Earth-Jupiter fast transit

I'm trying to figure out time-of flight for hyperbolic fast transits
from Earth to Jupiter, and it's not going well. Can anybody point me
towards either a table of such (with, say, varius departure V's at
earth, and the time of flight to Jupiter orbit), or perhaps a
spreadsheet or some such that does the same?

Alternatively... if you wanted to get to Jupiter in 3 months, or six
months, and you had an arbitrarily short thrust time (i.e. high thrust
system), how fast would you have to go? And how fast would you be going
at Jupiter?

Elliptical Hohmann orbits are *so* much easier...

--
Scott Lowther, Engineer
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In article ,
Scott Lowther wrote:
I'm trying to figure out time-of flight for hyperbolic fast transits
from Earth to Jupiter, and it's not going well...

'Tis a trifle messy. Starting from the beginning (although Scott
probably knows some of this):

Start with the vis-viva equation, `v^2 = GM*(2/r - 1/a)'. Assuming you're
leaving from parking orbit, plug in your post-burn `v' and the `r' of the
parking orbit, and solve for `a'; for an escape trajectory it will be
negative. Then set `r=infinity' and solve for `v', the Earth-relative
departure velocity after leaving Earth's gravitational field. You'll
naturally leave pointing "forward" along Earth's orbit, so add Earth's
orbital velocity to that. Now we switch coordinate frames (and `GM'
value) for Sun-centered orbits; we start at perihelion.

With that velocity and `r=1AU', vis-viva gives us `a', this time positive
(unless you're really departing fast...). `a = (aphelion+perihelion)/2',
so with `perihelion=1AU', check that `aphelion' gets you out as far as
Jupiter. Get eccentricity `e' from `perihelion = a*(1-e)'.

For an elliptical orbit, `r = a*(1 - e*cos(E))'. With `r' at Jupiter's
orbital radius, and the values of `a' and `e' for the orbit, solve for
`E', the eccentric anomaly. Note, angles should be in radians for these
calculations. Next, `M = E - e*sin(E)' (Kepler's equation), giving the
mean anomaly `M'.

Finally, `M = n*t', where `t' is time since perihelion, and `n' is the
mean motion, `n = sqrt(GM / a^3)'.

If you want arrival details, vis-viva will give you velocity, and
`sin(phi) = sqrt( a^2*(1-e^2) / (r*(2*a - r)) )' gives you the angle
between velocity vector and radius vector. Vector subtraction of that
from Jupiter's orbital velocity will give you Jupiter-relative velocity.

If the heliocentric `a' was negative -- velocity exceeds solar-system
escape -- hmm, let's see, less familiar case. Forget the aphelion check,
change `cos' and `sin' to `cosh' and `sinh', and I think `M' will come out
negative and you'll need to negate it before solving for `t'. Oh, and the
angle equation is `phi = pi/2 + arccos(sqrt(...the same insides...))'.

All of this, of course, assumes that you launch in the right launch
window, so Jupiter will be there when you cross its orbit. If not, things
get much uglier because now departure is not at perihelion in general.
That's called Lambert's problem and there are whole book chapters about
how to solve it.
--
MOST launched 30 June; science observations running | Henry Spencer
since Oct; first surprises seen; papers pending. |

Putting your formulae into a spread sheet, it makes sense initially,
but then goes wrong.

(Henry Spencer) wrote in message ...
In article ,
Scott Lowther wrote:
I'm trying to figure out time-of flight for hyperbolic fast transits
from Earth to Jupiter, and it's not going well...

'Tis a trifle messy. Starting from the beginning (although Scott
probably knows some of this):

Start with the vis-viva equation, `v^2 = GM*(2/r - 1/a)'. Assuming you're
leaving from parking orbit, plug in your post-burn `v' and the `r' of the
parking orbit, and solve for `a'; for an escape trajectory it will be
negative. Then set `r=infinity' and solve for `v', the Earth-relative
departure velocity after leaving Earth's gravitational field. You'll
naturally leave pointing "forward" along Earth's orbit, so add Earth's
orbital velocity to that. Now we switch coordinate frames (and `GM'
value) for Sun-centered orbits; we start at perihelion.

With that velocity and `r=1AU', vis-viva gives us `a', this time positive
(unless you're really departing fast...). `a = (aphelion+perihelion)/2',
so with `perihelion=1AU', check that `aphelion' gets you out as far as
Jupiter. Get eccentricity `e' from `perihelion = a*(1-e)'.

