Water Budgets of Tatooine Moisture Farms: Nocturnal Humidity, Unit Density and Servicing Interval as Determinants of Vaporator Yield in a 48-Farm Ledger Survey, 30–25 BBY

1. Introduction

Tatooine is a desert world of the Outer Rim, lit by two suns, and its settled population lives largely by moisture farming. Households such as the Lars family's homestead keep arrays of vaporators, tall condensing machines that cool night air below its dew point and deliver the condensate to a cistern. The Hutts dominate the Outer Rim, and Republic agencies have not extended their monitoring networks to Tatooine's settlements. We have found no Republic series on the water supplied by the vaporators of any Tatooine district. What exists is held by the farmers themselves: ledgers recording condensate delivered, units in service and servicing performed, kept because a farm that cannot account for its water cannot plan its year.

The first published reconnaissance of the settled basins treated atmospheric moisture as a fixed resource, to be divided among however many units a farm could afford (Outer Rim Planetary Survey, 29 BBY). Field ecologists have since documented that condensing arrays deplete the air column downwind of them, so that a unit standing in the lee of others meets drier air than the first unit in the row (Kevrin, 28 BBY). Earlier work by one of us on the physics of nocturnal condensation gives the expected sensitivity to humidity: condensation rate depends on how far the surface is cooled below the dew point, and the dew-point depression at the coldest hour of the night is the natural proxy for the moisture on offer (Ellesmar, 27 BBY). A third line of evidence concerns fouling. Blown sand loads the condenser surfaces and filter stages and cuts the flow of air across them (Orrelyn, 26 BBY). Each of these mechanisms is plausible and none has been placed on a common scale.

In practice the question is where the returns to another vaporator run out. A farmer with capital can always install a further unit. If yield per unit fell smoothly and slowly with crowding, additional units would remain worthwhile until the price of a unit exceeded its water. If instead yield per unit fell sharply beyond some density, the farm's water supply would stop growing and the additional unit would become pure cost. The two pictures imply different advice, and the ledgers are the only data that can distinguish them.

This paper reports a survey of 48 farms over the six standard years from 30 to 25 BBY, analysed at the Coruscant Academy of Letters and Sciences in 24 BBY, which is the vantage of all statements below. We estimate the dependence of yield per vaporator on dawn dew-point depression, unit density and servicing interval, and we ask specifically whether the density relationship has a threshold. We use the term water budget in a restricted sense. The ledgers record water delivered to the cistern; they do not record what a household consumed, sold or lost, and the budget presented here is therefore the supply side only.

2. Methods

Sources. The survey rests on two record types. Settlement ledgers of the kind kept by moisture farms enter, for each week, the condensate delivered to the cistern in litres, the number of vaporator units in service, and a dawn hygrometer reading of air temperature and dew point taken at the coldest hour. Maintenance logs enter each cleaning of condenser surfaces and filter stages by date. Both were read through the ledger series catalogued in the Tatooine Moisture Farm Ledgers (Tatooine Moisture Farm Ledger Keepers, 25 BBY), and the conventions for counting units in service, including how units under repair are treated, follow the published ledger guide (Pashtane, 27 BBY). Copies were consulted with the permission of the households concerned, and no farm is identified in this paper.

Farms. We approached 61 farms in six clusters, designated T-1 to T-6, three of them on the settled margin of the Dune Sea and three towards the edge of the Jundland Wastes. The designations are our own working labels and carry no administrative meaning. A farm qualified if its ledger covered at least four of the six standard years with no gap longer than one year, if its maintenance log was legible for the same years, and if its boundary had been surveyed so that tended ground could be computed. Forty-eight farms qualified, eight from each cluster. Thirty-seven contributed six farm-years, ten contributed five and one contributed four, giving 276 farm-years. Tended ground ranged from 9 to 31 hectares. Missing years are ledger gaps and were not imputed.

