Δv to a moon

[Loren Pechtel]

I found myself trying to figure out just how hard it would be to get to some of the outer moons.

There's that standard chart at Wikipedia but I find myself questioning it's accuracy in some situations. Specifically: aerobrakes to moons. You follow the arrows from Earth and add up every number you cross. To any planet, fine. By rockets, fine.

However, I was looking at Europa. It's showing the energy for dropping down to the main moons, and those are marked as being able to aerobrake. I can't make myself believe the numbers, though.

By rocket: Ok, the drop down energy is what it would take to put your periapsis there. The capture is what it would take to go from that orbit to Europa capture--but note that you're coming in at more than V_infinity so this isn't actually the true capture energy at Europa but it will yield the right answer as the chart is the additional energy to the objective.

However, by aerobrake: the ejection burn is aimed to skim Jupiter and at some point pass Europa. This could have a range of vectors depending on exactly where you set your apoapsis from the aerobrake, I would presume one is optimum. I can see no reason to think that the capture burn from this vector is the same as the capture burn from when the target is at the periapsis of a Jupiter capture. Furthermore, if you're going by rocket as you go down the moons you'll have a higher approach velocity for each moon in, but you will have a lower circularization energy if you're coming up from Jupiter. Thus I see no possible way that you can use the same numbers for the two scenarios.

Can anyone point me at something that actually addresses such matters?

 

[steve_bank]

Above my pay grade. What I can say is when I was confounded by a problem sometimes I'd go back to basics, conservation of energy and mass.

If you know the desired end energy you can work the vectors backwards. You could set it up in Python.

 

[lpetrich]

I've been doing similar calculations for a long time, and here are some sources for numbers.

Since we only get celestial bodies' masses from their gravity, what we find is

(gravitational constant) * (mass) =  Standard gravitational parameter - Wikipedia also has articles on the planets and their moons

At JPL:

• Astrodynamic Parameters - including planet masses
• Planetary Satellite Mean Elements - good for planning trips
• Planetary Satellite Physical Parameters - their masses and sizes

At NASA GSFC:

• Planetary Fact Sheet - Ratio to Earth
• Planetary Fact Sheet - metric units
• Planetary Fact Sheets - list of them, also has moons

 

[lpetrich]

As to the celestial mechanics, that is most simply treated by treating one celestial body's gravity as dominant, though that will vary over a trip. For each dominant body, one treats the spacecraft's path as an orbit around that body.

For going to one of Jupiter's moons, the dominant body is the Earth, then the Sun, then Jupiter, then that moon. One treats its path as a geocentric orbit, then a heliocentric one, then a jovicentric one, then one centered on that moon.

If one wants to be more precise, then one has to numerically integrate the spacecraft's equations of motion, and for best results, one will have to include planetary and lunar ephemerides, calculations of the positions of the planets and moons. They pull on each other, and that makes their orbits precess, and they can have resonance effects, which can make sizable speedups and slowdowns.

But there is no point in doing that if all one wants is a rough idea of the feasibility of some trip. In fact, one can treat the planets' and larger moons' orbits as circular to a first approximation.

 

[lpetrich]

Planets' orbital elements:

• Planetary Orbital Elements - at Reading University UK
• BAA Computing Section - British Astronomical Association
• Orbital elements - at University of Texas at Austin

Turning to rocket engines, we have two types:

• Chemical - high thrust, low exhaust velocity
• Electric - high exhaust velocity, low thrust

Chemical engines' thrust can usually be modeled as instantaneous impulses, while electric engines' thrust can be applied over several orbits, meaning that one will have to model them with numerical integration.

Tsiolkovsky's rocket equation for delta-V, the change in velocity:

\( \displaystyle{ \Delta v = v_e \log \frac{m_i}{m_f} } \)

ve = effective exhaust velocity, mi = initial mass, mf = final mass

The EEV is often reported as "specific impulse" Isp, and that as seconds, with conversion

ve = Isp*g0

g0 = 9.870665 m/s^2 - the average acceleration of gravity

That is because a rocket's thrust, the force that it makes, is often reported in English units as pounds, and a pound of force is g0 * a pound of mass.

The mass flux Q = (thrust) / (ve or Isp), so the mass flux is from (thrust in pound-force) / (Isp in seconds) = pound-mass per second.

In metric units, a rocket engine's thrust is reported either as newtons (kg*m/s^2) or as kilogram-force units.

