I’d like to share a technical paper about propellant depots and interplanetary mission orbital dynamics that I helped co-author this past year, with the help of two of my astrogator friends1, Mike Loucks and John Carrico of Space Exploration Engineering and The Astrogators Guild blog.
Backstory
By way of preface, this is a paper that Mike and I have been meaning to write for almost five years now2 as a rebuttal to some anti-LEO depot arguments that had started to come out back in the 2010 timeframe. You see, back when the FY2011 NASA Budget came out, those of us who had been advocating LEO propellant depots as a source-agnostic way of driving innovations in low-cost launch and Lunar/NEO ISRU thought we had finally won the day. Constellation had been cancelled. The president was proposing having NASA invest heavily in technology demonstrations for reducing to practice ideas like depots.
But then the idea of LEO depots started taking a lot of flack from many directions. Probably one of the most effective critiques of LEO propellant depots came from a NASA Flight Dynamics Office out of JSC who pointed out some orbital dynamics challenges of using LEO depots for doing interplanetary missions to places like asteroids. While I think the issues were raised in good faith–there are some legitimate challenges that LEO depots need to overcome–groups at NASA that didn’t want competition for their Monster Rocket used these arguments to “prove” that LEO depots weren’t really that useful after all, because you see, they weren’t useful for performing the asteroid missions NASA was now planning. It probably didn’t help that a certain one of its parent companies informed ULA’s depot advocates in no uncertain terms that depot was now a four-letter word that could be severely career limiting to use3.
The AAS Paper
Before I get into the orbital dynamics issue that was raised, and the solution we present in the paper, why don’t I share a link to the paper itself for those of you who would like to cut to the chase: AAS 17-696. This was presented at the AAS/AIAA Astrodynamics Specialist Conference which was held August 20-24, 2017 in Stevenson, Washington, U.S.A. If you’re interested, hard copies of this and the other presentations will soon be available in Volume 162 of Advances in the Astronautical Sciences. Fortunately AAS was fine with me sharing a copy of this on the blog so long as I gave proper credit.
The “Show Stopper”
So, unless you’re a real space nerd and happen to already know the answer, you’re probably wondering what the orbital dynamics issue was that NASA used as an excuse to ignore depots over the past five years. The tl;dr version of the issue is that while you can launch a LEO propellant depot in a way that lines up well for one specific interplanetary departure opportunity, it’s almost guaranteed to not be aligned well for most subsequent interplanetary missions you’d like to perform.
The longer version is driven by the concept of nodal precession. You see, because our planet spins, it’s a little… round about the middle. This bulge causes a “J2 perturbation” to orbits4, that basically causes the orbit plane to slowly precess around the earth’s rotational axis. I can’t remember exact numbers off the top of my head, but I think we’re talking 5-7 degrees per day for an ISS-like depot orbit. I could geek out on more facts about nodal precession, but here’s why that matters–once you launch a depot into LEO, you’ve established an orbital plane for that depot. That plane will precess over time in a very predictable way. The problem is that for a given interplanetary departure window, you need to leave earth on a specific departure vector (called the departure asymptote)5, and if the plane of your LEO parking orbit doesn’t intercept that departure asymptote vector at your departure date, you can’t do a coplanar one-burn departure, which means you have to deal with a very painful plane-change delta-V penalty that gets rapidly more painful the further the misalignment between your departure vector and your orbital plane. Another problem is that for weird destinations like some NEOs, the required departure asymptote is pointing off in a direction far from the equator–far enough that it might be higher than the inclination of a reasonable LEO depot, which means that you would never get a one-burn coplanar departure opportunity that didn’t require a debilitating plane change maneuver.
The good news is that since nodal precession is pretty easy to model, you can definitely place your depot into an orbital plane that will drift into alignment with a single departure opportunity. The problem is that the odds of it lining up with any given future arbitrary departure opportunity is pretty poor. Depending on how short the window is, and it tends to be really short for many NEOs, your odds might be less than 25% of having your depot in the right place at the right time. The odds of alignment are better if you’re talking planetary departures, as those tend to have longer windows at lower declinations (the angle from the equatorial plane to the departure asymptote vector), but still nowhere near the 100% you’d probably prefer for a critical piece of space infrastructure.
