How is the Lithium Depletion Boundary used to determine the age of a stellar cluster? According to my understanding of Soderblom et al. (2014), lithium ages of stars are determined as follows:


Determine lithium abundance from equivalent width measurement of Li$_{\mathrm{I}}$ transitions at 6708 angstroms for all stars in a coeval group.
Plot all stars in group on color-magnitude diagram, with each star labeled as Li-rich or Li-poor.
On plot, look for the boundary where the stellar population transitions from depleted to undepleted. That is the Lithium Depletion Boundary (LDB).
Use the luminosity of the LDB and the age-LDB luminosity relationship to determine the age of the stars in the group.


What is the age-LDB luminosity relationship, and what are its physical origins?
When low mass stars are very young, they are termed pre main sequence (PMS) stars. These PMS stars have larger radii than main sequence stars of the same mass, and energy transport in their interiors occurs primarily through convection. The convection ensures that the star is thoroughly mixed and chemically uniform.

As the PMS star radiates away its gravitational potential energy, it contracts. The virial theorem tells us that as it does so, its interior becomes hotter. Roughly speaking, the core temperature is proportional to $M/R$, where $M$ is the mass and $R$ the radius.

Nuclear fusion of hydrogen will not commence until the core temperature reaches more than 10 million K, however there are other fusion reactions that become possible at lower temperatures - namely deuterium burning at around 1 million K and then lithium burning at around 3 million K. The latter reaction is not energetically important in the star's life because there is not much lithium in the star to begin with (about 1 part in a billion), however this lithium can be observed in the photosphere of the PMS star (via the 670.8 nm Li I resonant absorption line) and the convective mixing means whatever we see at the photosphere also represents the abundance at the core.

The Li burning reaction is extremely temperature dependent (like $T^{20}$ or thereabouts), so it turns on like a switch once the core reaches the appropriate temperature (e.g. see Bildsten et al. 1997). The time it takes for a PMS star to reach this core temperature basically depends on its mass. More massive, and hence more luminous stars contract faster and reach the Li burning temperature quicker. Once they do so then the Li in the star is rapidly and thoroughly consumed by fusion. The relationship between the age at Li destruction and the luminosity of the PMS star at that time is the age-LDB luminosity relationship that you refer to.

The result is that if you look at a bunch of stars in a cluster (assuming they all have the same age), then the more massive, more luminous PMS stars will have destroyed their Li, whilst the lower mass, lower luminosity stars will still contain their original Li content. The luminosity at the reasonably sharp transition between these two regimes is known as the LDB.

LDB ages are arguably the most accurate way to find the ages of stars in clusters. All age determinations are to some extent dependent on what physical ingredients are in stellar evolution models, but the sensitivity of the age-LDB luminosity relationship to various uncertainties is quite weak (e.g. Burke et al. 2004) - we basically understand the physics of a contracting, fully convective ball of gas quite well. The LDB ages can also be precise, because the very sharp turn-on of Li burning and its rapidity should lead to a sharp transition between stars with low luminosities that have Li and stars with only slightly higher luminosities that don't.

Interestingly, in the last couple of years it has become apparent to us (and others) that there are some model ingredients that are not completely understood, namely the effects of dynamo generated magnetic fields and dark starspots. Both of these may lead to the suppression of heat transport, either throughout the star or just at the surface, slowing the PMS contraction so that at a given age, the PMS star has a cooler core temperature. This might delay the onset of Li burning and mean that the currently determined LDB ages are underestimates by 10-20% or so (e.g. Jackson &amp; Jeffries 2014; Somers &amp; Pinsonneault 2015 ).