 

[lpetrich]

Now for the two-body equations of motion. One can get velocity directly from conservation of energy:

v^2 = (G*M) * (2/r - 1/a)

where a is the semimajor axis of an orbit, the average of the minimum and maximum distances:

a = (rmin + rmax)/2

The orbit eccentricity is given as e = (rmax - rmin)/2 / a, and rmin = a*(1-e) and rmax = a*(1+e)

For hyperbolic orbits, a is negative, while for parabolic ones, a is infinite.

The escape velocity is the velocity for a parabolic orbit, where 1/a = 0:: it is sqrt(2 * (G*M) / r)

This is sqrt(2) times the orbital velocity for a circular orbit, where a = r: it is sqrt((G*M) / r)

For a hyperbolic orbit, the velocity at infinity is sqrt( - (G*M) / a ) = sqrt( v^2 - 2*(G*M)/r) for velocity v at distance r.

That makes velocity at r sqrt(vinf^2 + vesc^2) - not a linear sum. This is the Oberth effect, after its discoverer a century ago. To get a good velocity at infinity, one needs only little more than escape velocity. For the Earth, vesc = 11.2 km/s, and to go to Mars, one needs 3 km/s in interplanetary space, velocity at infinity relative to the Earth. That means that one needs only 11.6 km/s at the Earth, only 0.4 km/s more.

At the closest distances, the orbits have all transverse velocity; the radial velocity is zero. In between, one needs to calculate that velocity.

One can calculate the transverse velocity using conservation of angular momentum: sqrt( (G*M) * a * (1 -e^2) ) / r

 

[lpetrich]

A simple kind of interplanetary orbit is the Hohmann transfer orbit, an orbit with closest distance at one orbit and farthest distance at the other orbit, like closest distance at the Earth's orbit, and farthest distance at Mars's orbit.

For closest distance a and farthest distance a + da, where da << a, this gives us transfer velocities at both ends equal to
(1/4) * vcirc * (da/a)
with total
(1/2) * vcirc * (da/a)

Let us iintegrate over several orbits:

g = d(vsc)/dt = (1/2) * vcirc * (1/a) * (da/dt)

da/dt = 2 * g * a / vcirc

with spacecraft acceleration g. This gives us

t = (1/g) * ( vcirc(initial) - vcirc(final) )

and

vcirc(initial) - vcirc(final) = g * t = delv

for cumulative velocity change delv. For escape, one needs delv = vcirc.

For a high-thrust rocket, delv = vesc - vcirc = (sqrt(2) - 1) * vcirc ~ 0.414 * vcirc

So one needs more delta-V with an electric (ion) engine than with a chemical engine. This is because much of its thrust is applied where the Oberth effect is weaker than it is at one's starting distance.

One needs sqrt(2) + 1 ~ 2.414 times as much.

But that is not a very big problem for typical electric engines, because they can have very high exhaust velocity.

 

[lpetrich]

I'll now consider going to Venus, Mars, and Jupiter, with mean distances a = 0.723, 1.524, 5.203 AU -- the Earth's is 1 AU

The one-way travel time: 0.40, 0.71, 2.73 Earth years, or 4.8 mo, 8.5 mo, and 2 yr 9 mo. That's why Mars spacecraft take 9 months to make the trip.

The delta-V at each end. Departing: -2.50, 2.95, 8.79 km/s (the negative number means backward relative to the Earth's motion). Exploiting the Oberth effect gives 11.5, 11.6, 14.2 km/s from near the Earth, or 3.6, 3.7, 6.3 km/s from low Earth orbit.

For arriving, it is -2.7, 2.6, 5.6 km/s. I'll now consider exploiting the Oberth effect, with escape velocities 10.4, 5.0, 59.5 km/s. For arrival into a parabolic orbit, it is 0.35, 0.65, 0.27 km/s, while for arrival into a low orbit, it is 3.4, 2.1, 17.7 km/s.

That is why spacecraft have gone into low Venus and Mars orbits, but not into low Jupiter orbits. In fact, every spacecraft that has orbited Jupiter and Saturn has done so with a long elliptical orbit, with eccentricity close to 1, to keep the orbit-injection delta-V down.

 

[lpetrich]

For Jupiter, one may want to get there faster, and one can use a  Parabolic trajectory

One will need to go 12.3 km/s from the Earth (long-distance limit), or with the Oberth effect, 5.5 km/s more than escape velocity or 8.7 km/s more than LEO velocity.

The Earth (departure) - Sun - Jupiter (arrival) angle will be 128 d (Hohmann transfer: 180 d), and the travel time 1.11 years or 1 yr 1 mo.

