Saturday, January 12, 2013

PROFILE TO THE PLANETS









LIST OF TABLES
Mankind currently orbits to other planets.

It takes no fuel, but it takes a while.

Much quicker g-force will need fuel.

G-force acceleration will need a profile.


Line of Sight (LOS) Distance
For simplicity, Thought Experiment (TE) uses following typical values for a typical LOS distance from Earth to the planets.

KBONeptuneUranusSaturnJupiterMars
40 AU30 AU20 AU10 AU5 AU2 AU

 A g-force vessel could travel a LOS path much, much quicker then an orbiting vessel can travel a corresponding transfer orbit.



Table 1. Compare Travel Times:

Transfer Orbits vs. G-force Profiles
DestinationTransfer OrbitG-force Propulsion
SemiMajor Axis2Typical LOS3SO DistTravel Times4Constant5G-force Profile
aDdCT/2TYTdAcceltAcctDectTtl
1NEOn/a1 AUn/an/a n/a 2.00 dy1.41 dy1.41 dy2.83 dy
Mars1.52 AU2 AU5.54 AU0.71 Yr258 dy2.83 dy 2.00 dy2.00 dy4.00 dy
Jupiter5.2 AU5 AU12.33 AU2.73 Yr997 dy4.47 dy 3.16 dy3.16 dy6.32 dy
Saturn9.51 AU10 AU19.14 AU6.02 Yr2,200 dy6.32 dy4.47 dy4.47 dy8.94 dy
Uranus19.18 AU20 AU34.58 AU16.03 Yr5,854 dy8.94 dy6.32 dy6.32 dy12.65 dy
Neptune30.06 AU30 AU51.55 AU30.61 Yr11,179 dy10.95 dy 7.75 dy7.75 dy15.49 dy
Kuiper Belt40 AU40 AU67.47 AU46.42 Yr16,954 dy12.65 dy8.94 dy8.94 dy17.89 dy
ObservedAssumedπ(aτ2+bτ2)

√2
(1+aD)3/2

5.656
365.26×TY (2×d)

g
(2(d/2))

g
d

g
2d

g
1Near Earth Objects (NEOs) are a class of asteroids/comets which occasionally come within one AU of Earth.
²Line of Sight (LOS) distance; straight line distance from Earth to destination. G-force ships would be able to approximate a straight line path, a much shorter distance then a semi TO.
3SemiOrbit (SO) Distance. Typical transfer orbits must travel an elliptical path for one half of the entire transfer orbit; much greater distance then LOS.
4Constant Acceleration for entire trip increases inflight velocity way beyond usability. For useful operations at destination (orbiting and/or landing), vessel must decelerate from about half way point.
5G-force Profile: Acceleration + Deceleration. For constant g-force throughout the entire flight; acceleration time/distance must equal deceleration time/distance. Therefore, it makes sense to reverse propulsion vector at mid-point (or mid-time) of our trip.
Note:
2×(d/2)
=d
Thus,
(2×(d/2)/g)
= (d/g) = tAcc = tDec

Table 2. Compare Travel Velocities:

Transfer Orbits vs. G-force Profiles
DestinationTransfer OrbitG-force Propulsion
Transfer
Distance
Typical LOSSemiMajor AxisVelocitiesNo SlowdownG-force Profile
dτdaτvmaxvavevminvMaxvDeptvFinalvDest
NEO3.14 AU1 AUn/a AUn/a n/a n/a
1,728 k/s
01,222 k/s
0
Mars5.54 AU2 AU1.26 AU32.75 k/s26.25k/s21.54 k/s
2,444 k/s
01,728 k/s
0
Jupiter12.33 AU5 AU3.1 AU38.61 k/s14.85k/s7.43 k/s
3,864 k/s
02,732 k/s
0
Saturn19.14 AU10 AU5.255AU40.11 k/s10.65k/s4.22 k/s
5,464 k/s
03,864 k/s
0
Uranus34.58 AU20 AU10.09AU41.11 k/s7.23 k/s2.14 k/s
7,728 k/s
05,464 k/s
0
Neptune51.55 AU30 AU15.53AU41.48 k/s5.67 k/s1.38 k/s
9,465 k/s
06,693 k/s
0
K.B.67.47 AU40 AU20.5 AU41.65 k/s4.87 k/s1.04 k/s
10,929 k/s
07,728 k/s
0
TABLE-1Assumed1+aD

