Sunday, April 5, 2015

Potholes on the Interplanetary Superhighway.

Wikipedia describes the Interplanetary Transport Network as "… pathways through the Solar System that require very little energy for an object to follow." See this Wikipedia article. They also say "While they use very little energy, the transport can take a very long time."

Low energy paths that take a very long time? I often hear this parroted in space exploration forums and it always leaves me scratching my head.

The lowest energy path I know of to bodies in the inner solar system is the Hohmann orbit. Or if the destination is noticeably elliptical, a transfer orbit that is tangent to both the departure and destination orbit. Although I think of bitangential transfer orbits as a more general version of the Hohmann orbit.

Bitangential Transfer Orbit
The transfer orbit is tangent to both departure and destination orbit.
The Hohmann transfer is the special case where departure and destination orbits are circular.
Illustration from my pdf on tangent orbits.


In the case of Mars, a bitangential orbit is 8.5 months give or take a month or two. Is there a path that takes a lot longer and uses almost no energy? I know of no such path.

L1 and L2

The interplanetary Superhighway supposedly relies on weak stability or weak instability boundaries between L1 and/or L2 regions. Here is an online text on 3 body Mechanics and their use in space mission design. The authors are Koon, Lo, Marsden and Ross. Shane Ross is one of the more prominent evangelists spreading the gospel of the Interplanetary Super Highway.

The focus of this online textbook is the L1 and L2 regions. From page 10:


L1 and L2 are necks between realms. In the above illustration the central body is the sun, and orbiting body Jupiter. L1 and L2 are necks or gateways between three realms: the Sun realm, the Jupiter Realm and the exterior realm.

Travel between these realms can be accomplished by weak stability or weak instability boundaries that emanate from L1 or L2. From page 11 of the same textbook:



My terms for various Lagrange necks

First letter is the central body, the second letter is the orbiting body.

Earth Moon L1: EML1
Earth Moon L2: EML2

Sun Earth L1: SEL1
Sun Earth L2: SEL2

Sun Mars L1: SML1
Sun Mars L2: SML2

Since I'm a lazy typist that is what I'll use for the rest of this post.

EML1 and 2

I am very excited about the earth-moon Lagrange necks. They've been prominent in many of my blog posts and I plan to devote another blog post to EML2 soon.

EML1 and 2 are about 5/6 and 7/6 of a lunar distance from earth:


Both necks move at the same angular velocity as the moon. So EML1 moves substantially slower than an ordinary earth orbit would at that altitude. EML2 moves substantially faster.

It takes only a tiny nudge and send objects in these regions rolling about the slopes of the effective potential hills. Outside of the moon's influence they tend to fall into ordinary two body ellipses (for a short time).

Here's the ellipse an object moving at EML1 velocity and altitude would follow if the moon weren't there:

An object nudged earthward from EML will fall into what I call an olive orbit.
It's approximately 100,000 x 300,000 km.

In practice an EML1 object nudged earthward will near the moon on the fifth apogee. If coming from behind, the moon's gravitational tug can slow the object which lowers perigee.

Here is an orbital sim where the moon's influence lowered perigee four times:



I've run sims where repeated lunar tugs have lowered perigees to atmosphere grazing perigees. Once perigee passes through the upper atmosphere, we can use aerobraking to circularize the orbit.

Orbits are time reversible. Could we use the lunar gravity assists to get from LEO to higher orbits? Unfortunately, aerobraking isn't time reversible. The atmosphere can't increase orbital speed to achieve a higher apogee. And low earth orbit has a substantially different Jacobi constant than those orbits dwelling closer to the borders of a Hill Sphere.

So to get to the lunar realm, we're stuck with the 3.1 km/s LEO burn needed to raise apogee. But once apogee is raised, many doors open.

There are low energy paths that lead from EML1 to EML2. EML2 is an exciting location.

Without the moon's influence, an object at EML2's velocity and altitude
would fly to an 1,800,000 km apogee. This is outside of earth's Hill Sphere!


In the above illustration I have an apogee beyond SEL2. But by timing the release from EML2, we could aim for other regions of the Hill Sphere, including SEL1.

