Want to see the GJ 876 planets?

Robert Zubrin cannot be accused of thinking small. With the hat-tip to Reynolds, now he has a telescope plan. This one would knock down the (kilo)grams needed to do space astronomy.

As for other factors, Teh Zoob seems to be competing against the Webb. This isn't a scanner; it's there to be pointed at a specific location, and to send data home. Probably not the Planet Nine solution.

Imagine, though: several Webb-like space telescopes that Rocket Lab (say) may send up into space for the same cost as the one, all working in tandem. That could be the Planet Nine scanner.

Thinking bigger (as Zubrin would) I don't know if he perused Stross but, yes, the Big Dumb Booster allows for that weight to be scaled right back up again, without risking astronauts' lives. This boils off some of the commenters' cold water on the lower-radius telescope options; raising the radius would correct the errors. So yes: I am recommending that SpaceX start here, rather than with human tourists.

A 'scope of this scale would be... impressive. Webb has only been rated for Gliese 832 b and for transits. For Zubrin's proposal, noted in the comments is Epsilon Eridani, whose planets are presently visible as... pixels. This telescope might even be able to see, say, the GJ 876 system being only 15 light years away as it is. Not to mention Alpha Centauri. Or such planets right here as we seem to be done sending probes at, namely Uranus and Neptune.

The 'scope does require a boost to pull the parabola back. That implies (to Zubrin) several sorts of mission: a 'scope spun on a tether, a propellant-limited mission, and a sail. (I'd add a statite high in Venus umbra.) For the propellant mission he'd have this thing out 3.1 AU from here: between Ceres and Hilda, more like 2530 Shipka. A mission with a sail means there is solar wind, which can interfere with the results; not, though, for results outside the Sun's electromagnetic range, as Zubrin notes for radio.

When to launch 2L4

Today we use the Hop David spreadsheet, to get Cycler 2L4's launch window after Year 2000.

Probably the 2004 or 2012 transits of Venus would have been better but on a 243 Julian year scale, as you will see, Why Two-Kays is close enough. For more precision in later synods the Manifold Thesis had a decent "brute force" writeup on how to estimate departure moments. Overall this blog shall leave the Pork Chop Plots to the unparalleled experts in bacon consumption, namely NASA.

G-d and His Spreadsheet deliver B30, δW; and B32, δAngle @J2000... Themself. Since I'm going Venus outbound B30 is -225.1878256° and B32 is -81.5153°. (Earth-to-Venus, these are positive.) Our job is to find B24.

For me that starts with B20, the travel-time in sidereal years of the destination (Earth). We just got that: 159 / 365.25636. B22 tells us that in degrees: 156.5. B24 just orients that in the 180° semicircle for 23.5° angle at launch window.

The 2000 date of departure comes from adding 2000 to that fraction of Earth sidereal.

Cycler 2L4 at Earth's L1

I spent the last 24 hours (sleep and work permitting) learning what mine own code could do. First thing I had to figure out was that left way / right way didn't always correspond with the short way / long way... in general, fewer synods mean the longer periods.

Another thing: I gave up finding what equation could get Aldrin's ass to Mars. Instead I just iterated from initial-velocity to the distance I wanted, using Newton's law of gravitational acceleration. This got me the time it took and the position and velocity vectors. Although that's the best algorithm we're ever likely to find, Microsoft's System.Numerics stuck me with 23-bit decimal places for accuracy. I could not improve on δT = a millionth of the inner planet's year. A half-minute here on Earth.

Soooo: from Venus 2L4's time to aphelion will be 159 days. I wasn't expecting a large velocity at the end given it is aphelion.

Where aphelion falls short, Longuski and McConaghy were subtracting planet speed from cycler aphelion speed. Why, I don't know - probably because of Mars' ellipse, so they quit caring. Venus has no ellipse (pretty much), and Earth's isn't a problem. For circular(ish) orbits I target SPL1 velocity which, covering a shorter circumference at the same time as planet P's year, will be slower than P. V_∞ at STL1 is about 1950 m/s.

I confess une petite surprise that the 2L4 Venus-Earth time was (slightly) more than the usual Hohmann suspects and, indeed, than 5S10[-right] which is telling me 115 days (and 600 m/s V_∞ at Earth, let alone STL1...). I note it's 53.3% of the unpatched period - and, as noted, a Left Branch where Hohmann turns Right. It looks like it dips to perihelion before running to STL1. Fry-by, one might say. Better shield it.

Local news stories

Virgo's GJ 486 / Wolf 437 - 26 light-years away - has a transiting planet. 45 light-years away is LHS 3844; this one also transits, and now has delivered enough data for some Bern Uni scholars (very proud to be at Bern) to run some simulations.

Transits are generally found where the planet rides very close to the star. The closer the better. Although, not so better for any potential colonists; as noted, neither of these planets are fit for human habitation.

I admit I find difficult to comprehend the Bernais press-release for LHS 3844 b. It doesn't seem based on observations so much on simulations based on those observations. The Wolf 437 b article is much better written although I cringe a bit at its title (hot new neighbour!!).

As for Wolf 437, here I ask why it has taken so long for a transit to be noticed from here, given that it is so close, and low-mass at that (so radial spectrometry should have noted any planets' wobble). Better late than never. Time to pre-cover its data, I think.

