Binary systems emit gravitational waves in a well-known pattern ; for binaries in circular orbits , the emitted radiation has a frequency that is twice the orbital frequency .
Systems in eccentric orbits , however , emit gravitational radiation in the higher harmonics too .
In this paper , we are concerned with the stochastic background of gravitational waves generated by double neutron star systems of cosmological origin in eccentric orbits .
We consider in particular the long-lived systems , that is , those binaries for which the time to coalescence is longer than the Hubble time ( \sim 10 Gyr ) .
Thus , we consider double neutron stars with orbital frequencies ranging from 10 ^ { -8 } to 2 \times 10 ^ { -6 } Hz .
Although in the literature some papers consider the spectra generated by eccentric binaries , there is still space for alternative approaches for the calculation of the backgrounds .
In this paper , we use a method that consists in summing the spectra that would be generated by each harmonic separately in order to obtain the total background .
This method allows us to clearly obtain the influence of each harmonic on the spectra .
In addition , we consider different distribution functions for the eccentricities in order to investigate their effects on the background of gravitational waves generated .
At last , we briefly discuss the detectability of this background by space-based gravitational wave antennas and pulsar timing arrays .