Here is a sim where slightly different nudges send payloads from EML2:


See how the sun bends the path as apogee nears the Hill Sphere? From EML2 there are a multitude of wildly different paths we can choose. In this illustration I like pellet #3 (orange). It has a very low perigee that is moving about 10.8 km/s. And it got to this perigee with just a tiny nudge from EML2. Pellet # 4 is on it's way to a retrograde earth orbit. Most of the other pellets are saying good bye to earth's Hill Sphere.

I am enthusiastic about using EML1 and EML2 as hubs for travel about the earth-moon neighborhood. But a little less excited about travel about the solar system.

We've left Earth's Hill Sphere. Now what?

Recall that EML1 and 2 are ~5/6 and 7/6 of a lunar distance from the earth. SEL1 and 2 are much less dramatic: 99% and 101% of an A.U. from the sun. Objects released from these locations don't vary much from earth's orbit:



Running orbital sims gets pretty much the same result pictured above.

Mars is even worse:


Are there weak instability boundary transfers leading from SEL2 to SML1? I don't think this particular highway exists.

To get a 1.52 A.U. aphelion, we need a departure Vinfinity of 3 km/s. To be sure EML2 can help us out in achieving this Vinfinity. In other words we could use lunar assists to depart on a Hohmann orbit. But a Hohmann orbit is different from the tube of weak instability boundaries we're led to imagine.

And once we arrive at a 1.52 aphelion. we have an arrival 2.7 km/s Vinfinity we need to get rid of.

Pass through SML1 at 2.7 km/s and you'll be waving Mars goodbye. The Lagrange necks work their mojo on near parabolic orbits. And an earth to Mars Hohmann is decidedly hyperbolic with regard to Mars.

What about Phobos and Deimos? The Martian moons are too small to lend a helpful gravity assist. We need to get rid of the 2.7 km/s Vinf and neither SML1 nor the moons are going to do it for us.

Mars ballistic capture by Belbruno & Toppotu

Edward Belbruno is another well known evangelist for the Interplanetary Superhighway (though he likes to call them ballistic captures). Belbruno cowrote this pdf on ballistic Mars capture.

Here is a screen capture from the pdf:



The path from Earth@Departure to Xc is pretty much a Hohmann transfer. In fact they assume the usual departure for Mars burn. Arrival is a little different. They do a 2 km/s heliocentric circularization burn at Xc (which is above Mars' perihelion). This particular path takes an extra year or so to reach Mars.

So they accomplish Mars capture with a 2 km/s arrival burn. At first glance this seems like a .7 km/s improvement over the 2.7 km/s arrival Vinf.

Or it seems like an advantage to those unaware of the Oberth benefit. If making the burn deep in Mars' gravity well, capture can be achieved for as little as .7 km/s.

Comparing capture burns it's 2 km/s vs .7 km/s. So what do we get for an extra year of travel time? 1.3 km/s flushed down the toilet!

What About Ion Engines?

"What about ion engines?" a Belbruno defender might object. "They don't have the thrust to enjoy an Oberth benefit. So Belbruno's .7 km/s benefit is legit if your space craft is low thrust."

Belbruno & friends are looking at a trip from a nearly zero earth C3 to a nearly zero Mars C3. In other words from the edge of one Hill Sphere to another.

So to compare apples to apples I'll look at a Hohmann from SEL2 to SML1. I want to point out I'm not using Lagrange necks as key holes down some mysterious tube. They're simply the closest parts of neighboring Hill Spheres.

"Wait a minute..." says Belbruno's defender, "We're talking Hall thrusters. So no Hohmann ellipse, but a spiral."

Low earth orbit moves about 4º per minute. So a low-thrust burn lasting days does indeed result in a spiral. But Earth's heliocentric orbit moves about a degree per day while Mars' heliocentric orbit moves about half a degree per day. At this more leisurely pace, a 4 or 5 day burn looks more an impulsive burn. The transfer between Hill Spheres is more Hohmann-like than the spiral out of earth's gravity well.

Instead of a 1 x 1.524 AU orbit, the new Hohmann is  a 1.01 x 1.517 AU ellipse. The new Hohmann's perihelion is a little slower, the new aphelion a little faster.

Moreover, SEL2 moves at the same angular velocity as earth. So it's speed is about 101% earth's speed. Likewise SML1 moves at about 99.6% Mars' speed.

With this revised scenario, aphelion rendezvous delta V is now more like 2.4 km/s. Still, Belbruno's 2 km/s capture burn saves .4 km/s.