Variables. The outcome was yield per unit, defined as the farm-year's delivered condensate divided by the units in service and by the days in the year, in litres per unit per standard day. Dawn dew-point depression was the mean over the farm-year of air temperature minus dew point at the recorded hour, in degrees of the ledger hygrometer, the standard graduation of the instruments, where larger values mean drier air. Unit density was the mean units in service per hectare of tended ground. Servicing interval was the mean number of standard days between successive cleanings in the year. Ledgers record volume in litres and area in hectares, their standard units, and we keep them. Yield entries are rounded to the nearest litre per week, hygrometer readings are entered by the farmers on instruments never compared with one another, and conventions for counting units in service differ between ledgers. Derived quantities should therefore be read as good to about two figures, and the further digits in the coefficients are given only so that the fit can be reproduced.

Model. Yield is positive and its spread grows with its level, so we analysed its natural logarithm. The primary model was a linear mixed-effects model with a random intercept for farm, fitted by restricted maximum likelihood, in which log yield depended linearly on dew-point depression and servicing interval and on density through a broken-stick term (Lodrace, 31 BBY). Below a threshold density τ the density coefficient gives the slope of log yield; above τ the slope changes by a second coefficient. We estimated τ by profile likelihood over a grid of 0.05 units per hectare between 0.9 and 2.6, refitting by maximum likelihood at each value, and took as its interval the values whose log-likelihood fell within 1.92 of the maximum (Ysmarel, 30 BBY). Coefficients are reported at the selected τ with Wald intervals, which treat τ as known and are therefore somewhat optimistic. With 276 observations and five fixed effects the normal approximation to the Wald statistic is adequate. Because the null distribution of a likelihood ratio for a threshold is not standard, we compare the broken-stick model with a model linear in density by Akaike's criterion, counting τ as an estimated parameter, and report no p-value for the existence of the break.

Marginal yield. Farm output per hectare is density times yield per unit. The marginal yield of an added unit is the derivative of that product with respect to density, which equals yield per unit multiplied by one plus density times the slope of log yield. We report it as a fraction of the mean yield of units already in place, so that a value of 1 means an added unit yields as much as its predecessors, 0 means it adds no water, and a negative value means the farm's output falls. Intervals come from 20,000 draws from the estimated sampling distribution of the coefficients at the selected τ. Because density is measured per hectare of tended ground, the 20 of the 48 farms whose density changed during the survey, through added units or a changed count of units in service, contribute within-farm contrasts as well as between-farm ones.

3. Results

Yield per unit had a median of 3.0 litres per standard day (interquartile range 2.4–3.7; range 1.2–6.3). Dawn dew-point depression averaged 15.2 degrees (range 7.0–23.5), mean servicing interval had a median of 62 standard days (range 22–95), and density ranged from 0.70 to 2.91 units per hectare with a farm mean of 1.74. Of the 276 farm-years, 116 (42.0%) lay above the estimated threshold; 22 farms were above it in at least one year, 18 in every year, and four crossed it as they added units. Sample composition across density bands is given in Table 2.

The log-likelihood profile for the density threshold had a single well-defined maximum at 1.85 units per hectare, a value on the 0.05-unit grid and not a finer estimate and fell below the maximum by more than 1.92 at 1.45 and below and at 2.05 and above, giving a 95% profile interval of 1.5 to 2.0 units per hectare. The broken-stick model had an Akaike criterion 19.1 lower than the model linear in density, with the threshold counted as a parameter, and its maximised log-likelihood was higher by 11.5.

Table 1 gives the fixed effects. Drier dawns reduced yield. Each additional degree of dew-point depression lowered yield by 4.4% (coefficient −0.045, 95% CI −0.051 to −0.038; z = −13.8, p < .001), so a farm-year whose dawn air was 5 degrees farther from saturation delivered 20% less per unit (multiplier 0.80, 0.77–0.82). Longer servicing intervals also reduced yield: a further 30 standard days between cleanings was associated with a multiplier of 0.91 (0.89–0.92), or 9.5% less (z = −9.7, p < .001).

