  We present the thermal evolution and emergent spectra of solidifying
terrestrial planets along with the formation of steam atmospheres. The lifetime
of a magma ocean and its spectra through a steam atmosphere depends on the
orbital distance of the planet from the host star. For a type-I planet, which
is formed beyond a certain critical distance from the host star, the thermal
emission declines on a timescale shorter than approximately $10^6$ years.
Therefore, young stars should be targets when searching for molten planets in
this orbital region. In contrast, a type-II planet, which is formed inside the
critical distance, will emit significant thermal radiation from near-infrared
atmospheric windows during the entire lifetime of the magma ocean. The Ks and L
bands will be favorable for future direct imaging because the planet-to-star
contrasts of these bands are higher than approximately 10$^{-7}$-10$^{-8}$. Our
model predicts that, in the type-II orbital region, molten planets would be
present over the main sequence of the G-type host star if the initial bulk
content of water exceeds approximately 1 wt%. In visible atmospheric windows,
the contrasts of the thermal emission drop below $10^{-10}$ in less than $10^5$
years, whereas those of the reflected light remain $10^{-10}$ for both types of
planets. Since the contrast level is comparable to those of reflected light
from Earth-sized planets in the habitable zone, the visible reflected light
from molten planets also provides a promising target for direct imaging with
future ground- and space-based telescopes.