.4 km/s is better than chopped liver, right? Well, recall ion engines with very good ISP. I'll look at an exhaust velocity of of 30 km/s.

e2.4/30 - 1 = .083
e2/30 - 1 = .069

So given a 100 tonne payload, rendezvous xenon is 8.3 tonnes for Hohmann vs 6.9 tonnes for Belbruno's ballistic capture.

108.3/106.9 = 1.0134

We're adding a year to our trip time for a one percent mass improvement? Sorry, I don't see this a great trade-off.

Summary

The virtually zero energy looooong trips between planets are an urban legend.

I'll be pleasantly surprised if I'm wrong. To convince me otherwise, show me the beef. Show me the zero energy trajectory from an earth Lagrange neck to a Mars Lagrange neck.

Until then I'll think of this post as a dose of Snopes for space cadets.

I'd like to thank Mike Loucks and John P. Corrico Jr. I've held these opinions for awhile but didn't have the confidence to voice them. Who am I but an amateur with no formal training? But talking with these guys I was pleasantly surprised to find some of my heretical views were shared by pros. Without their input I would not have had the guts to publish this post.







Friday, April 3, 2015

A spiral of tethers

First off let's look at the great granddaddy of vertical tethers, the Clarke tower.

For a vertical tether in circular orbit, there's a point where the net acceleration is zero. Above that point, so called centrifugal force exceeds gravity. Below that point, gravity exceeds so-called centrifugal force. If a payload is released on this point of on the tether, it will follow a circular orbit alongside the tether. This point I call the Tether Center.

In this case, the tether center is at geosynch height, about 42,000 km from earth's center. I set 42,000 km to be 1. What path does a payload follow if released from the tether below the center?

It will be a conic section. Call the conic's eccentricity e. Call the distance from tether point r.

If dropped from below center, r  = (1-e)1/3.
If released from above center, r  = (1+e)1/3.

Here's my derivation. Mark Adler also gives a nice demonstration in the comments on that post.

This is true of any vertical tether in a circular orbit.

If there are two prograde, coplanar vertical tethers at different altitudes, there's an elliptical path between them where the perigee velocity matches a point on the lower tether and apogee velocity matches a point on the upper tether.




If a payload is released from the lower tether at the correct time, it will rise to the upper tether which will be moving the same velocity as the payload at apoapsis. Rendezvous can be accomplished with almost no delta V. Cargo can be exchanged between tethers with almost no reaction mass.

Let r for the release point above the tether be (1+e)1/3 and release point below the tether be (1-e)1/3. Then both the larger and smaller ellipse will be the same shape.


Center of the above tether is 8000 km. I tried to place it above the dense orbital debris regions of low earth orbit. The tether is 461.6 kilometers long. Dropping from the foot will send a payload to a 150 km attitude perigee. Throwing a payload from the tether top will send a payload to a 9780 km apogee.

From a 150 km altitude orbit, it takes about .33 km/s to send a payload to the tether foot.

Both ellipses have the same eccentricity, about .0864

I repeatedly clone, scale by 126% and rotate 180º:




By ascending and playing catch with a series of tethers, a payload might make it's way from LEO to the vicinity of the moon:



But there's a problem with this scheme. A tether loses orbital momentum each time it catches a payload from below. Ascending and throwing to a higher orbit also saps orbital momentum. How do we keep these tethers from sinking?

Imagine resources parked in lunar orbit. Maybe propellent mined from the lunar poles. Or perhaps platinum from an asteroid parked in a lunar DRO. To send cargo to earth's surface or low earth orbit would entail catching from a higher orbit, descending and dropping to a lower orbit:



If cargo is moved down as well as up, momentum boosting maneuvers can be balanced with momentum sapping maneuvers.

Thus mass in high orbits are sources of up momentum. This itself could be a commodity, a way to preserve orbits of momentum exchange tethers.

This tether spiral scheme cuts tether length, especially in regions of high debris density and the Van Allen Belts.

In this illustration successive ellipses vary by a factor of 21/3. Other rates of expansion are possible. Let k be the ratio of one ellipse apogee to the apogee below. k = (1+e)4/3/(1-e)4/3. Thus we can wind the spiral tighter or loosen it by choice of ellipse eccentricity.