2
√(
2μ
-
μ

aτ
)
dτ ÷ tτ
√(

aDest
-μ

aτ
)
864 kps

day
×(2d)

g
864 kps

day
×d

g
Standard gravitational parameter (μ)
is the product of the gravitational constant and the mass of a relevant dominant astronomical body.
If this body is the Sun (☉), it may also be called the heliocentric gravitational constant (μ
).
G = 6.67428 × 1011 m3/(kg-sec2)
M= 1.98892 × 1030 kg
μ
=6.67428 ×1011m3

kg-sec2
× 1.9889×1030 kg =1.32746×1020 m3

sec2
=39.49 AU3

yr2
)
=
1.15×1010 mm

 sec
=6.28 AUAU

yr

Transfer Orbit Velocities
From Kepler's Laws, vis-viva equation shows that an orbiting object's velocity depends on
rτ, distance from Sol, varies for every point on the orbit.
aτ, semimajor axis, stays the same value for entire orbit.
vτ =[μ(
2

rτ

-
1

aτ
)]


MAXIMUM VELOCITY. For any given Transfer Orbit (TO), an orbiting object attains maximum speed when closest to the Sun, orbit's perihelion. Furthermore, these TOs always depart from orbit of Earth; so, perihelion is always 1 AU, semimajor axis of Earth ( a).
qτ = a= 1 AU
vMAX=[μ(
2

qτ

-
1

aτ
)]=[μ(
2

1 AU

-
1

aτ
)]=[μ(
2
-
1

aτ
)]

MINIMUM VELOCITY. For any given Transfer Orbit (TO), an orbiting object is at minimum speed when farthest from Sol, orbit's aphelion. Given TOs is destined for orbit of destination object; thus, TO's aphelion is also semimajor axis of destination orbit (aDest)
vMIN=[μ(
2

Qτ

-
1

aτ
)]=[μ(
2

aDest

-
1

aτ
)]

AVERAGE VELOCITY. Divide transfer distance by transfer time to determine average velocity. As a cursory, cross check, the average velocity should be between min and max velocities.
dτ ÷ tτ
G-force Propulsion


G-force propulsion gets us there via a straight line (i.e. "line of sight (LOS)"), and it attains very high velocities (much higher then orbital speeds.


Diagram shows that constant g-force ever increases velocity; in turn, distance per day increases.
TE assumes acceleration = g;
g ≈ .5 AU/day per day = .5AU/day2


To determine days required to g-force any give distance, use Newtonian expression
t =(2×d)

g
=1.414×2.5AU

.5AU/dy2
=3.16 day
To determine g-force velocities in kilometers per second (kps), recall that one day of g-force can accelerate our ship to about 864 km per second.
Thus, another value for g:
g ≈ 10 m/sec per second = 864 kps per day.

VFinal =g × t=864 kps

day
×(2d)

g
Recall Newtonian expression: Final Velocity = acceleration × time

Gotta Slowdown well prior to Destination
As a matter of fact, if we applied a consistent g-force vector all the way to Jupiter, TABLE-5 shows that ship velocity would achieve almost four thousand kilometers per second. EXAMPLE: A typical LOS distance between Earth and Jupiter could be 5 AU (depends on relative orbital positions).
Time required to g-force accelerate for 5AU is
t =(2×5AU)

.5AU/dy2
=4.472 days
After 4.5 days of g-force, vessel would travel entire distance (5 AUs), but velocity would exceed 3,800 km/sec.
VFinal =864 kps

day
×4.472 day=3,864 kps
This would be way too fast. Jupiter itself orbits the Sun at about 13 km/sec; at 3,864 km/sec, ship could not land, orbit or do anything except glimpse at Jupiter as it quickly passes by.
OPTIMAL SLOWDOWN
To maintain constant g-force for entire flight, reverse propulsion vector at midpoint.
G-force Profile: Acceleration + Deceleration.
To choose the appropriate SLOWDOWN point, consider that we want constant g-force throughout the entire flight; thus, we want acceleration time/distance to equal deceleration time/distance. Therefore, it makes sense to reverse propulsion vector at midpoint (or midtime) of our trip.

