, this lets us compute percent Take Off Gross Weight, %TOGW, portion of ship's initial gross weight needed for fuel.
%TOGW = F/MShip

MOMENTUM EXCHANGE

Rocket Exhaust = Ship Propulsion
mfuel× vExh = Mship × Vship
mfuel : Mass of fuel consumed.
vExh: Velocity of fuel particles exiting exhaust.
Mship : Mass of Spaceship.
Vship : Velocity of Spaceship.
CONVERT TO RATES

On each side of equation, divide selected term by one second.
mfuel

1 sec
× vExh = Mship ×Vship

1 sec
Select VShip = 10 meters per sec.
VShip

1 sec
= g = 10 m/sec

1 sec

Fuel flow per second
ffsec =mfuel

1 sec
MShip × g =  ffSec × vExh
EXAMPLE: Let particle exhaust speed, vExh = 30 million m/sec.
Mship
=ffsec × vExh

g

Mship
=1 kg/sec × 30,000,000 m/sec

10 m/sec2
From Earth, there are
many useful targets within 1 AU.
Many scientists would love a g-force spaceship
to travel in this area within a few days
vs months at orbital velocity.
 
TABLE-1. Constant g-force Acceleration
Short Range: 0.2 to 1.0 AU.
INITIAL FUEL PROFILE: No slowdown.
Assume fuel flow per second: ffsec = 1 kg
Thus, also assume daily fuel flow:
ffDay = 86,400 kg .
Givendist.(d)0.2 AU0.4 AU0.6 AU0.8 AU1.0 AU
t=(2d/g)time (t)0.89 dy1.26 dy1.55 dy 1.79 dy 2.0 dy
 864 kps / day = g  = 10 m/sec² =  g =  .5 AU /day  
VFin=t×gVel.(VFin) 769kps1,089kps1,339kps1,547kps 1,728kps
F=ffDay×tfuel (F)
77,279kg
109,228kg
133,850kg
154,557kg
172,800kg
vExh
mega-m/s
MShip
megakg
%Take Off Gross Weight (%TOGW)
101
7.73%
10.93%
13.39%
15.46%
17.28%
20 2
3.86%
5.46%
6.69%
7.73%
8.64%
303
2.58%
3.64%
4.46%
5.15%
5.76%
1kg/s×vExh

10m/sec2
=MShip
%TOGW
=F

MShip
Recall Effects of Constant Acceleration
Let d equal straight line distance from departure to destination.

g10 m

sec2
g(86,400 sec)2

sec2 × day2
×10m × AU

1.5x1011 m
g0.5 AU

day2
From Newton's laws of motion:
d = 1/2 × g × t2


Solve for t: 
t = (2 d / g )

Let g be same acceleration as experienced in free fall near Earth's surface.
For convenience, assume g closely approximates 10 m/sec2
Straightforward conversions allow us to further assume g approximates 0.5 AU/day2.

  EXAMPLE:
Let d=0.4 AU, then calculate trip time for constant acceleration for entire distance:

= (2 × 0.4 AU / 0.5 AU/day2) = 1.26 days

One can compute travel distance with trip time as independent variable. Thus, solve for d:
d =(1/2) × g × t2
 EXAMPLE:
Accelerate at 0.5 AU/day2 for .9 days; then, determine distance traveled:
d = 0.5 × 0.5 AU/day2 × (.9 day)2 = 0.202 AU
WARNING!!!!!!!!!!!!!
After only one day of g-force acceleration, spacecraft attains enormous velocity:
vFin = 864 kilometers per second!




Assume velocity increases 10 m/sec
for every second of g-force propulsion.

vFin = tsec × g = 86,400 sec ×10 m/sec²
vFin = 864,000 m/sec = 864 km/sec




STATE ANOTHER WAY.
For every day of constant acceleration, g;

spacecraft velocity increases 864 km/sec:




EXAMPLE: After two days of g-force spaceflight:
vFin = g × t = 864 km/sec /day × 2 days = 1,728 km/sec






