Context. Astrometric monitoring of directly-imaged exoplanets allows the study of their orbital parameters and system architectures. Because most directly-imaged planets have long orbital periods (>20 AU), accurate astrometry is challenging when based on data acquired on timescales of a few years and usually with different instruments. The LMIRCam camera on the LBT is being used for the LEECH survey to search for and characterize young and adolescent exoplanets in L band, including their system architectures. Aims. We first aim to provide a good astrometric calibration of LMIRCam. Then, we derive new astrometry, test the predictions of the orbital model of 8:4:2:1 mean motion resonance proposed by Gozdziewski & Migaszewski, and perform new orbital fitting of the HR 8799 bcde planets. We also present deep limits on a putative fifth planet interior to the known planets. Methods. We use observations of HR 8799 and the Theta1 Ori C field obtained during the same run in October 2013. Results. We first characterize the distortion of LMIRCam. We determine a platescale and a true north orientation for the images of 10.707 +/- 0.012 mas/pix and -0.430 +/- 0.076 deg, respectively. The errors on the platescale and true north orientation translate into astrometric accuracies at a separation of 1 of 1.1 mas and 1.3 mas, respectively. The measurements for all planets are usually in agreement within 3 sigma with the ephemeris predicted by Gozdziewski & Migaszewski. The orbital fitting based on the new astrometric measurements favors an architecture for the planetary system based on 8:4:2:1 mean motion resonance. The detection limits allow us to exclude a fifth planet slightly brighter/more massive than HR 8799 b at the location of the 2:1 resonance with HR 8799 e (~9.5 AU) and about twice as bright as HR 8799 cde at the location of the 3:1 resonance with HR 8799 e (~7.5 AU).
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