Density acted differently on either side of the threshold. Below 1.85 units per hectare the coefficient was −0.107 per unit per hectare (95% CI −0.205 to −0.010; z = −2.15, p = .032), which corresponds to a multiplier of 0.95 (0.90–1.00, the upper limit lying just below 1) for each added half unit per hectare, a fall of 5.2% in yield per unit. The slope changed by −0.407 at the threshold (95% CI −0.570 to −0.244), so that above it the slope of log yield was −0.514 (−0.618 to −0.410) and each added half unit per hectare cost 22.7% of yield per unit (multiplier 0.77, 0.73–0.81). These intervals are conditional on the selected threshold and are not a test that a threshold exists; the Akaike comparison above carries that question. The farm random intercept had a standard deviation of 0.168 on the log scale against a residual standard deviation of 0.106, so that 72% of the variance left after the fixed effects lay between farms, a matter taken up in the Limitations.

The consequences for farm output follow from the marginal yield. At 1.0 unit per hectare an added unit contributed 0.89 of the mean yield of units already in place; at 1.6 units per hectare, 0.83 (interval from resampling 0.67–0.98). Once past the threshold, the picture changed. At 2.0 units per hectare the marginal yield was −0.03 (−0.24 to 0.18), indistinguishable from zero; at 2.4 it was −0.23 (−0.48 to 0.01), and at 2.8 it was −0.44 (−0.73 to −0.15), meaning that on these estimates a farm crowded to 2.8 units per hectare would deliver less water in total than the same farm at a lower density. Predicted output per hectare, expressed relative to its value at the threshold, was 0.70 at 1.2 units per hectare, 0.89 at 1.6, and about 0.99, 0.96 and 0.91 at 2.2, 2.6 and 2.9. The observed band means in Table 2 show a plateau rather than a decline: output per hectare rose from 3.0 litres per standard day in the sparsest band to 6.2 in the band containing the threshold, and stayed near 5.6–6.0 in the two densest bands. The fitted model predicts a decline beyond 2.2 units per hectare that these two bands do not show, although the difference between them and the 6.2 band is small relative to the spread between farms. These band means are the partial, unadjusted version of a total-output check; no other test of total output against density was made.

Where the threshold is placed shifts the size of the slope above it but not its character. Refitting at the ends of the profile interval, τ = 1.5 gave a slope above the break of −0.439 (−0.524 to −0.353) and τ = 2.0 gave −0.544 (−0.664 to −0.424). In both, the marginal yield of an added unit 0.3 units per hectare past the break was low and fell with further crowding, although at τ = 1.5 it remained modestly positive (0.21) at 1.8 units per hectare and at τ = 2.0 it was −0.25 at 2.3.

4. Discussion

Three statements about how Tatooine moisture farms convert night air into water follow from the survey. Yield per unit falls steeply as the dawn air moves away from saturation, at 4.4% per degree of dew-point depression, which is what a condensation mechanism would predict and is compatible with the hygrometer reading kept in the ledgers being a useful proxy for the moisture on offer. Yield per unit falls with the time since the condensers were last cleaned, at about 9.5% per additional 30 standard days, so that a farm which services its units every 30 days instead of every 90 would expect on these estimates a yield roughly 22% higher, other things equal. Yield per unit also falls with crowding, and it does so in two regimes.

Two regimes make up the main result. Below about 1.85 units per hectare, crowding costs each unit a little, and the farm's output rises almost in proportion to the number of units installed. Above it, each added half unit per hectare removes nearly a quarter of every unit's yield, and the extra unit cannot make up the loss. The marginal yield falls from between 0.8 and 0.9 of an average unit's yield to about zero across the threshold, at about 2.0 units per hectare at the selected break and between 2.0 and 2.3 across its interval, and is negative in the fitted model beyond it. In practical terms the ledgers indicate that a farm holding more than about 2 units per hectare is paying for machines that return little or no additional water. The fitted model, though not the band means, indicates that at the densest end of the sample a farm may deliver less than it would with fewer units. Because the profile interval for the threshold spans 1.5 to 2.0 units per hectare, we would not advise farmers to treat 1.85 as a boundary of engineering precision. The dependable finding is the shape: a modest penalty up to the region of 1.5 to 2 units per hectare, and a heavy one beyond it.