I assume a launch at 10km/s from the edge of Earth's gravity well.

Mass (Sun) 2E+30
G 6.65E-11

GM 1.33E+20

HEO Launch V 10 km/s
10000 m/s

V 39790 m/s
V^2 1583244100

V^2/GM 1.19E-11

r0 1.50E+11 (=1AU)

1/a 1.43E-12
a 7.00E+11

R 5.2 AU (Jupiter's distance)
7.8E+11 m

V 12,286 m/s (velocity at target (Jupiter))
12.286 km/s

Aphelion 1.25E+12 m
8.33 AU

eccentricity 0.79

This all seems logical.

For an elliptical orbit, `r = a*(1 - e*cos(E))'. With `r' at Jupiter's
orbital radius, and the values of `a' and `e' for the orbit, solve for
`E', the eccentric anomaly. Note, angles should be in radians for these
calculations. Next, `M = E - e*sin(E)' (Kepler's equation), giving the
mean anomaly `M'.

Finally, `M = n*t', where `t' is time since perihelion, and `n' is the
mean motion, `n = sqrt(GM / a^3)'.

Cos(E) -0.146
Eccentric anomoly, E 1.717460878 rad
98.40325977 degrees

Mean anomoly, M 0.940 rad
53.87416513 degrees

Mean motion, n 1.97054E-08

t 2733983665 sec
t 31643.33 days
t 88.89 years

This is too long, surely. Am I missing something obvious? Scott - I
can send you the spreadsheet if you want.

Alex

In article ,
Alex Terrell wrote:
Putting your formulae into a spread sheet, it makes sense initially,
but then goes wrong.

Okay, let's take a look...

I assume a launch at 10km/s from the edge of Earth's gravity well.

Okay, simple enough. That puts us near solar escape velocity at 1AU, but
not quite at it.

Mass (Sun) 2E+30
G 6.65E-11
GM 1.33E+20

Close enough. In fact, it is GM that is measured, and for the Sun it's
known to a ridiculous number of decimal places (1.32712440018e20), while
the uncertainty in G means the Sun's *mass* is relatively poorly known.

V 39790 m/s
V^2 1583244100
V^2/GM 1.19E-11
r0 1.50E+11 (=1AU)
1/a 1.43E-12
a 7.00E+11
R 5.2 AU (Jupiter's distance)
7.8E+11 m
eccentricity 0.79
This all seems logical.

Yep, all looks right.

Cos(E) -0.146
Eccentric anomoly, E 1.717460878 rad
98.40325977 degrees
Mean anomoly, M 0.940 rad
53.87416513 degrees
Mean motion, n 1.97054E-08

Yep, I check all that.

t 2733983665 sec

Nope. You used the degree value of M; the equations all want radians.
With radians, `t' is 4.77e7 seconds, 552 days. Which is plausible.
--
MOST launched 30 June; science observations running | Henry Spencer
since Oct; first surprises seen; papers pending. |

Scott Lowther wrote in message ...
I'm trying to figure out time-of flight for hyperbolic fast transits
from Earth to Jupiter, and it's not going well. Can anybody point me
towards either a table of such (with, say, varius departure V's at
earth, and the time of flight to Jupiter orbit), or perhaps a
spreadsheet or some such that does the same?

Alternatively... if you wanted to get to Jupiter in 3 months, or six
months, and you had an arbitrarily short thrust time (i.e. high thrust
system), how fast would you have to go? And how fast would you be going
at Jupiter?

Elliptical Hohmann orbits are *so* much easier...

It isn't exactly what you're looking for, but the paper by W.E.
Moeckel entitled "Comparison of Advanced Propulsion Concepts for Deep
Space Exploration" (Journal of Spacecraft, Vol. 9, No. 12, Dcember
1972, pp. 863-868) gives some simple equations and charts to quickly
assess the sort of system performance you'd need to enable extremely
rapid deep space missions (e.g. one month trips to Jupiter). Though it
assumes free-field space and charcterises propulsion systems into two
general types - Isp limited and thrust limited - it does give a nice
insight as to what sort of missions various advanced propulsion
systems may, or may not, enable.

There's also a paper by Smith & Mead entitled "Mission Profiles for
the Exploration of the Outer Solar System with Nucear Pulse Rockets"
(Annals New York Academy of Sciences, 4 September 1969, pp.442-452)
that has graphs of characterisic vlicity against trip time, which may
also be of some interest... though I suspect you've already seen it.