TABLE-5: Three "G-force Profile" columns:
  1. vDept At start of flight, departure velocity(vDept) is assumed to be zero for simplicity.
  2. vFinal At midway, spacecraft attains final velocity, vFinal. At this point, vessel's crew must reverse g-force vector to stop acceleration and start deceleration.
  3. vDest At end of flight, TE also assumes destination velocity (vDest) to be zero for simplicity.

There's no doubt that constant acceleration will make interplanetary travel much quicker.
For this and other benefits of constant acceleration propulsion see below.

SIDEBAR. ⇑Pros and ⇓Cons of Two Space Travel Concepts

Transfer Orbit
G-force Profile
Flight Path

Shape
⇓ No enroute propulsion constrains vehicle to the path of an elliptical orbit.⇑ Constant g-force propulsion enables vehicle to closely approximate a straight line.
Length
⇓ Longer⇑ Shorter
Control
⇓ No enroute propulsion means unable to adjust flight path prior to intercepting destination orbit.⇑ Adjust propulsion vector as required to maintain flight path.
Target Aspect
⇓ Due to everchanging direction of flight, destination remains obligue throughout most of flight.⇑ Line of Sight (LOS) really applies to this profile, target remains straight ahead throughout entire flight.
Timing
Launch Window must be considered. Since nonpowered flight must fly an orbit from dept to dest, duration will take years, thus considerable effort and waiting time required to sync prop dept from Earth to arrive at dest.⇑ Launch Window is not important. Powered flight enables vehicle to fly LOS path from dept to destination at any time. Very slight lead required, which can be accomplished enroute.
Flight Duration
Orbital Speeds Take
Months to Years
G-force Flights Reduce to
Weeks and Days
Training
Time
⇑ Lots of time to prepare for mission at destination.
Enough time to study entire college courses.
⇓ No time to train; crew will be extremely busy getting ready for subsequent flight phases.
Social
Time
⇑ Lots of time to establish essential relationships.
Marry, have babies and many other things.
⇓ Much shorter durations barely give passengers enough time to know each other.
Inflight Morale
⇓ Cabin Fever! If ship's population is too few, lack of social interaction will likely cause severe depression amongst many of those onboard.
⇑ No time for cabin fever.

Crew and pax are far too busy
Public
Interest
⇓ Eventually public will lose interest.

Media will turn to other matters.
⇑ With impending mission accomplishment, public will more likely maintain interest for first few missions. Eventually, interplanetary g-force flights will become routine and will cease to be "news".
Fuel
⇑ Extremely fuel efficient.

Virtually no fuel is required other then small amount needed to change orbits.
⇓ Large Quantity Needed. Relative amount is small (very small %TOGW due to near light speed exhaust particles); however, even a small percentage could be thousands of tons).
Large
Populations
⇓ Extremely difficult!
Quantity of food storage nearly impossible.
Alternatives: Inflight agriculture;
Stasis: (inflight hibernation) is currently a risky proposition
⇑ Still challenging, but transporting large pop for short durations has got to be easier then for long durations.
Small
Populations
⇓ Quantity of food to be stored is more practical but still challenging.
As pop shrinks, socializing opportunities correspondingly decrease.
⇑ Even for short durations, smaller populations have less problems then large pops.
Radiation
Hazards
⇓ Longer durations mean more REMs to hit ship and likely affect crew/pax.
Must add shielding to infrastructure which affects construction process and performance.
⇑ Shorter durations mean fewer REMs.

However, g-force ships will still need to shield occupants from inflight radiation.
Need to Bleed Speed
(Deceleration Requirement)

⇑ No need.


