  Short-orbital period small-eccentricity binary pulsars can, in principle,
experience substantial advance of periastron. We explore the possibility of
measuring this effect by implementing a timing model, ELL1k, in the popular
TEMPO2 pulsar timing package. True secular variations in the Laplace-Lagrange
parameters, present in our ELL1k model, can lead to measurable timing residuals
while pursuing decade-long timing campaigns using the existing ELL1 timing
model of Lange et al. (2001), especially for binaries exhibiting significant
periastron advance. We also list the main differences between our approach and
various implementations of the ELL1 model present in both TEMPO and TEMPO2
packages. Detailed TEMPO2 simulations suggest the possibility of constraining
the apsidal motion constant of pulsar companions in certain observed binary
pulsars with minuscule eccentricities such as PSR J1719-1438. Fortunately, the
ELL1k timing model does not pose any challenges to the on-going Pulsar Timing
Array campaigns that routinely employ the ELL1 timing model. Detailed TEMPO2
simulations suggest the possibility of constraining the apsidal motion constant
of pulsar companions in certain observed binary pulsars with minuscule
eccentricities such as PSR J1719-1438. Fortunately, the ELL1k timing model does
not pose any challenges to the on-going Pulsar Timing Array campaigns that
routinely employ the ELL1 timing model.