Nothing in the data identifies the mechanism behind the break. Our preferred hypothesis is that units share a finite supply of air. Below a certain spacing, wind refreshes the column between machines faster than they strip it, and beyond that spacing each unit begins to draw on air already worked by its neighbours, as the depletion documented downwind of arrays would predict (Kevrin, 28 BBY). A threshold of this kind is what one expects when a replenishment rate is exceeded, and the density at which it is exceeded should depend on wind and array geometry, neither of which the ledgers record. We flag this as a hypothesis for measurement with anemometers and paired-unit trials, and not as a result of this survey.

Our fouling estimate is compatible with the account of sand loading on condenser surfaces (Orrelyn, 26 BBY), though the ledgers cannot show whether the loss is progressive or arrives in steps after particular storms. The dew-point coefficient accords in sign with the physics of nocturnal condensation set out in earlier work by one of us (Ellesmar, 27 BBY), which is therefore not an independent check. We did not test interactions among the three predictors. It is possible, for example, that crowding costs more on dry nights, when the air has less moisture to share, and the present model would not detect it.

For a farmer the results are modest in ambition and concrete in effect. Servicing schedule and unit count are both under the farm's control, and the two effects are large enough to matter for a household's yearly water. Shortening the servicing interval yields a gain that is available at any density. Adding units yields a gain only up to the neighbourhood of the threshold. A farm at 2.4 units per hectare that removed a fifth of its units, bringing it near 1.9, would on the fitted model lose no output while saving the upkeep of the machines it removed. We have not tested that prediction on any farm.

5. Limitations

The data are observational and archival. Farmers choose their unit density and servicing interval, and they may do so in response to conditions the ledgers do not show, such as the depth of a cistern, a poor season or the plan to sell water. Within-farm density changes help by allowing each farm to act as its own comparison, but only 20 farms changed density and only four crossed the threshold, so the location of the break rests mainly on between-farm contrasts, and the very large farm-level variance (72% of what remained after the fixed effects) warns that unmeasured farm characteristics, such as the siting of the ground and the local wind, could be masquerading as density effects.

Yield per unit and density share a denominator, the count of units in service, so error in that count would produce a negative association between them even if none existed. Ledger conventions for units under repair (Pashtane, 27 BBY) reduce but do not remove that risk. The output-per-hectare figures in Table 2, which do not divide by the count, show a rise and then a plateau and so are not open to the same artefact, but they are a descriptive check, not a formal test. The model also has no term for year and does not model autocorrelation within farms, so shocks common to all farms in a standard year, such as a run of dry nights, and persistence from one year to the next within a farm, are not separated from the effects reported.

Density is computed per hectare of tended ground, not per unit of air volume or by spacing, and the ledger boundaries do not record neighbouring farms. Units on an adjacent holding can draw on the same air and are invisible to our measure. The dew-point depression is a single reading at the coldest hour, entered by the farmer, and the instruments were not calibrated against one another. Noise in the predictor will bias its coefficient towards zero, so the humidity effect is more likely understated than overstated.

The threshold was estimated from the same data used to describe it. Its interval is a profile interval and its coefficients are conditional on the selected value, so the standard errors above the break are somewhat too small. We did not attach a p-value to the existence of the break, and the Akaike comparison is a measure of relative fit, not a formal test. The two sensitivity refits show that the qualitative shape survives across the profile interval, but they are not independent confirmations.

Our 48 farms come from six clusters that qualified by the completeness of their records, and farms with the best ledgers may be the best managed. The survey ends at 25 BBY, and conditions since then, including any change in the length of dry spells, are not covered. Finally, yield is supply only. Nothing here speaks to consumption, sale or loss from the cistern, and we make no claim about the sufficiency of any farm's water.