Transfer orbital velocity stays within a relatively close range.
⇓ Indeed, need is great. Considerable energy is needed to g-force accelerate to enormous speeds at midpoint. Same amount of energy needed to bring velocity back to operational levels. Trade off for extremely short duration of flight.
Inflight Navigation
⇓ Moot point! Without viable inflight propulsion, unable to adjust course. Even though destination will not be "straight ahead" for most of flight, there is no choice but to stand fast and wait for orbit change.⇑ Flying nearly straight line course, destination stays within sight for entire flight. (Thus, Line of Sight (LOS)!)

If course adjustments must be made, there is plenty of available propulsion capability to do so.
Gravitational Environment
Free
Fall
⇑ Continuous free fall can be fun! It might even prove useful, for example, takes less effort to move heavy loads.⇓ No free fall until g-force is turned off.

If that happens, ship stays at that velocity until g-force starts again.
Earth
Like
⇓Long term effects of gravity deprivation can be substantial: muscles atrophy, bones get brittle, and other medical dysfunctions.⇑ Maintaining Earth like gravity facilitates transitions to/from residing on Earth via residing in space.
Technology
⇑Current.

TO has been used many times
⇓ Achievable but not there yet. Particle accelerators are old news and ion drives are new news, but maintaining a tightly controlled, consistent flow of kilograms per second for extended periods (days) is very much future news. Thus, a spaceship with such a propulsion system remains a TE for now.
TE is now absolutely convinced that g-force powered flight is much better for human occupied interplanetary flight.
However, powered flight requires fuel.






Table 3: Fuel to Planets

Typical LOSG-force ProfileTravel
distancetimefuel
dtf
NEO
1 AU
2.83 dy
0.462% TOGW
Mars
2 AU
4.00 dy
0.654% TOGW
Jupiter
5 AU
6.32 dy
1.033% TOGW
Saturn
10 AU
8.94 dy
1.461% TOGW
Uranus
20 AU
12.65 dy
2.067% TOGW
Neptune
30 AU
15.49 dy
2.531% TOGW
Kuiper Belt
40 AU
17.89 dy
2.923% TOGW
Given2(d/g)t × %TOGWDay
Recall:
 g = .5 AU/day²

%TOGWDay0.283% /day

(mr²-1)
Assume: Exhaust particle speed: VExh = .866c;
then, dc = .866 and mr = 2 = 1÷√(1-dc²)
TE assumes propulsion vector to be a stream of near light speed particles exiting the exhaust port.
--For first half of flight, propulsion vector points opposite to direction of flight (in the same manner that current rocket propulsion systems work).
--For adequate slowdown, the spaceship must reverse the propulsion vector's direction at midpoint.
SUMMARY: G-force profile uses constant stream of accelerated exhaust particles throughout the flight.
TE assumes water as source of plasma (ionized particles to form propulsion stream), because it's plentiful, readily obtained and easily heated to steam, thence to plasma (just heat it!!). Water has many other useful qualities.
Momentum Exchange
From previous work, TE assumes mass and speed of exhaust particles directly affect propulsion capacity. Thus,
--Expelling one gram per second at 10% light speed can propel a ship of 3 metric tons (mTs) with g-force.
--1,000 grams per second (1 kg/s) can propel a 3,000 metric Tonne (mT) vessel (size of small ship or yacht).
--1 kg/s at 20% light speed can g-force propel 6,000 mTs.

TE farther assumes relativistic growth also contributes to propulsion effect of exhaust particles. For example, if original mass of 1 kg is accelerated to .866 c, then Lorentz Transform shows a doubling in size to 2 kg of exhaust particles. Thus, 1 kg/s accelerated to 86.6% light speed can g-force 50,000 mTs.
From previous work, TE further simplifies by determining fuel requirements as a percentage of ship's Take Off Gross Weight (TOGW).
Thus, TE makes hypothesis:
  • Accelerate 1% of ship's initial mass (TOGW) to 86.6% light speed can propel ship for a g-force profile of 5 AU or 6 days.
  • 2% of ship's TOGW can propel ship to 10 days.
  • 3% TOGW can propel ship to the Kuiper Belt.
TE assumes efficiency factor, ε, to account for ineffective use of exhaust particles. This is due to inevitable design flaws as well as peripheral energy needs.