At Jupiter, one will be moving at total 18.5 km/s (Sun-relative), radial 16.6 km/s, tangential 8.1 km/s (Sun-relative) or - 5.0 km/s (Jupiter-relative), or total 17.3 km/s (Jupiter-relative).

 

[lpetrich]

Jupiter's four large moons have these orbital velocities and escape velocities for them:

• Io: 17.334 km/s, 2.558 km/s
• Europa: 13.743 km/s, 2.025 km/s
• Ganymede: 10.880 km/s, 2.741 km/s
• Callisto: 8.204 km/s, 2.441 km/s

I'll now turn to what performance that one can get from various kinds of rocket engines.  Comparison of orbital rocket engines -- I'll look at those that have flown.

Chemical ones:

• LH2/LOX: 4.56 km/s - 3.6 MN
• CH4/LOX: 3.73 km/s - 2.4 MN
• RP-1/LOX: 3.52 km/s - 8.9 MN
• UDMH/N2O4: 3.10 km/s - 6.7 MN
• Solid fuel: 2.89 km/s - 16 MN
• Hydrazine: 2.2 km/s

Electric ones (ion ones):

• Dawn: 3 of NSTAR electrostatic: 30.4 km/s - 90 mN
• BepiColombo: 4 of Kaufman electrostatic: 42.2 km/s - 145 mN
• Psyche: 4 of SPT-140 Hall-effect: 17.6 km/s - 280 mN

 

[Loren Pechtel]

A simple kind of interplanetary orbit is the Hohmann transfer orbit, an orbit with closest distance at one orbit and farthest distance at the other orbit, like closest distance at the Earth's orbit, and farthest distance at Mars's orbit.

When your destination is a planet the math is simple enough, the issue is moons. And KSP-level gravity calculations are good enough to see what the situation is.

I expect the answer is nobody has addressed the problem.

 

[lpetrich]

A simple kind of interplanetary orbit is the Hohmann transfer orbit, an orbit with closest distance at one orbit and farthest distance at the other orbit, like closest distance at the Earth's orbit, and farthest distance at Mars's orbit.

When your destination is a planet the math is simple enough, the issue is moons. And KSP-level gravity calculations are good enough to see what the situation is.

I expect the answer is nobody has addressed the problem.

What problem?

I'll work out the gravitational-dominance boundary for a satellite of some object, its primary. They have masses Msat, Mpri, and distances rsat, rpri

Gravitational force of satellite must be comparable to the tidal force made by the primary. Yes, tidal force, the differential gravitational force, because the primary pulls on the satellite and one's test object.

\( \displaystyle{ \frac{M_{sat}}{r_{sat}^2} \sim \frac{M_{pri} r_{sat}}{r_{pri}^3} } \)

giving us

\( \displaystyle{ \frac{M_{sat}}{r_{sat}^3} \sim \frac{M_{pri}}{r_{pri}^3} } \)

which gives us (unperturbed period of test object around satellite) ~ (period of satellite around primary)

and also

\( \displaystyle{ r_{sat} \sim r_{pri} \left( \frac{M_{sat}}{M_{pri}} \right)^{1/3} } \)

I have thus derived the size of the  Hill sphere

 

[lpetrich]

 List of extraterrestrial orbiters - the only two orbiters of Jupiter are Galileo and Juno, but JUICE is headed outward to join them, and Europa Clipper will be launched to join them on October 10 or thereabouts.

• Gravitational parameter: GM
• Pericenter and apocenter distances: q, Q
• Distance: r
• Major axis: (q+Q)/2
• Total velocity: sqrt(GM*(2/r-1/a)) -- parabolic: sqrt(2GM/r)
• Tangential velocity: (1/r) * sqrt(GM*q*Q/a) -- parabolic: sqrt(2GM*q)/r

For Jupiter's big moons, I find these flyby velocities:

Perijove = (Io's distance)/2, apojove = (Callisto's distance)*2 and infinity (parabolic)
15.9, 14.54, 12.11, 8.73 km/s
17.33, 16.34, 14.37, 11.75 km/s

Perijove is instead 1.1*(Jupiter's equatorial radius), same two apojoves
21.73, 17.87, 14.12, 9.87 km/s
23.11, 19.58, 16.26, 12.77 km/s

Using the first one, the delta-V for going into a low orbit is
14.29, 13.25, 10.48, 7.34 km/s

So one would need ion engines for first matching velocities, and then going into orbit.

If one wants to land on the moons's surfaces, those delta-V's are
16.1, 14.68, 12.42, 9.06 km/s

Very big delta-V, and no spacecraft has ever been flown that is designed to survive such an impact speed on a solid surface. I've found proposals for penetrator spacecraft, which are designed to penetrate top layers by doing just that, but such spacecraft are still at the proposal stage.