To accommodate these enormous speeds
 interplanetary g-force space travel needs
FLIGHT PROFILE:
Slowdown at midpoint.
Operational range to 5.0 AUs expands options of notional spaceship:
1) anywhere on Martian orbit, 2) large portion of asteroid belt, 3) selected portion of Jovian orbit.
Mix Units.
To conveniently avoid large cumbersome numbers.
Mix velocity units:
  • Express exhaust particle speed, VExh. as decimal light speed, dc.
    % cdc*cmeters/sec
    10%.1 c30,000,000
    20%.2 c60,000,000
    30%.3 c90,000,000
  • Express velocity incremental increase per second as g, VShip/sec = g = 10 m/s2
    g = 86,400 sec/day ×10 m/sec2 = 864 kps/day
    g = 86,400 sec/day×864 kps/day = 0.5 AU/day2
Mix mass units:
  • Use metric Tonnes (mT) for ship size, MShip
    1 mT
    1,000 kilograms
    1,000 mT
    1,000,000 kg
  • Use kilograms for fuel flow per second, ffsec.
    Mship
    =ffsec × dc × c

    g
    Let ffsec = 1 kg/sec
    then replace constant c/g with:
    K = 30,000
    Mship= ffsec × dc × 30,000
    Let ffsec = 1 kg/sec
    vExh(dc × c)Mship
    .1 c
    3,000 mT
    .2 c
    6,000 mT
  • metric Tonnes in fuel flow per day, ffDay. While we now set fuel flow to only 1 kg/sec, this will quickly add up to many thousands of kgs.
    ffsec =
    1 kg/sec
    ffDay=86,400 kg/Day
    ffDay =
    86.4 mT/day
    F
    =
    t × ffDay
  • Thus, express total fuel, F, in mT, metric Tonnes.
Adjust Flight Profile
A better flight profile would accelerate to halfway then slowdown. Thus, expression for total time of travel changes to
 t = 2(d/g)
which is explained further below. To accommodate enormous velocity gains and still maintain g-force throughout the entire flight, accelerate at g-force until the halfway point; then, decelerate at g-force for remaining half of time/distance. Consider it axiomatic that the two halves of the flight duration are equal. The main difference between these two flight phases is that the ship's propulsion vector will point in opposite directions.

TABLE-2. Accelerate to Halfway
Midrange: 1 to 5 AU
Adjust Flight Profile to accommodate
slowdown from midpoint of trip.
Givendist. (d)1 AU2 AU3 AU4 AU5 AU
t = 2(d/g)time (t)2.8dy4.0 dy4.9 dy 5.66 dy6.32 dy
F = ffDay × tfuel (f)244mT346mT423mT489mT546mT
vExh = dc cMShip%Take Off Gross Weight (%TOGW)
.1 c3,000 mT 8.15%11.52%14.11%16.29%18.21%
.2 c6,000 mT 4.07%5.76%7.05%8.15%9.11%
.3 c9,000 mT 2.72%3.84%4.70%5.43%6.07%
.4 c12,000 mT 2.03%2.88%3.53%4.08%4.55%
g≈0.5 AU/day2
ffDay=86.4mT/dy
c≈3×108 m/sec
30,000×dc%TOGW = F/ MShip
If g-force vessels accelerates for entire distance from departure to destination, determine travel time as shown in TABLE-1:
 t = (2 ×  d / g )
However, if the acceleration distance (dAcc) is half of total distance (d/2); then, determine acceleration time as shown.
 tAcc = (2 ×  dAcc / g )
Substitute d/2 for dAcc.
  tAcc = (2 × d/2 / g )
tAcc= (d/g)

To gracefully intercept destination,
ship must decelerate during 2nd half of trip.
To maintain Earth like gravity, ship must decelerate at g-force.
Thus, apply equal but opposite propulsion vector for same distance/time as for acceleration leg.

Thus, let deceleration time equal acceleration time. 
tAccel = tDecel

Thus, we can double the above acceleration time to halfway and determine entire 1-way travel time.
tTtl = tAccel + tDecel= 2(d/g) = t
EXAMPLE. For trip distance = 5 AU, we can determine trip as follows:
2(5 AU / 0.5 AU/day2) = 6.32 days
 
Relativistic Units
THOUGHT EXPERIMENT'S MAIN ASSUMPTION
Exhaust particles gain enormous velocity, VExh, prior to exiting spacecraft.
VExh is a significant portion of c, speed of light.
Thus, VExh adds enormous momentum to propulsion particles as they exit the spacecraft.