TABLE 4. Inefficiency

Efficiency factor increases fuel burn requirement. 
(assume: ε = 2)
Typical LOSG-force ProfileTravel
distancetimefuel
dtf
NEO
1 AU
2.83 dy
0.924%TOGW
Mars
2 AU
4.00 dy
1.307%TOGW
Jupiter
5 AU
6.32 dy
2.067%TOGW
Saturn
10 AU
8.94 dy
2.923%TOGW
Uranus
20 AU
12.65 dy
4.133%TOGW
Neptune
30 AU
15.49 dy
5.062%TOGW
Kuiper Belt
40 AU
17.89 dy
5.846%TOGW
Given2(d/g)t × %TOGWDay

%TOGWDayε × 0.283%/day

(mr2-1)
For simplicity, TE arbitrarily chooses 50% efficiency to consider energy required for nonpropulsion use. This includes: power generation, life support, and general inefficiencies bound to occur in any human designed system.
  • Particle collisions with beam guide and/or other particles.
  • Energy is needed to power accelerator's huge array of magnets and electric apparatus.
  • Energy is needed for numerious peripheral devices as well as life support for onboard humans.



Thus, TE arbitrarily assumes ε = 2 to include all energy needed to insure adequate g-force propulsion. To further clarify, TE adjusts a couple of statements from Table-3 as follows:
To propel a ship of 3 metric tons (mTs) with g-force, ship's power system must consume 2 grams per second.
  • Expelling one gram per second at 10% light speed.
  • Allow for other gram to power peripheral devices and be consumed by design flaws.
  • 2% of ship's mass is needed to propel ship for a g-force profile of 5 AU or 6 days.
  • 1% of ship's mass actually accelerates t0 .866c and exits exhaust system.
  • Other 1% is needed for nonpropulsion uses.

Actual efficiency will need to be determined;
it will no doubt improve as g-force technology advances.

TABLE 5: 2-Way Fuel

Gotta go AND come back.
Typical LOSG-force Profile2-way
distance1-way timefuel
dtf
NEO
1 AU
2.83 dy
1.848%TOGW
Mars
2 AU
4.00 dy
2.614%TOGW
Jupiter
5 AU
6.32 dy
4.134%TOGW
Saturn
10 AU
8.94 dy
6.846%TOGW
Uranus
20 AU
12.65 dy
8.266%TOGW
Neptune
30 AU
15.49 dy
10.124%TOGW
Kuiper Belt
40 AU
17.89 dy
11.692%TOGW
Given2(d/g)2×t×%TOGWDay
TWO WAY TRAVEL:DEPARTURE AND RETURN
TE assumes it'll be a while before mankind learns to harvest fuel from spaceborne resources; thus, initial g-force missions will have to carry enough fuel to go and return.

TE assumes that scope of interplanetary missions includes a return leg. Circumstances compell TE to further assume that fuel for return must be included in ship's Take Off Gross Weight (TOGW). Thus, TE must double fuel requirements from corresponding cells in Table 4.



Onboard fuel must suffice to travel to destination and return since there are no other fuel supplies available.



Very little fuel will be required for in orbit operations while at destination; we'll assume it to be small enough to disregard. Perhaps our interplanetary vessel will immediately discharge an orbiting payload and quickly return.



Good judgment dictates considerable margin in addition to fuel planned for consumption during the two way trip. However, our thought experiment will live life on the edge and just determine how much fuel is needed to travel to our interplanetary destination and return.
Other considerations:
  • Incremental model (this one) is only an approximate model of fuel consumption but it's "good enough" for the few days required for interplanetary flights.
  • Logarithmic model of fuel consumption is much more accurate, becauses it accounts for decreasing "gross weight" of spacecraft. For longer space flights, this must be used.
Discussed in more detail in subsequent chapters.



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



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