Ground launches don’t have the same problem because unlike an orbital plane that precesses very slowly, a ground launch site rotates with the earth 360 degrees every 24hrs. And a ground site can easily launch into a parking orbit with any inclination higher than the latitude of the site. Basically, you can pick the parking orbit of your departure to line up with your target every time.
I’m glossing over some details, but it all seems pretty damning when you think of it like that. Why build an expensive piece of infrastructure that you can’t count on being usable for any given mission?
Previous Proposed Solutions
Fortunately, there are some potential solutions that have been proposed in the past that still allow you to use LEO depots but get around this problem6. One option would be to have multiple depots instead of just one. If you have somewhere around 4-6 evenly-spaced depots, you should be able to always have a depot that will be aligned with a given destination at the right time, and you’ll know far enough in advance that you can logistically schedule different missions out of different depots. The problem is that this probably only makes sense if you have a really high flight rate. Another option proposed is to phase the depot orbit between missions–you can change your nodal precession rate by raising your apogee or lowering your perigee. If you know where you need to be far enough in advance, you don’t have to tweak your orbit too far to have it either precess faster or slower so it ends up in the right place at the right time. The problem with this approach is that it only works with really low flight rates, since it can take a long time to phase into the right orbit, especially if radiation or drag limits prevent you from changing your apogee or perigee too much. And neither of these solutions solve the problem if your departure declination is higher than the inclination of of your orbit, because the only solution at that point would be to do a plane change for your depot to get into a higher inclination orbit–and those are really painful maneuvers from a delta-V standpoint.
3-Burn Departures
While both of those solutions kind of work, Mike Loucks and I came up with a potentially better solution based on a MXER tether paper by someone who many Selenian Boondocks readers might recognize. You see, the best place to put a MXER tether is in equatorial orbit, because that maximizes the frequency of launch opportunities you have. But as I mentioned above, if the declination of the departure vector is higher than the inclination of your orbit, you’ll never have a coplanar one-burn departure. So what was Kirk’s solution? Use a 3-burn trajectory!
But first I need to explain something about departure orbits. Interplanetary departure orbits are by definition hyperbolic trajectories. And hyperbolic trajectories don’t behave quite like circular or elliptical ones. With a circular orbit, if you do an instantaneous (impulsive) burn, that point in your orbit where you did the burn becomes the perigee of an eliptical orbit, and the apogee will occur on a line that goes from that perigee through the center of the planet (that line is called the line of apsides if you’re wondering). So long as you’re still in an elliptical orbit this will always be the case. If you hit exactly escape velocity (i.e. have no excess velocity above escape), you’re in a parabolic orbit, which will point asymptotically as it approaches infinity in that same direction (i.e. parallel with the line between your perigee and the center of the earth). But as soon as you go a little faster, you’re in a hyperbolic trajectory, and the trajectory you end up asymptotically approach at infinity no longer aligns with that same line–the angle that a hyperbolic orbit has relative to that line between the perigee and the departure asymptote is I believe called the turning angle. And the turning angle gets bigger and bigger the faster you’re going relative to escape velocity. Combine that with the fact that any plane that is coplanar with the desired departure asymptote can allow a hyperbolic departure, and you get a famous drawing like that in the astrodynamics book by Bate, Mueller, and White (Figure 2 from both of the referenced papers) shown below:
Hyperbolic Departures (credit: Bate, Mueller, and White)
What you may notice is that there’s actually a ring, or “locus” of injection points (aka locus of periapses) that all allow you to get to the same departure asymptote. This ring, which is centered on the axis formed by the desired departure asymptote gets wider as the required injection velocity increases (higher C3), and narrower as it decreases.
Ok, so how does this enable a 3-burn departure that gets around the constraints described in the earlier section?
In order to do a coplanar departure without penalties, you need your perigee to be on that locus of injection points at the time of the final departure burn, you need the plane to be coplanar with the departure asymptote, and you need to be going in the right direction in your orbit. You can meet those three criteria if your LEO depot orbit plane happens to line up with the departure asymptote at the right time, but it turns out there’s a 3-burn trick you can do that allows you to meet those criteria for your final burn so long as your depot’s orbital plane ever crosses through any point on that locus of periapses at a point prior to your desired launch window.