Thought experiment further assumes that relativistic mass growth imparts even more momentum which adds even more mass capacity to the spaceship. This growth can be quantified by the Lorentz Transform.
LORENTZ TRANSFORM
① 
Compare exhaust particle's original mass (mO) with increased relativistic mass (mr).
mr
= mO

(1-v2Exh/c2)
 ②
Let mO = 1; then,
mr
= 1

(1-v2Exh/c2)
 ③ 
Substitute dc × c for vExh;
then, reduces to:
mr
=
1

(1 - dc2)
 ④ 
 Let mO = ffsec
Let original mass be the fuel consumed at rest.  To indicate this quantity, TE uses "ffsec", fuel flow per second.  To indicate mass of exhaust particles per second, TE uses "ffExh".
 ⑤
Show relativistic growth of fuel particles; consumed fuel,  ffsec, becomes high momentum
exhaust particles, ffExh.
ffExh
=
ffsec

(1 - dc2)
 ⑥ 
 Expressed another way: ffExh = mr × ffsec
Example: Let VExh
= d× c = .3 c
mr =1.0/(1 - d2) = 1.0/(1-.32)
ffExh mr × ffsec  
 = 1.048 ffSec
 ⑦
To account for relativistic effects on exhaust particles, momentum exchange equation should use  ffExh.
Mship=
ffExh × vExh

g
 ⑧ 
Make suitable substitutions:
vExh dc × c
ffExh = mr × ffsec
Mship=
mr ffsec×dcc

g
 ⑨
c ≈ 300,000,000 meters/sec
g ≈ 10 meters/sec/sec
Express ffsec in kg; Mship in mTs;
Conversion: 1kg = 10-3 mT
Replace c/g with 30,000
Mship=
mr×ffsec×dc×30,000
 ⑩
dc ffExhMship
.1
1.005 kg
3,015 mT
.2
1.021 kg
6,124 mT
.3
1.048 kg
9,435 mT
vExh/c
1 kg/(1-dc²)
ffExh × dc × 30,000

TABLE-3. 2-Way Fuel Plan

distanceGiven(d)5 AU10 AU15 AU20 AU
20 AU range includes orbits of Jupiter and Saturn
as well as large portion of orbit of Uranus.
time4(d/g)(t)12.65days 17.89days 21.91days 25.30days
To accommodate two way trip,
double the time factor from 2(d/g) to 4(d/g).
Fuel ffDay×t(F)1,093 mT1,546 mT1,893 mT2,186 mT
ffDay = 86.4 mT /day
vExh(dc×c)ffExhMShip%Take Off Gross Weight (%TOGW)
2-Way Flight Profile:
assumes four phases of equal duration
as described below.
Phase 1: Accel from dept to midway, t= (d/g).


Phase 2: Decel from midway to dest, t= (d/g).


Phase 3: Accel from dest to midway, t= (d/g).


Phase 4: Decel from midway to dept, t= (d/g).

.3c1.048kg
9,435 mT
11.59%16.39%20.06%23.17%
.4c1.091kg
13,093 mT
8.35%11.81%14.46%16.70%
.5c1.155kg
17,321 mT
6.31%8.93%10.93%12.62%
.6c1.250kg
22,928 mT
4.77%6.74%8.26%9.53%
.7c1.400kg
29,965 mT
3.65%5.16%6.32%7.30%
.8c1.667kg
40,760 mT
2.68%3.79%4.64%5.36%
GivenffSec

(1-dc2)
ffExh×dc×30,000 %TOGW = F / MShip
Derive Mass to Motion
How Much Mass Does Motion Take???
From direct inspection of prior tables, we conclude that VExh directly affects %TOGWDay which in turn determines overall %TOGW.
A
%TOGWDay=ffDay

MShip
Axiomatic: Define %TOGWDay
One day's fuel flow (ffDay) divided by initial mass of the ship will give us the proportion of ship's mass needed for that daily fuel.
B
ffDay=86,400 secs

Day
×ffSec
Axiomatic: Define ffDay
Multiply one second's fuel flow (ffsec) times 86,400 seconds/day to closely approximate one day's fuel flow.
C

MShip=mFuel/Sec × vExh

VShip/sec
MShip=ffExh* vExh

g
MShip=mrffsec× dcc

g
Recall following identities from previous work.
--Let VShip =10 m/sec; then, VShip/sec = g, acceleration due to Earth's surface gravity.
--mFuel/sec = ffExh = mr*ffsec, where mr is the factor from Lorentz Transform.
--vExh = dcc, decimal coefficient times light speed.
D
%TOGWDay=g × 86,400 ffSec

mrffsec × dcc
Rewrite %TOGWDay
Percent of TakeOff Gross Weight (%TOGW) for first day can be rewritten as shown with substitutions for ffDay and MShip.
E
mr=1

(1-dc2)
Recall mr identity
from Lorentz Transform.

Rewrite to identify dc,
light speed coefficient.
dc=(mr2-1)

mr
F
%TOGWDay=g×86,400

c
×1

mr × dc
%TOGWDay=g×86,400

c
×mr

mr×(mr2-1)
Rearrange D;
then, substitute
dc identity
from E.
G
%TOGWDay=0.283%/day

(mr2-1)
Given precise values:
g = 9.80665 m/s2;
c = 299,792,458;
day = 86,400 sec;
86,400g/c=.002826 ≈ 0.283%
and F becomes G.

