How long can you hold a pace? What the 1979 Daniels and Gilbert equations say

The equation behind every VDOT calculator

Jack Daniels and Jimmy Gilbert published two curves in their 1979 booklet Oxygen Power. One gives the oxygen cost of running at a speed. The other, the one this guide is about, gives the fraction of VO2 max a runner can sustain for an effort lasting t minutes:

%max = 0.8 + 0.1894393 x e^(-0.012778 x t) + 0.2989558 x e^(-0.1932605 x t)

A race time goes into both curves, and the ratio of the two is the fitness score Daniels calls VDOT (his term, used here descriptively; this site is not affiliated with him). The training pace calculator, the race time predictor and the race option of the VO2 max calculator all use these coefficients, as quoted by Larry Simpson from the original booklet. The numbers in this guide are computed from the equations, not taken from any printed table in Daniels' books.

What it predicts at each duration

The model crosses 100% at 11 minutes (11:02 to be exact): shorter efforts can run above VO2 max because they also draw on anaerobic energy. It reaches 95% at 21 minutes, 90% at 50 minutes and 85% at 1 hour 44 minutes. After that the curve is close to flat: 82% at 2 hours 56 minutes, 81% at 3 hours 50 minutes.

Why the curve flattens: a floor at 80%

The formula is a constant plus two terms that decay at different rates. The second term, 0.2989558 x e^(-0.1932605 x t), is a fast one: it is worth 11.4 points at 5 minutes and has vanished by the hour. The first term decays slowly: 17.8 points at 5 minutes, 8.8 at one hour, 1.9 at three hours and 0.9 at four. What is left is the constant 0.8.

That constant is a floor. However long the effort, the equation never predicts below 80% of VO2 max. Physiologically that cannot hold forever (nobody runs a day at 80%), and the equations were fitted to race performances, not to all-day efforts. Within the marathon range it means each extra hour of running costs very little in the model.

A 3:00 and a 4:30 marathoner, side by side

Here is the arithmetic for both, done the way the calculators do it.

3:00:00. 42,195 m in 180 minutes = 234.4 m per minute. Oxygen cost = -4.60 + 0.182258 x 234.4 + 0.000104 x 234.4² = 43.8 ml/kg/min. Share held for 180 minutes = 81.9%. VDOT = 43.8 / 0.819 = 53.5.

4:30:00. 42,195 m in 270 minutes = 156.3 m per minute. Oxygen cost = 26.4 ml/kg/min. Share held for 270 minutes = 80.6%. VDOT = 26.4 / 0.806 = 32.8.

Across the whole table, from 2:30 to 5:00, the share held only moves from 82.8% to 80.4%. The model treats a five-hour marathon as almost exactly as hard to sustain, relative to fitness, as a 2:30 one. The fitness score differs hugely (66.3 against 28.7); the relative intensity barely changes.

What that means for predictions

This flat tail is why the Daniels-Gilbert model gives more optimistic marathon times for slower runners than Riegel's formula. A runner with a 35:00 5K holds about 92% in the model for that 5K; the model then asks them to hold just over 80% for five and a half hours. The model's implied fatigue exponent falls as runners get slower, which I worked through, with computed predictions, in why race predictors disagree. A study of 2,303 recreational runners found that even Riegel predicted marathons too fast for most of them, so the model's optimism lands on the side that the real-world data already found too fast.

The flip side: for races up to about an hour, where the curve is steep and well supported by race data, the model and Riegel's formula agree closely. A runner's VDOT 45 profile shows how intensity falls with distance:

Reading your own races against the curve

Because every race distance maps to a duration, and every duration to a share of VO2 max, two of your own races give a quick check of where you sit on the curve. Run each one through the VO2 max calculator (race option) and compare the two scores.

Worked example. A 10K in 45:13 gives a VDOT of 45.0, and for that score the model expects a half marathon of 1:40:13. Suppose the same runner's half was 1:45:00. That half gives a VDOT of 42.6, not 45.0. Put another way: 21,097.5 m in 105 minutes is 200.9 m per minute, which costs 36.2 ml/kg/min, and 36.2 / 45.0 is 80.5% of the 10K score, where the curve expects 85.0% for a 105-minute effort. The runner held less than the model assumes for that duration.

A gap like that suggests endurance, not speed, is the limiter, and it is a warning for the marathon: from the 10K the model predicts 3:28:13, from the half 3:37:50. The longer race is the better guide, which matches what the curve's flat tail would lead you to expect.

How the training paces use it

The training pace calculator turns a race into VDOT, then sets each pace as the speed whose oxygen cost equals a share of that VDOT: easy running at 59 to 74%, threshold at 83 to 88%, intervals at 95 to 100%, using the %VO2max bands reported by secondary sources for Daniels' system. Read against the table above, those bands line up with the duration curve in a useful way. Threshold, 83 to 88%, is what the model says you could hold for roughly one to two and a half hours (88% at 67 minutes, 83% at 2 hours 24 minutes), so a tempo run of 20 to 40 minutes at that intensity is well inside what you could race. Interval pace, 95 to 100%, matches an all-out effort of 11 to 21 minutes, which is why intervals are broken into repeats of a few minutes with recoveries.

Marathon pace in the calculator is simply the marathon time the model predicts for your VDOT, so it inherits the flat tail. If your marathon pace from a short race looks ambitious, the curve above is the reason.

Limits

The two equations describe an average fitted relationship, not your body. Running economy, the share of slow-twitch muscle, fuelling and heat all change how long you can hold a given intensity, and none of them appear in the formula. Percentages of VO2 max above 100% for short efforts are a modelling convention for the extra anaerobic contribution, not a measured oxygen uptake. VO2 max itself can only be measured in a laboratory with gas analysis; VDOT is an estimate from a race. Use the numbers to understand why the calculators behave as they do, and test a marathon pace in long training runs before trusting it on race day. Lab-style estimates of VO2 max come from tests such as the Bruce protocol.

Sources: Larry Simpson, The Daniels/Gilbert formula, percent-max equation (from Daniels and Gilbert, Oxygen Power, 1979); Larry Simpson, The Daniels/Gilbert formula, oxygen cost equation; Ray Boardman, Jack Daniels running intensity, Coach Ray (2023), %VO2max bands; Vickers AJ, Vertosick EA (2016), An empirical study of race times in recreational endurance runners, BMC Sports Sci Med Rehabil 8:26. Rules checked 7 October 2026. Found a problem? Tell us.

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