Total-power radiometry with individual meter-wave antennas is a potentially
effective way to study the Cosmic Dawn ($zsim20$) through measurement of sky
brightness arising from the $21$~cm transition of neutral hydrogen, provided
this can be disentangled from much stronger Galactic and extra-galactic
foregrounds. In the process, measured spectra of integrated sky brightness
temperature can be used to quantify the foreground emission properties. In this
work, we analyze a subset of data from the Large-aperture Experiment to Detect
the Dark Age (LEDA) in the range $50-87$~MHz and constrain the foreground
spectral index $beta$ in the northern sky visible from mid-latitudes. We focus
on two zenith-directed LEDA radiometers and study how estimates of $beta$ vary
with local sidereal time (LST). We correct for the effect of gain pattern
chromaticity and compare estimated absolute temperatures with simulations. We
develop a reference dataset consisting of 14 days of optimal condition
observations. Using this dataset we estimate, for one radiometer, that $beta$
varies from $-2.55$ at LST~$<6$~h to a steeper $-2.58$ at LST~$sim13$~h,
consistently with sky models and previous southern sky measurements. In the
LST~$=13-24$~h range, however, we find that $beta$ fluctuates between $-2.55$
and $-2.61$ (data scatter $sim0.01$). We observe a similar $beta$ vs. LST
trend for the second radiometer, although with slightly smaller $|beta|$, in
the $-2.46<beta<-2.43$ range, over $24$~h of LST (data scatter $sim0.02$).
Combining all data gathered during the extended campaign between mid-2018 to
mid-2019, and focusing on the LST~$=9-12.5$~h range, we infer good instrument
stability and find $-2.56<beta<-2.50$ with $0.09<Deltabeta<0.12$.
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