TABLE-4. Longer Range: 10 to 40 AU
dist.Given(d)10 AU20 AU30 AU40 AU
time4(d/g)(t)17.89 days 25.30 days30.98 days35.78 days
vExh
(dc×c)
mr%TOGWDay%TOGW
.866 c20.16% 2.92%4.13%5.06%5.85%
.943 c30.10% 1.79%2.53%3.10%3.58%
.968 c40.07% 1.31%1.85%2.26%2.61%
.980 c50.057% 1.03%1.44%1.78%2.04%
Given1

(1-dc2)
0.283%

(mr2-1)
t × %TOGWDay
Adjust Fuel Profile to directly determine daily decrease in ship's mass due to loss of propulsion particles. Thus, we no longer need to compute daily fuel requirement (ffDay) and compare to ship size (MShip). We can directly determine percentage mass decrease of ship's mass. Thus, delete fuel mass row from TABLE-3, and %TOGWDay column replaces TABLE-3's ship size (MShip)column. Tables focus on %TOGW, a relevant parameter of space flight. They show that TOGW varies according to speed of exhaust particles. TOGW does not depend on distance, time or fuel which our table keeps static. (Subsequent blogs consider effects of varying fuel consumption; eventually, we'll consider effects of varying g.)
As vExh grows, ship size (MShip)grows dramatically; thus, ship size depends on particle momentum but total fuel consumption still depends on distance/time. Thus, for the same distance, %TOGW (= Fuel / MShip) decreases dramatically as vExh grows. For increasing distances, %TOGW as distance increases and decreases monotonically as vExh increases.
Thus, we conclude... %TOGW is the amount of ship's mass that must convert to kinetic energy (i.e., "motion") to g-force accelerate it throughout the entire trip.
Furthermore, the converted energy does other things. TE spaceship must use the accelerated particles for propulsion, but some energy will be needed for ship's life support, navigation, communication, entertainment. Even the propulsion system draws energy which doesn't directly propel the vessel; the particle acceleration process requires energy for the many components (magnets, particle detectors, etc.) of the accelerator which need lots of energy.
%TOGWDay =ffDay


MShip
=86,400 × ffsec


30.57 (mr2-1) × 106 × ffsec
=0.283%/day

(mr2-1)
%TOGWTtl = t × %TOGWDay = t × 0.283%/day

(mr2-1)


%TOGW depends on mr, relativistic growth factor and time, t. As shown in above equation, %TOGW is inversely proportional to mr. As it grows, %TOGW shrinks which means less matter needs to convert to energy for same days performance. Thus, more of ship's GW can be devoted to payload, a good thing.

Father of Modern Rocket Propulsion

Sidebar: Robert Goddard
Electric Propulsion
RHG's" early career as a young academic physicist was divided between his official research on electricity and his personal passion for propulsion . This would naturally lead him to think of Electric Propulsion (EP).

In particular, he posed the question: “At enormous potentials can electrons be liberated at the speed of light, and if the potential is still further increased will the reaction increase (to what extent) or will radio-activity be produced?"
Why was Goddard more concerned with the electrostatic acceleration of electrons rather than ions? (Recall that ions are much more massive and contribute far more to momentum exchange between the rocket and exhaust particles.)
An online paper suggests five reasons:

  1. The nature of cathode rays was still debated at that time and the ionization physics underlying the production of electron-ion pairs was not clear.
  2. There was the implicit belief in these early writings that high accelerating voltages (and not high beam currents) were the main technical difficulty. This, consequently, favored electrons as the propellant needed to reach extremely high velocities.
  3. There was still a lack of appreciation of the immense difficulty, stipulated by the laws of special relativity, in accelerating a particle having a finite rest mass to a speed very near that of light.
  4. It is doubtful that Goddard, at this early time, had fully appreciated the practical (i.e., system-related) penalty incurred by an electric rocket with an exceedingly high exhaust velocity.
  5. There is another system-related penalty that was likely far from Goddard’s mind. Electrostatic acceleration of lighter atoms, let alone electrons, although less demanding on the voltage, results in beam currents which, because of space charge limitation, incur adverse demands on the required area (and therefore size and mass) of the accelerator.
As in many instances in the career of this ingenious and practical scientist, his ideas culminated, by 1917, in two inventions whose importance to the history of EP has been largely unrecognized.
SLIDESHOW



VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR



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