Dave

In article ,
Dave Salt wrote:
...gives some simple equations and charts to quickly
assess the sort of system performance you'd need to enable extremely
rapid deep space missions (e.g. one month trips to Jupiter). Though it
assumes free-field space...

With enough propulsion performance, that's a reasonable assumption.
For example, solar gravity is 0.6mG at 1AU, so if you have continuous
acceleration of 10mG or more, you can pretty much ignore the Sun. The
resulting trajectories are integral signs -- nearly-straight lines with
little curved bits at the ends to match the planetary orbits.
--
MOST launched 30 June; science observations running | Henry Spencer
since Oct; first surprises seen; papers pending. |

On Mon, 29 Dec 2003 04:12:28 GMT, Scott Lowther
wrote:

.... if you wanted to get to Jupiter in 3 months, or six
months, and you had an arbitrarily short thrust time (i.e. high thrust
system) ....

How high is "high"? If you accelerate at 1G for half the trip, coast
for a bit, flip over, and deccelerate at 1G for the remainder of the
voyage (and engines with such capabilities are theoretically
possible), you'd have your three month travel time while the crew
could live in relative comfort of 1G for most of the trip.

.... how fast would you be going
at Jupiter?


A better question is, How fast would you WANT to go? If youre
hypotheical spacecraft is bound for orbit of Jupiter, then you don't
want to be going much faster than Jupiter's orbital velocity when you
arrive. That -- along with the relative positions of the planet at
arrival and departure -- would determine for how long the engine is
firing to accelerate and deccelerate the vehicle.

(Henry Spencer) wrote in message ...
In article ,
Dave Salt wrote:
...gives some simple equations and charts to quickly
assess the sort of system performance you'd need to enable extremely
rapid deep space missions (e.g. one month trips to Jupiter). Though it
assumes free-field space...

With enough propulsion performance, that's a reasonable assumption.

Yes, there's a section of Moeckel's paper that assesses just how far
such free-field simplification can be taken – the conclusion being
that they're good for missions that don't involve escape from or
descent to low orbits about the major planets. However, even for
missions to Jupiter, he shows that results from simplified analyses
are off by less than 10% if the final orbit is greater than 16 Jupiter
radii.


Dave

"Michael Gallagher" wrote in message
...
On Mon, 29 Dec 2003 04:12:28 GMT, Scott Lowther
wrote:

.... if you wanted to get to Jupiter in 3 months, or six
months, and you had an arbitrarily short thrust time (i.e. high thrust
system) ....

How high is "high"? If you accelerate at 1G for half the trip, coast
for a bit, flip over, and deccelerate at 1G for the remainder of the
voyage (and engines with such capabilities are theoretically
possible), you'd have your three month travel time while the crew
could live in relative comfort of 1G for most of the trip.

Your paragraph jumped out as excessive trip time to me.
BOTE gives 37M kilometers the first day at that accelleration.
Three days to 333M kilometers. Then an equal time to distance
to stop seems to give 666M kilometers total travel in 6 days.
I will check this for correct velocity and actual distance later,
just wanted to mention it now. BTW, what engines have that
theoretical capability at this time?

.... how fast would you be going
at Jupiter?


A better question is, How fast would you WANT to go? If youre
hypotheical spacecraft is bound for orbit of Jupiter, then you don't
want to be going much faster than Jupiter's orbital velocity when you
arrive. That -- along with the relative positions of the planet at
arrival and departure -- would determine for how long the engine is
firing to accelerate and deccelerate the vehicle.

JRS: In article , seen in
news:sci.space.policy, Michael Gallagher posted at
Wed, 31 Dec 2003 09:58:54 :-
On Mon, 29 Dec 2003 04:12:28 GMT, Scott Lowther
wrote:

.... if you wanted to get to Jupiter in 3 months, or six
months, and you had an arbitrarily short thrust time (i.e. high thrust
system) ....

How high is "high"? If you accelerate at 1G for half the trip, coast
for a bit, flip over, and deccelerate at 1G for the remainder of the
voyage (and engines with such capabilities are theoretically
possible), you'd have your three month travel time while the crew
could live in relative comfort of 1G for most of the trip.


1 G is about 70 Gm/d^2; you'll get there in about a week, not three
months, unless you take a detour.

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