Here’s how:
- At any time your orbit crosses through that locus of injection points, you do a large apogee raising burn that has an orbital period timed so that you return to perigee at the exact time you want to do the interplanetary departure burn7. This solves having your perigee on the right locus of injection points at the right time even if your LEO depot orbit has long since precessed out of optimal alignment. This burn also is not wasted as all of the energy you put into raising your apogee has come back as kinetic energy when you’re back at perigee for the final burn.
- Once you’re at apogee you’ll probably need to do a plane change maneuver to rotate your plane so that when you get back down to perigee your plane is coplanar with the desired departure asymptote, and you’re headed in the right direction. Since plane change costs are proportional to your velocity at the point in your orbit that you do them, they’re cheapest at apogee, especially for a high apogee near escape velocity.
- When you drop down to the perigee, you’re now lined up for your third and final burn which sends you to your destination.
If you’re having a hard time visualizing it, our AAS paper has a bunch of illustrations of what the trajectory looks like, including some practical mission examples, including an excellent acid test case provided by Josh Hopkins of Lockheed Martin (a mission to NEO 2007 XB23, which has a really crazy departure declination of -72 degrees).
Now there are all sorts of subtle nuances that can complicate this or make things better. For instance, because of how vectors add, you almost certainly want to split your plane change up between burns one, two, and three, because that’ll lower the overall penalty. Also, if your apogee is too high, you might start running into lunar perturbations that would have to be compensated for. Also, you might want to lower your perigee when you’re at apogee, to get a little more boost from the Oberth effect. Also, while your nodal precession rate drops off dramatically for a highly elliptical orbit, it’s not exactly zero. But all of these perturbations can be modeled and planned for when designing your 3-burn departure trajectory.
It’s worth mentioning that you can do this process in reverse to rendezvous with a LEO depot when coming in from an arbitrary interplanetary trajectory8.
While this is more complex than a single-burn departure, look what this does for you:
- You now can always hit your desired departure window even if your depot orbit itself is very misaligned with the asymptote.
- You can hit a departure asymptote even if the declination is higher than your inclination–it now just has to be lower than the sum of your inclination and turning angle, meaning that the higher the required departure velocity, the lower the inclination of your depot can be to hit a given departure declination. If your departure C3 is >16km^2/s^2, you can hit any departure declination from an ISS-like 51.6 degree depot orbit.
- It can also allow you to do missions that would require more propellant than your depot can handle. Basically, you can launch one tanker into this highly elliptical parking orbit that has your perigee on the desired injection point, long in advance of your desired departure window. The nodal precession rate of this highly elliptical orbit will be almost zero, so the depot orbit will rotate into alignment with it approximately once every 2-2.5 months. Each time it aligns with this parking orbit with the tanker in it, you can launch another tanker to rendezvous with, and add propellant to the first one, until it’s all the way full. Once that’s done, you can expend, or better yet aerobrake these depleted tankers back into LEO for reuse. Then when you’re at your last perigee before departure, you can launch the actual mission stack into the highly elliptical phasing orbit, rendezvous with the now full tanker, transfer propellant from the tanker, and then do your plane change burn at apogee, and your departure burn when you’re back down to perigee. In this way, or with variations on the theme, you can do really impressive missions using relatively modest sized depots9.
- And you get all of these benefits without having to do large numbers of depots and without having to move the depot, so you could theoretically park the depot in something like a resonant orbit that makes refueling logistics much easier10.
There are a few drawbacks or complexities, but they’re mostly minor:
- You do get two or more extra passes through the Van Allen Belt per departure mission. Not the best thing in the world, but not the end of the world either.
- You do still have some plane change penalty11, unless you can perform the first burn at a time when your LEO depot orbit is coplanar with the the departure asymptote12.
- You add a non-trivial amount of time to the mission, possibly on the order of weeks.
- You now have to do at least two burns a very long time after the first burn. Most current rocket stages are only designed for mission durations of 14hrs or less. Which means if you don’t have a long-lived stage (like ULA’s planned ACES stage), you probably need to do the second and third burn using something storable, which does cause a slight performance hit. In most cases the first burn is the biggest of the three though, so you still get some benefit from having your higher performance stage even if you have to have a kick stage of sorts for the final departure.
- You have more mission complexity and three important burns rather than just one for departure. But the nice thing is if either the first or second burn fails, you can probably abort the mission.
