The only discovery of Earth Trojan 2010 TK$_7$ and the subsequent launch of OSIRIS-REx motive us to investigate the stability around the triangular Lagrange points $L_4$ and $L_5$ of the Earth. In this paper we present detailed dynamical maps on the $(a_0,i_0)$ plane with the spectral number (SN) indicating the stability. Two main stability regions, separated by a chaotic region arising from the $ u_3$ and $ u_4$ secular resonances, are found at low ($i_0leq 15^circ$) and moderate ($24^circleq {i_0}leq 37^circ$) inclinations respectively. The most stable orbits reside below $i_0=10^circ$ and they can survive the age of the Solar System. The nodal secular resonance $ u_{13}$ could vary the inclinations from $0^circ$ to $sim 10^circ$ according to their initial values while $ u_{14}$ could pump up the inclinations to $sim 20^circ$ and upwards. The fine structures in the dynamical maps are related to higher-degree secular resonances, of which different types dominate different areas. The dynamical behaviour of the tadpole and horseshoe orbits, reflected in their secular precession, show great differences in the frequency space. The secular resonances involving the tadpole orbits are more sensitive to the frequency drift of the inner planets, thus the instabilities could sweep across the phase space, leading to the clearance of tadpole orbits. We are more likely to find terrestrial companions on horseshoe orbits. The Yarkovsky effect could destabilize Earth Trojans in varying degrees. We numerically obtain the formula describing the stabilities affected by the Yarkovsky effect and find the asymmetry between the prograde and retrograde rotating Earth Trojans. The existence of small primordial Earth Trojans that avoid being detected but survive the Yarkovsky effect for 4.5,Gyr is substantially ruled out.
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