We have measured electron-ion recombination for Fe$^{9+}$ forming Fe$^{8+}$ and for Fe$^{10+}$ forming Fe$^{9+}$ using merged beams at the TSR heavy-ion storage-ring in Heidelberg. The measured merged beams recombination rate coefficients (MBRRC) for relative energies from 0 to 75 eV are presented, covering all dielectronic recombination (DR) resonances associated with 3s->3p and 3p->3d core transitions in the spectroscopic species Fe X and Fe XI, respectively. We compare our experimental results to multi-configuration Breit-Pauli (MCBP) calculations and find significant differences. From the measured MBRRC we have extracted the DR contributions and transform them into plasma recombination rate coefficients (PRRC) for astrophysical plasmas with temperatures from 10^2 to 10^7 K. This spans across the regimes where each ion forms in photoionized or in collisionally ionized plasmas. For both temperature regimes the experimental uncertainties are 25% at a 90% confidence level. The formerly recommended DR data severely underestimated the rate coefficient at temperatures relevant for photoionized gas. At the temperatures relevant for photoionized gas, we find agreement between our experimental results and MCBP theory. At the higher temperatures relevant for collisionally ionized gas, the MCBP calculations yield a Fe XI DR rate coefficent which is significantly larger than the experimentally derived one. We present parameterized fits to our experimentally derived DR PRRC.
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