The universe is expanding, at a rate called the "Hubble Constant". But that rate of expansion *(probably)* changes over time --- i.e., the Hubble Constant isn't constant! The name for that cosmological slowing-down is the "Deceleration Parameter".

I have my own personal Deceleration Parameter, however, rather more physical than astrophysical. In running *(as in living!)* pace is crucial. The most efficient racing tactic, in theory and as confirmed by general experience, is to maintain a constant speed. In practice this is extraordinarily difficult. The almost irrestible temptation is to go out fast near the beginning of an event --- *" Hey, I feel fresh ... I can get ahead of schedule and put some time in the bank!"* But almost always that early enthusiasm leads to a much worse slow-down later.

Quanto-freak that I am, after a big race I like to take the mile-by-mile times (splits) that I've captured in my watch and then run them through a least-squares (linear regression) calculation *(using the trusty old HP-11C that has served me well since 1980 --- talk about overdesign!)*. The resulting slope of the curve is my "Deceleration Parameter" for that event.

But in truth, this approach may be massive overkill. A simple approximation gives almost the same answer: take your time at the halfway point of the race, double it, and subtract that from your finishing time. *(Or equivalently, subtract your first-half-of-the-race time from your second-half time.)* Then divide the result by the square of the length of the race and multiply by 4. That's roughly the average amount of **deceleration** that you have experienced during the event. In symbols:

d = 4*(T - 2*T_{1/2})/L^{2} |

... where d = deceleration, T = total time, T_{1/2} = halfway-point time, and L = length. *(The derivation is straightforward, based on the assumption that each half of the race is run at a constant rate.)* If **d** comes out less than zero then you've done the second half of the race faster than the first and have achieved "negative splits" ... and I envy you!

For quick reference, here's what the formula simplifies into for various distances when you calculate the split-difference S = T - 2*T_{1/2} in minutes and ask for the deceleration parameter d(S) in units of seconds/mile^{2}:

race distance | approximate d(S) |

5 k | 25*S |

8 k or (5 mile) | 10*S |

10 miles | 2.4*S |

half marathon | 1.4*S |

marathon | 0.35*S |

So an even simpler rule of thumb for the marathon deceleration is "take the difference in minutes between the second half and the first half, and divide by three". Testing this against my three marathonic experiences yields:

event | approximate d(S) | least-squares d(S) |

Marine Corps Marathon 2002 | 7 sec/mi^{2} | 6.6 sec/mi^{2} |

Marathon in the Parks 2002 | 12 | 11.3 |

Marathon in the Parks 2003 | 5 | 4.2 |

Not bad, much faster and simpler to compute, and perhaps as accurate as the data justify.

*( ... but note that when I did preliminary tests for shorter distances the errors were apparently much larger, for reasons which remain unclear to me ... see also Bless the Leathernecks (28 Oct 2002), MarineCorpsOrdnance (1 Nov 2002), Rocky Run (17 Nov 2002), Marathon in the Parks 2003 (11 Nov 2003), MarathonGraphs (17 Nov 2003), ... )*

TopicRunning - TopicScience - 2003-12-28

*(correlates: 2007-08-31 - Lap, Dog, HandicapJogging, ZimmermannEnvironmental, ...)*