mass and initial entropy (for a given age). These models would imply much higher planet masses than hot-start models by up to a factor of two (or perhaps more). For planet b, a fully self-consistent solution for the planet properties remains elusive once spectroscopic constraints are combined with infrared photometry, likely due to limitations of the models [@marley12]. Dynamical constraints on planet masses would be valuable, but they are generally difficult to obtain for wide-separation planets that are impractical to monitor over many orbital periods.

For HR 8799, analysis of the long-term stability of the system offers one form of dynamical mass constraint on the planets. The HR 8799 multi-planet system was investigated numerically by @gotberg16 who found that nominal models with 5–7 M$_{\rm Jup}$ planets are unstable on timescales much shorter than the system age. One natural solution to this problem is if the planet masses are substantially lower than estimated from infrared observations. Lower mass planets have larger separations in terms of Hill radii, and the system stability persists for much longer times. If standard cooling models underpredict the planet luminosities, as some calculations suggest [e.g. @mordasini13], then lower planet masses – more like 2–3 M$_{\rm Jup}$ – 