Question #2: The Coastdown Recovery Problem (hard).

Given a machine with stable running at L=L_Peak loses the ability to continue trickle injection. Luminosity decays at some rate corresponding to some luminosity lifetime. If decay happens for only a very short time, trickle recovery would be appropriate. If the decay happens for a very long time, filling at high rate (under filling rather than trickle conditions) without data logging would be appropriate (at least initially...see question #4 below).

Consider the picture to the right: At time t=0 ability to inject is lost. At time t=1 Luminosity has decayed to a level L=Lmin but injection is again available. Injection proceeds at time t=t1 with two courses shown. In the course shown in green, trickle injection is used, and data logging continues. The trickle luminosity increase rate is Rtrickle. At time t=t3 the peak luminosity is again achieved. The green dashed line shows the running sum of integrated luminosity while under deliverable conditions for the green course. In the course shown in yellow, data collection is abandoned and beams are injected under (faster) filling conditions. A luminosity increase rate Rfill occurs. At time t=t2 the peak is again achieved and logging begins. The yellow dashed line shows the running sum of integrated luminosity while under deliverable conditions for the yellow course. The specific conditions shown in the picture would indicate that less integrated luminosity is lost by choosing the trickle (green) course.

Can you show that if the luminosity decays below the level L_min=L_peak(1-2R_trickle/R_fill) then logging should be abandoned and refilled with the higher rate R_fill?

Solution to Question #2

One way to approach this problem is to write down an expression for the luminosity lost during each of these two courses. Finding the value for L_min for which the two courses have equal luminosity loss solves the problem.

The algebra is messy, but I'll get you started:

The Luminosity lost in the period between t=0 and t=1 is the same in both courses, so we don't need to include a term for that interval. This means the luminosity lifetime is not a (direct) factor.

Recognize that for the last step I substituted for the (t₃-t₁) and (t₂-t₁) terms expressions for the luminosity increase rates as in the example earlier on this page.

Task is to solve this equation for L_min.

This equation is quadratic in L_min. Quadratic equations have two solutions (roots). For this problem the non-extraneous root is:

L_min=L_peak(1-2R_trickle/R_fill)

The savvy reader will recognize immediately from this what the extraneous root is. (The less-than-savvy of us see the extraneous root as confirmation that we've done the algebra right.)