 

[Tigers!]

For going to one of Jupiter's moons, the dominant body is the Earth, then the Sun, then Jupiter, then that moon. One treats its path as a geocentric orbit, then a heliocentric one, then a jovicentric one, then one centered on that moon.

This is where I got lost as my eyes glazed over and the world spun on a Tigers!centric orbit.

 

[Loren Pechtel]

A simple kind of interplanetary orbit is the Hohmann transfer orbit, an orbit with closest distance at one orbit and farthest distance at the other orbit, like closest distance at the Earth's orbit, and farthest distance at Mars's orbit.

When your destination is a planet the math is simple enough, the issue is moons. And KSP-level gravity calculations are good enough to see what the situation is.

I expect the answer is nobody has addressed the problem.

What problem?

I'll work out the gravitational-dominance boundary for a satellite of some object, its primary. They have masses Msat, Mpri, and distances rsat, rpri

Gravitational force of satellite must be comparable to the tidal force made by the primary. Yes, tidal force, the differential gravitational force, because the primary pulls on the satellite and one's test object.

\( \displaystyle{ \frac{M_{sat}}{r_{sat}^2} \sim \frac{M_{pri} r_{sat}}{r_{pri}^3} } \)

giving us

\( \displaystyle{ \frac{M_{sat}}{r_{sat}^3} \sim \frac{M_{pri}}{r_{pri}^3} } \)

which gives us (unperturbed period of test object around satellite) ~ (period of satellite around primary)

and also

\( \displaystyle{ r_{sat} \sim r_{pri} \left( \frac{M_{sat}}{M_{pri}} \right)^{1/3} } \)

I have thus derived the size of the  Hill sphere

The question was energy to moon capture when aerobraking on the parent body is possible. The charts always simply list it as aerobrake possible but the energies are clearly wrong.

 

[Loren Pechtel]

For going to one of Jupiter's moons, the dominant body is the Earth, then the Sun, then Jupiter, then that moon. One treats its path as a geocentric orbit, then a heliocentric one, then a jovicentric one, then one centered on that moon.

This is where I got lost as my eyes glazed over and the world spun on a Tigers!centric orbit.

What did you expect to find sticking your nose into a thread about rocket math?

(Although it is more cryptic than it should be--his equations are not rendering. I've already reported it as a bug.)

 

[bilby]

For going to one of Jupiter's moons, the dominant body is the Earth, then the Sun, then Jupiter, then that moon. One treats its path as a geocentric orbit, then a heliocentric one, then a jovicentric one, then one centered on that moon.

This is where I got lost as my eyes glazed over and the world spun on a Tigers!centric orbit.

Aww, c'mon mate, it's not rocket science.

Oh, wait.

Shit.

 

[Tigers!]

For going to one of Jupiter's moons, the dominant body is the Earth, then the Sun, then Jupiter, then that moon. One treats its path as a geocentric orbit, then a heliocentric one, then a jovicentric one, then one centered on that moon.

This is where I got lost as my eyes glazed over and the world spun on a Tigers!centric orbit.

What did you expect to find sticking your nose into a thread about rocket math?

(Although it is more cryptic than it should be--his equations are not rendering. I've already reported it as a bug.)

To broaden my horizons and improve my knowledge

 

[Loren Pechtel]

For going to one of Jupiter's moons, the dominant body is the Earth, then the Sun, then Jupiter, then that moon. One treats its path as a geocentric orbit, then a heliocentric one, then a jovicentric one, then one centered on that moon.

This is where I got lost as my eyes glazed over and the world spun on a Tigers!centric orbit.

Aww, c'mon mate, it's not rocket science.

Oh, wait.

Shit.

Actually, it's about orbits rather than rockets per se. But it definitely is math heavy.

 

[lpetrich]

British engineers report successful test of space penetrator - July 15, 2013

In the test, the penetrator was fired at a 10 tonne block of ice—it struck the block moving at approximately 340m/s, which is of course nearly the speed of sound. While the block of ice was reduced to a giant snow-cone, the electronic instruments inside the probe remained intact and in fact, continued to operate as planned, thanks to a spring mechanism engineers crafted to help soften the blow.

...
The researchers report the test penetrator experienced 24,000g as it came to a rest.

That would be suitable for most of the larger moons of the outer planets, because most of them have mostly water ice on their surfaces. Among Jupiter's four big moons, Io is an exception, with a rocky surface.

But the test penetration speed, 340 m/s, is much less than what one would need for an outer-planet moon.