Thursday, March 26, 2015

Partial vs full reusability

This is a topic suggested by Doug Plata. What impact will partial reusability have on efforts to settle and exploit space?

Reusable Booster stages.

SpaceX is working on a reusable booster stage. This has potentially enormous savings.

Why is reusing a booster such a big deal? Some might think getting above the atmosphere is a minor challenge compared to achieving orbital velocity. Lets take a look at hows and whys of vertical ascent.



The flight path angle is the angle between horizontal and the velocity vector.


If earth were an airless world, horizontal launches would be optimal. In other words the flight path angle would be zero.

But earth has an atmosphere. To avoid a long trip through the atmosphere a flight angle closer to vertical is called for.


Taking off from the earth at 8 km/s, a nearly vertical flight path angle vs horizontal take off.

Low earth orbit velocity is about 8 km/s. If a spacecraft achieved this velocity at earth's surface with a zero flight path angle, nearly a quarter of it's orbit (about 10,000 kilometers) would be through earth's atmosphere.

Most meteorites burn up in the mesosphere about 70 km up. Air density at this altitude is less than a thousandth of sea level. Orbital velocity at sea level would subject the rocket to extreme temperatures.

Dynamic pressure is another quantity to consider. Dynamic pressure is often denoted with the letter q. The maximum dynamic pressure a spacecraft endure is referred to as max-Q. The max-Q of the space shuttle was about 33 kilo-pascals. A severe hurricane has a dynamic pressure of 3 kilo-pascals.

8 km/s at sea level would give a dynamic pressure of about 40,000 kilo-pascals.

Before making the major horizontal burn to achieve orbital velocity, we must get above the dense lower atmosphere. The shortest path through the atmosphere is a vertical ascent.

But a vertical ascent incurs gravity loss.




Earth's surface gravity is 9.8 meters/sec^2. Each 102 seconds spent in vertical ascent costs 1 km/s delta V. Gravity loss is a major expense associated with ascent.

To minimize ascent time, a high thrust to weight ratio (T/W) ratio is desirable. The more oomph a booster stage has, the less time gravity loss is incurred.

A booster stage with more rocket engines will have a higher thrust to weight ratio. The Falcon 9 booster has 9 Merlin engines as compared to the second stage which has only 1.

Since a booster has 9 engines and the upper stage 1, would reuse mean 90% savings?

The upper stage also needs avionics, a power source, propellent tanks etc.. So I'd be surprised if the upper state is 10% of the expense. My guess would be more like 1/6. Still a 5/6 savings would be substantial.

But even a 5/6 savings wouldn't be realized by re-use. Still unknown are refurbishment costs. Also unknown is how many times a booster can be re-used.

I give better than even odds SpaceX's reusable booster will cut launch costs by 50%.


Reusable Upper Stage

After the booster stage has lifted the spacecraft above the atmosphere, the upper stage provides the horizontal burn to achieve orbital velocity. This take about 8 km/s.

Tsiolkovsky's rocket equation and an 8 km/s delta V budget mandate the upper stage is about 90% propellent and 10% dry mass. The smaller dry mass fraction means more tenuous structure and less thermal protection. It is hard to see how an upper stage could endure the extreme conditions of an 8 km/s re-entry into earth's atmosphere.

I would bet against SpaceX achieving a reusable upper stage.

Reusable Capsule

A capsule doesn't need a huge delta V budget. Just enough to lower it's perigee so it passes through the upper atmosphere. With a delta V budget less than 1 km/s, a capsule can have robust structure as well as a substantial heat shield.

I give SpaceX better than even odds at achieving a reusable Dragon capsule.

What does re-use do to economies of scale?

An item can be much cheaper if many units are mass produced on an assembly line. With mass production, design and development is amortized to a marginal expense.

If the average rocket engine is re-used 10 times, we would need at least a ten fold market increase to maintain economies of scale.

Could re-use lower prices enough to boost the market ten fold or more? I am not sure this would happen. What's the market for launch vehicles? Communication sats, surveillance and weather sats, occasionally ferrying passengers to the I.S.S. It's not clear cutting launch costs by half or even two-thirds would explode this market.


Economies of Scale with Re-use

The are possible new markets such as space tourism or mining. I don't expect those markets to take off so as a launch costs millions.