One “turning lemons into lemonade” advantage of this approach is that you get more time to check out your vehicle before its committed to interplanetary space. Once you’ve done that final departure burn, most systems really have no way to abort if something goes wrong. And with the bathtub reliability curve most complex systems have, it might actually be better to have an extra 2-3 weeks of checkout time during the elliptical phasing orbit to make sure everything is really ready for commit, with the option of aborting the mission if its not. Another side benefit of the elliptical orbit is that they have much lower (likely 10x lower) cryogenic boiloff rates, since you spend more time far away from nice warm planetary bodies. And you spend most of your time away from LEO where the MMOD environment is much, much better, which means that if you want to build up a mission stack over time (in a similar manner to the tanker concept discussed above), you can do so with less worry of MMOD damage to your mission stack while it waits than if you built it up in LEO.
Conclusions
While there’s still a lot of work to better evaluate optimizations and tradeoffs, and to explore various architectures enabled by this 3-burn departure (or arrival) method, we were able to identify and demonstrate a method that allows you to reuse a LEO depot for multiple missions, in a way that can always hit the desired departures at minimum penalty, even in spite of the previously raised issues due to LEO depot nodal precession. I wish we had been able to present this four or five years ago when we first discovered the solution. Maybe that would’ve at least dispelled the notion that orbital dynamics are a show-stopper for LEO propellant depots.
Jonathan Goff
Latest posts by Jonathan Goff (see all)
- NASA’s Selection of the Blue Moon Lander for Artemis V - May 25, 2023
- Fill ‘er Up: New AIAA Aerospace America Article on Propellant Depots - September 2, 2022
- Independent Perspectives on Cislunar Depotization - August 26, 2022
- And yes, I think it’s awesome that there are people who legitimately have the job title of astrogator.
- Unfortunately, one of the challenges of running a startup is that it leaves you with less time than you’d desire for writing projects that don’t have the potential for getting your startup investment or more paying work. As you all have probably noticed from how rarely I’m able to post on this blog anymore. It’s even harder to do conference-quality technical papers. And even harder when your coauthors are also running their own startups.
- The less said about that, the better.
- Kudos to the first astrogation geek who makes a t-shirt cracking a joke about how it isn’t a beer gut, it’s just a J2 perturbation
- Think of this as the leftover velocity vector relative to earth that your spacecraft would have once you leave earth’s gravitational sphere of influence
- I should probably also note that this problem is not a problem for LEO depot missions to the Moon. Since the moon orbits around the earth approximately every 28ish days, and orbits in a direction opposite to the direction of LEO orbit nodal precession, it means you get coplanar lunar departure opportunities for most LEO orbits approximately every 7-9 days. And if we ever get to the point where we’re flying to the Moon so often that that isn’t frequent enough, we’ve basically already won, haven’t we?
- Theoretically, you could do a lower apogee orbit whose period is some integer fraction of the time between this first burn and the final injection burn. You’d do two or three orbits and then at the end of the last orbit do your injection burn
- Or a LLO depot coming from Earth, which is a likely issue, because the depot on the receiving end will almost never be aligned correctly when the planes align for a departure from the departing depot
- I should probably run some numbers to illustrate, but that’s an exercise for another day
- Previously discussed here and here.
- The analyses we performed were all less than 200m/s penalty vs a coplanar single-burn departure.
- We haven’t flexed the models enough to be sure, but my guess is that it’s a trade of hang-time versus plane change penalty
This is seriously useful tech. Works well with a couple of my hobby horses. Propellant to depot can be an automatic secondary payload for launch vehicles with lighter than max primaries. And dedicated tankers should be cheaper than standard LVs with the payload in the regular stretch tanks and have a higher payload for the same reason.
Here’s hoping some decision makers notice this capability to incorporate into future missions. I’ll have to reread this post and the links a couple of times to get a better understanding.
This wouldn’t save much if at all, but are there some occasions when Lunar perturbations can be used in place of the second burn? And what are the destination limits on a craft fueled at apogee for a 6 km/sec Oberth burn at perigee with your method?
If one is planning to reuse interplanetary vehicles, then on return you park them in the highly elliptical orbit, not LEO.
