What Are The Minimum Periods Of Hydrogen Rich Bodies? (Planetary Science)

Orbital periods in systems containing white dwarfs (‘WDs’) can be extremely short, especially if both of the stars are H-exhausted objects. The best examples of this are WD+WD binaries with periods of 7 and 9 minutes. Such systems almost certainly involve one or more phases of mass transfer. But what will be the minimum allowed orbital periods when at least one of the stars is still H-rich? Take BD+WD binaries, for instance (BD- Brown dwarf). Thats what, Rappaport and colleagues answered in their recently published paper on Arxiv.

Figure 1. Mass-radius relation for H-rich bodies spanning masses from Saturn to stars on the ZAMS up to 1 M (red curve). For the region between Saturn’s mass and the end of the brown-dwarf range (at ∼0.074 M) they used equation (9) given in paper with X = 0.7. For stars on the ZAMS they used the simple expression given by equation (15) given in paper. The blue and cyan dots are an empirical sample of planets and main-sequence stars. The green points are their compilation of brown dwarfs. The black curve is an approximation to the red curve which has been smoothly blended near the transition between the ZAMS stars and brown dwarfs. Note how both the red and black curves hug the lower locus of measured objects – as desired. © Rappaport et al.

They have used radius-mass relations, R(M) displayed in fig 1, in conjunction with function:

to derive the minimum allowed orbital period vs. the body’s mass. The results are shown in fig 2 below.

Figure 2. Minimum allowed orbital period of H-rich bodies as a function of their mass. The various closely spaced colored curves (red, orange, … blue, purple) are for different masses of the host star ranging from 0.3 M to 1.4 M, respectively. The dotted red curve is the limit obtained for incompressible fluid bodies. Heavy, filled, colored circles refer to fiducial-mass objects detailed in Table 1. © Rappaport et al.

As we can see, there is a general trend of decreasing minimum orbital periods, Pmin from 620 min (10.3 hr) for Saturn-mass objects (red circle in Fig. 4), to 430 min (7.2 hr) for Jupiters (orange circle), to 104 min (1.7 hr) for objects on the boundary between super-Jupiters and brown dwarfs (blue circle), all the way down to 37 min (0.62 hr) for the coldest and most massive brown dwarfs (purple circle), respectively. These values are summarized in Table 1 below.

Table 1: Minimum Orbital Periods of H-Rich Bodies © Rappaport et al.

Their work also aimed, towards distinguishing brown dwarfs from planets that are found transiting the host white dwarf without recourse to near infrared or radial velocity measurements. For objects with orbital periods ≲ 100 minutes they concluded that we are observing a brown dwarf (or second WD) rather than a gas-giant planet.

Reference: S. Rappaport, A. Vanderburg, J. Schwab, L. Nelson, “Minimum Orbital Periods of H-Rich Bodies”, Arxiv, pp. 1-10, 2021. https://arxiv.org/abs/2104.12083

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