The Venus-to-Earth 2L4 Cycler

Now we got an approximation to a viable Venus-Earth, more exactly Venus-STL1, let's fine-tune it. Or at least mark where someone else can; I didn't get that chance today.

First, that 72 degree angle per synod is, I think, a bit less in practice. Tweaking that (but keeping the time-of-flight) the V_∞ contributing to delta-V is (usually) closer 3864 km/s. Bit less than 3876.6 km/s. Angle's down to 108° and aphelion up but still around 0.979 AU. This doesn't take into account inclination, unfortunately usually high, so as to take away all our gainz here.

To get back to Venus - making up for the lost angles - the cycler got 1168 days all within 1 AU of the Sun, so sails should do it. As compared with Hop David's 5S10 Hohmann cycler, 2L4 gets more daylight overall, but it also spends a higher fraction of its time fully laden. Maybe twice as much: he's 1/20, I'm a bit over 1/8.

As to where this km/s and 108° can be shifted: when we get to Venus, Oberth's dive into low Venus will help; and for the rest I'm looking at that tether.

TAKING THE L 3/4: Its orbit's a long-period.

Lambert, cont'd

It turned out my maths were incomplete yesterday. I blame Longuski and McConaghy for not Showing Their Work. But with Aldrin's help I got "Vinf" and the angle at the beginning. Also of course we need trajectories that get close to Earth. TL;DR - the Venus / Earth cyclers are possible but (besides Hohmann 5S10) infeasible. With one intriguing exception, the 2L4.

There are no Right cyclers below 5S10, likewise 1P3+ and 2P7+ don't exist; and my calculator wasn't good for *P5 because it kept using Venus' own orbit for that (which doesn't intersect Earth's, duh). For the others, the best I got was 2L4 "left branch". Semimajor 0.841 earth AU; unpatched period 282 days (right branch was 244 - 2S4 - but it's terrible). Eccentricity 0.163; aphelion 0.979 AU. V_∞ at Venus 3876.6 m/s. Turn angle at Venus 109 degrees. (Compare 5S10 rightbranch: 2705 m/s with no estimated turn angle.) All this assumes planar inclination and that the synods work like clocks. UPDATE 3/3: Fixed the clocks.

There should be a reverse cycler that gets from Earth to Venus about the same time. Then the thing goes around by itself until its two synods are up - bit more than three Earth years - and it is at Venus again. Also (as Hop David on the 5S10) since it is a 2P*, there will be two of these things on the move, so the other one can be boarded on the other synod.

Here's the bad news: the turn angles were frankly prohibitive for all the trajectories I looked at. For that I got none to blame but Lucifer his own dark self. The Venus / Earth Pentagram means the angles are in fifths rather than in Martian sevenths. Baseline 72 degrees, not 51.

Earth's perihelion is 0.98329 AU although, yes, the mutual ecliptic angles are a pain. In the best years of the 243 Earth/Venus cycle, I expect 2L4 will enter STL1 halo centred 0.0098 AU inward of that. There she can use the manifold for some deltaV help. But she can never use Earth's well directly and unassisted. I suppose sails are up on whatever revolutions are made after Earth.

Lambert 1, 2, 3, 4

Last Friday night (properly, Saturday morn) we looked at Lambert problems for the 13/5 Earth synod in Venus years. Since here (unlike Mars) we get a Hohmann at metonic n=5, which is awesome, I am interested in the nonideal solutions where n is 1, 2, 3, or 4. These are such Earth trajectories as might return to Venus before the metonic runs out. Maybe the Earth-Venus stretch is even under the 146 day Hohmann; maybe that's the Venus-Earth stretch, it just cannot be both.

Note that with n=2, double-boarding from Venus is possible as with the Earth-Mars S1L1. This can lower delta-V: lowering my initial launch cost and, on its cycles, how far I must dip into either planet's well. For whatever best-case scenario we might lower launch-cost further from the inner planet's L2 or maybe the outer's L1.

First: uh, what's a Lambert. Johann Heinrich Lambert found in 1776 that time-of-flight aka delta-T, for a trajectory in-system, depends only on its semimajor axis, a few constants, and some vicious mathematics. Per Matthew Peet's pdf: the equation starts with Math.Sqrt(Math.Pow(a,3.0)/μ * (α-β - (Math.Sin(α) - Math.Sin(β)))). Nobody solves Lambert directly. Peet has a graph on slide 19 (of 30), or use Gooding's code. McConaghy et al. are looking to find semi-major or period (by Kepler, the one is had from th'other) such that delta-T is a synod. Best I could do was to get the velocity vectors at r1 and r2. Unfortunately that was enough for my project.

Maybe around 7:00 this morning before work (yeah, I still do that) I tested the solver for n=5 and r=5, which is the five-synod run with five returns. When I got initial velocity near identical to final velocity, I declared victory on my code.

With that, this week, I am evaluating results for the one- and two-synod periods, five revolutions right to five revs left.

V-E DAY 3/3: Well, I'd kicked this post down the road to 1 March and its posting was premature even then. I'd only got aphelion (still progress!); and r=5 was always dumb for Venus' 72°. Real victory was achieved at, like, 6:30 AM, when I wrote a C# Forms app for Earth/Mars and checked it against Longuski-McConaghy's table. And for V/E n=5 and r=10. Let it stand as a promissary note. UPDATE 3/4: promise kept - finally. Only having weeknights and 6 AM wakeups does take a toll.