John,
Re: flying propellant as a secondary payload. This might be doable in some situations, but there are probably ones it doesn’t fit well with. I could see two general scenarios where that idea could work:
1- If they’re launching into a popular LEO orbit, such as going to ISS (or some future space facility), or possibly either sun-synchronous, or near polar orbits. In that case if you had a co-orbiting LEO depot, after the main payload leaves and heads to ISS (or after it gets dropped off in sun synch), it might be possible for the upper stage to rendezvous with the depot and top it off. We have a technical concept for enabling that, but it’ll almost certainly require at least some light modifications to the upper stage doing the drop-off.
2- If you have a launcher with spare capacity that isn’t going to a popular LEO destination, you might be able to have the upper stage get to one of the staging ellipses I spoke of. But it’ll take some real work to plan that out in advance. You’d need to have enough performance to get all the way up to the desired final elliptical orbit, and you’d have to make sure your perigee is at the right place relative to the desired injection point. For a LEO orbit, probably mostly a question of if you have enough left over propellant to get to the elliptical phasing orbit. But in the case of GTO or such, you’d have to have your GTO injection take place at the same point as the desired injection point. Which probably only will work for missions where the locus of injection points crosses the equator, since usually I think GTO injections take place where the initial parking orbit crosses the equator. Not impossible, but you’d definitely need to plan ahead–my guess is you could probably have a decent company just handling the planning logistics of making something like that work.
Re: using lunar perturbations to reduce the plane change dV–might be possible, but you’d need someone who really knew their stuff to plan that out, and every time you add an extra constraint, it comes at a cost. My guess is that you’d develop some automated departure planning tools that would allow you to check what the dV penalty would be without it, and would do a quick evaluation of if it is likely that the Moon is in the right place to help or not. And only if it passed the screening check would you bother with doing the detailed planning work to make that happen. People like Mike Loucks, and John Carrico, and others have done tricks like this, but from the sound of it, it’s a lot more complicated to pull off than it sounds. But when you’re talking about saving even $1M, I guess that throwing an expert at the problem full-time even for a full year isn’t ridiculous.
I’m not sure I have enough info to answer your last question.
~Jon
Paul,
I think that could work in some cases. You’d need to know what the next interplanetary mission is going to be before your ship got back from the previous one in order to get into the right highly-elliptical staging orbit. I’m guessing here, since I haven’t done a lot of interplanetary arrival work, but I think that when you arrive you have an arrival asymptote vector, and there’s probably a locus of planetary insertion points where you’d want to do your capture burn (or center your aerocapture braking). So long as that locus overlaps with the locus of injection points for the next planned mission, I would think you could do a minor course correction to come in from the right angle, do your planetary capture burn/aerocapture, and capture into the planned ellipse for the future mission. You’d probably want to do a plane change at that point to prep things, then have the part that returns to LEO break down from there at the next perigee. Or I guess you could split the part that stays and the part that is going back to LEO, and have the LEO-bound part directly aerocapture/aerobrake down to LEO, and have the part you want to leave in orbit do the breaking maneuvr and plane change to prep for the next mission.
I hope that answer makes any sense. I’m fighting off a cold or fever or something.
~Jon
Jon,
You’re correct on the interplanetary arrival. Your periapsis (or closest approach, if we dislike that term for hyperbolic trajectories) on arrival should automatically land on the locus of planetary insertion points.
But that locus doesn’t hold still. Even neglecting the small precession for an elliptical orbit, Earth’s orbital movement means that even a few weeks later, your depot’s locus will shift to increasingly higher energies until it crosses the Earth-Sun “noon” line. Then the transfer energies will decrease again, but they’ll correspond to asymptotes pointing the opposite direction. For instance, if your depot or spacecraft is correctly aligned for a given Mars departure window on a given date, three months later it will be correctly aligned for a direct Solar escape, and another three months later it will be correctly aligned for a Venus crossing orbit (all neglecting precession).
I suppose it would be worthwhile to do a per-orbit correction to keep the locus fixed relative to the terminator? Or maybe to tune the orbit’s ellipticity so that the nodal precession works in your favor? Actually, that might be a good idea, picking a particular locus and an ellipticity that results in just under 1 degree of nodal precession per day. That may not be ideal for the later plane change burn for your spacecraft, but it does allow destination-dedicated depots and reusable craft parking orbits, and it’s better for the mission than the LEO depot. Really, it would be for a class of destinations: NEOs with SMA greater than Earth’s, for instance, would generally have similar-enough transfer energy requirements that the locus associated with the Mars-oriented depot’s perigee could be close enough.