But what if the entire package was re-usable? The upper stage as well as booster and capsule? Reducing the cost by another order of magnitude opens many new markets: orbital hotels, lunar and asteroid mining, bases on the moon and Mars, etc..

But for upper stage re-use we would need propellent sources other than from the bottom of earth's gravity well. We would need orbital infra-structure: ferries between the various orbits and regions in our earth moon neighborhood: LEO, GEO, EML1, EML2 and DRO. 

Establishing this mining and transportation infra-structure could provide the initial market. Once infra-structure is established, development of space would proceed like a snow ball rolling down a hill.

In my opinion partial re-use isn't sufficient to get the ball rolling. But it's an important step toward achieving full re-use. What happens after full re-use? If we can cut expenses down to the point where propellent is the dominant cost, I'd expect the market to explode at an exponential rate.

Saturday, February 28, 2015

Clive Cussler - Two Thumbs Up

I was lamenting to a friend that science fiction is ignoring robotic advances and the near term possibilities they create. He replied "You must read Clive Cussler. I wouldn't call him a science fiction writer but he uses ROVs a lot. DARPA, the military and mining entities show up in most of his stories. I know you will like the characters from NUMA."

I took his advice and am now 3 books into the adventures of Dirk Pitt and friends. Not high literature by any means. Just entertaining, satisfying adventure yarns. And Cussler does his homework. The printed words in his books are the tip of a large iceberg. It is obvious many hours of research lie beneath the surface of each story. Cussler's interests are eclectic. He likes to study engineering and technology. Also biology, oceanography, chemistry, geology, archeology, art, history, culture, food, religion, etc., etc. Each book I've learned new stuff from many different fields.

Why am I so fired up about ROVs, AUVs, etc? It is my belief advancing robotics will be the game changer that opens the door to space, the final frontier. Cussler's stories are more relevant to space exploration than most current science fiction.

Already remotely operated robots are doing work in places too dangerous or hard to reach for human workers. This technology is being advanced by many players: DARPA, NOAA, British Petroleum, Rio Tinto, the military and others.

It is interesting that Google bought up the best performers in a recent DARPA robotics competition. Google has also invested in Planetary Resources and SpaceX as well as funded the Google Lunar X-Prize. Dot com billionaires opening the door to space could be a rich vein for story tellers.

On my wish list: Cussler taking a look at the void that lies between us and our neighbors in the solar system. Asteroids and planets are islands and continents in an ocean that extends past all horizons.







Thursday, January 29, 2015

Will I be banned from Nasa Space Flight Forum?

The largest space forum I know of is forum.nasaspaceflight.com. There are a lot of knowledgeable people who participate including a number of professional aerospace engineers.

I enjoy this forum but in my opinion there is a bias for NASA sponsored HLV.

For example, the forum has subsections devoted to manned missions to Mars, the Moon and Near Earth Asteroids. There are a number of ways such missions could be accomplished. But evidently Chris Bergin feels SLS is the only option:



Notice all the HSF (Human Space Flight) missions come under the HLV/SLS/Orion/Constellation heading.

I mentioned to Chris Bergin that there may be other routes. For example an architecture based on propellent depots might get us to the moon. Chris retorted that some of the ULA depot guys participate in NSF and they don't bad mouth SLS. Well, one of ULA's parent companies is Boeing. Umm, Chris, maybe there's a reason ULA employees would be hesitant to criticize SLS.

I posted a cartoon to NSF:



Chris found my use of the "pork" offensive. He also didn't like didn't like my portrayal of Senator Shelby.

There are a few things Apollo, Ares and SLS have in common:
1) They're very large rockets
2) They're completely expendable.

Since they're big, that means big expense. Since they're not reusable, that big expense will be incurred each and every trip.

If every mission is going to cost a few billion, we are not going to colonize the Moon, Near Earth Asteroids or Mars. Settlement would take a long, sustained effort and this sort of expense just isn't sustainable.

If Shelby et al are trying to sell SLS as a way to open a new frontier, they are committing fraud.

So what is the goal of NASA's human space flight program? The occasional flag & footprints publicity stunt doesn't justify the expense. If NASA's human space flight program is all about jobs in Florida, Texas and Alabama, it should be axed.


I make no apologies for my cartoon.


Thursday, July 10, 2014

Space topics from Dr. Plata

Doug Plata recently suggested some possible space exploration topics. All of them are very interesting.

