We present the analysis of a deep Chandra observation of a ~2L_* late-type galaxy, ESO 137-002, in the closest rich cluster A3627. The Chandra data reveal a long (>40 kpc) and narrow tail with a nearly constant width (~3 kpc) to the southeast of the galaxy, and a leading edge ~1.5 kpc from the galaxy center on the upstream side of the tail. The tail is most likely caused by the nearly edge-on stripping of ESO 137-002s ISM by ram pressure, compared to the nearly face-on stripping of ESO 137-001 discussed in our previous work. Spectral analysis of individual regions along the tail shows that the gas throughout it has a rather constant temperature, ~1 keV, very close to the temperature of the tails of ESO 137-001, if the same atomic database is used. The derived gas abundance is low (~0.2 solar with the single-kT model), an indication of the multiphase nature of the gas in the tail. The mass of the X-ray tail is only a small fraction (<5%) of the initial ISM mass of the galaxy, suggesting that the stripping is most likely at an early stage. However, with any of the single-kT, double-kT and multi-kT models we tried, the tail is always over-pressured relative to the surrounding ICM, which could be due to the uncertainties in the abundance, thermal vs. non-thermal X-ray emission, or magnetic support in the ICM. The H-alpha data from SOAR show a ~21 kpc tail spatially coincident with the X-ray tail, as well as a secondary tail (~12 kpc long) to the east of the main tail diverging at an angle of ~23 degrees and starting at a distance of ~7.5 kpc from the nucleus. At the position of the secondary H-alpha tail, the X-ray emission is also enhanced at the ~2 sigma level. We compare the tails of ESO 137-001 and ESO 137-002, and also compare the tails to simulations. Both the similarities and differences of the tails pose challenges to the simulations. Several implications are briefly discussed.
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