By the way, your description was very easy to visualize, even without the diagram. Thank you.
I really like your idea, at any rate. Even without reusable craft, Mars is such a frequent target that dedicated depots tuned to nodally precess in a useful way to maintain the correct locus might be superior to generic depots in LEO, but even the LEO depots can save a lot on mission requirements.
Apparently, I should have read the paper before commenting:
“3. Depot Orbit Phasing: This technique would use maneuvers to increase or decrease the semimajor axis to change the nodal precession rate.”
Still, there’s a large class of missions for which, if the depot SMA were chosen to correspond to the terminator’s movement (0.98 degrees per day), with a locus corresponding to the energy required for a Mars crossing transfer, the depot wouldn’t have to do too much maneuvering besides station keeping. Of course, I also haven’t checked whether such an orbit would regularly cross the inner radiation belt threshold. If its apogee is parked at 3,000 km, with perigee of 300-400 km, that’s problematic.
John Hare’s 3rd question:
“what are the destination limits on a craft fueled at apogee for a 6 km/sec Oberth burn at perigee with your method?”
Hopefully I’m understanding the question correctly. You fuel from a tanker after the first two burns, and the third burn doesn’t have to have a plane change component.
If you start in a highly elliptical orbit your perigee velocity can be just under escape, say 11 km/s. You then add 6 km/s in a perigee burn to go 17 km/s. Then your C3 is 17^2-11.2^2 = ~160 m^2/s^2. Browsing the web, it appears that would get you to at least Saturn.
Bob,
160km^2/s^2 would get you a lot more than to Saturn. A direct to Saturn launch is on the order of half of that (80km^2/s^2). 160km^2/s^2 is almost as high as the New Horizons C3 (which was ~170km^2/s^2). But in this case with a *much* bigger payload.
~Jon
“I think that could work in some cases. You’d need to know what the next interplanetary mission is going to be before your ship got back from the previous one in order to get into the right highly-elliptical staging orbit.”
It would also be possible to rearrange the orbit by boosting out to very large distance and doing something with solar gravity, I think, so it could then come back into a highly elliptical orbit pointed in a different direction. The delta-Vs required need not be high.
Is there a particular orbit for a depot that covers both interplanetary 3 burn departure and transfer orbits to the Moon (maybe assume L1/L2 shenanigans) ? Collary, is a there a 3 burn depot orbit that lines up nicely with a terminator riding SSO orbit as well (For an atmosphere scooper tanker like PROFAC to deliver propellant to the depot)?
N/A,
Depots with good orbits for the Moon are relatively easy, I think. Just about any orbit is going to intersect with the Moon’s orbit plane multiple times per month. So I’d pick a convenient medium inclination orbit (somewhere in the 30-60degrees range). You want something low enough to minimize delta-V penalties, but high enough inclination to enable as many launch sites to access it as possible.
Re: SSO, I’m not sure it would make a good depot orbit. Thinking it through, your plane in SSO is only nodally precessing at the same rate as the earth’s rotation around the sun (ie ~1 deg/day). While you’d have no trouble hitting any departure asymptote (since you’re nearly polar), your orbit is precessing so slowly, that you might have to depart a *long* time before a mission, and probably go with a multi-pass phasing orbit (which means lots of van Allen Belt passes). Also lunar mission windows would be less frequent because the nodal precession rate is so slow that basically only the lunar orbit rate (in the opposite direction) is what’s driving when your plane lines up for a lunar departure. That’d suggest you’d likely only have windows every ~14 days (plus or minus) instead of the every ~9 days IIRC you get with a more normal departure plane.
On the gripping hand, if you’re using this as a depot to feed prop to “roving depots” in multiple high orbits, maybe it’s not as big of a problem. You could in theory have another depot in an easier to reach lower inclination orbit serve as your depot for boosting people or cargo to the assembly orbit, and use the SSO atmospheric gathering depot to launch most of the prop mass for the final departure… Intriguing.
Paul,
Re: moving your departure HEEO around using lunar or solar perturbations, that might be possible–but it’s above my paygrade so to speak. I don’t know if you can realistically use Weak Stability Boundary tricks to realign HEEOs like that in a way that is controllable and predictable. I’d ask Mike Loucks or John Carrico. That’s more their wheelhouse than mine.
~Jon
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