Dr. Plata has good ideas and invests a great deal of time and effort looking at them. He's involved in two excellent websites: LunarCOTS.com and CisLunarOne.com .

July and August is a slow period for the Ajo Copper News, the weekly newspaper my sister and I publish. Most people with money and good sense leave Ajo, Arizona for the summer months. Hopefully I will have time to examine some of Plata's topics in my blog over the next few months.

Here they are:

Partial vs full reusability
Falcon 9 has nine engines on the first stage and one engine on the second stage.  So, if only the first stage is reused, it would seem to me that 9/10 engines would be recovered.  That's got to be a huge reduction in launch cost right there, yes?  Just how much?  Certainly achieving even partial reusability would make SpaceX even more competitive that it already is.  If the Falcon Heavy were to be partially reusable, reusing the lateral boosters would mean only 18 out of 28 engines would be recovered unless the central core could be reused as well.

Propulsion service options
For a cis-lunar transportation system we most often think of fuel depots in LEO.  One problem with this is the need for fuel depots in multiple LEO planes with those depots being used only occasionally.  However, if propellant were coming from ice harvesting operations at the lunar poles, then conceivably an OTV could bring propellant into any LEO inclination just prior to a launch into that orbit from Earth.  However, in this scenario, do we even need LEO depots?  Why couldn't the OTV dock with the launched satellite and then use its own engines to boost the satellite to GTO or even GEO?  Do we need fuel depots or could propulsion service be enough?

Power options for lunar mining
Say you are wanting to do ice harvesting operations in a lunar polar permanently shadowed crater with the rim of such having a peak of eternal light (PEL).  Great, but there's still potentially kilometers of distance between the source of power and the ice harvesting operations site.  How to deal with that gap?  RTGs?  Laser beaming of power?  Drive a rover laying a cable down the side of the crater?  Hop the lander from the rim to the floor while draping a (superconducting) wire?  Or forget a solar panel farm at the PEL and crack the water at a fuel depot in orbit?  An interesting trade analysis.

Aerobraking
OTVs tend to be painted as broad, turtle-shaped craft.  But how do you launch and assemble such a thing?  Can aerobraking be done about as easily with a cylindrical-shaped OTV?  How about heat flaps popping out giving more surface area and control?  Necessary?  If one skims high enough in the atmosphere does one even need a heatshield?  What about using a lifting body form?

Travel times further out into the solar system
So if we develop the ability to safely send humans to Deimos, how much longer would it take to send them to Vesta, Ceres, and a Moon of Jupiter?
 
---------
ROCKETRY
Heavy Lift vs Single Stage vs Reusable vs Gun
Air launch
Partial vs Full reusability
Chemical versus liquid rockets

CIS-LUNAR TRANSPORTATION INFRASTRUCTURE
Propulsion service options
Power options for lunar mining
Aerobraking

ORBITAL DYNAMICS
The orbital dynamics of a Phobos vs Deimos vs surface mission

HUMAN FACTORS
Mass calculations of open-loop, vs closed chemical, vs ECLSS
How could an RP5 be provided?
How do the space radiation numbers compare between locations (i.e. LEO, free-space, lunar surface, Phobos, Mars?)
Animal studies
Partial gravity options

COLONIZATION
O'Neillian vs lunar colony - Where first?
Travel times further out into the solar system

Wednesday, July 2, 2014

Kerbal Space Program

Lately this blog has been getting some hits from the Kerbal Space Program forum.

This looks like a good game. It seems based on the patched conics approach to orbital mechanics. It's good to see a popular game teaching users concepts like Hohmann or bi-elliptic transfers, sphere of influence, etc.

The art is appealing. Descriptions are entertaining. I purchased a copy for $27.00. It might be a way to become acquainted with some folks who share my orbital mechanics hobby. Hope it's a good investment!

Using the Kerbal Wiki I whomped up a HohmannKSP Spreadsheet. A few people like my spreadsheets for our solar system. Hopefully I'll be making some useful spreadsheets for this game.

Usual disclaimers apply:

My spreadsheets assume circular, coplanar orbits. Some of the game orbits are inclined and eccentric.

I occasionally make mistakes -- data entry as well as arithmetic errors. I'd be grateful if users check my efforts.