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Figure 1: Envelope mass solution manifold. Environmental parameters for this manifold are set to a=5.2 AU, and T=123 K. Each point on the surface gives the mass of the protoplanet's envelope for a given  and gas density at the core surface, . Models with different initial parameters generally connect to different nebulae. Several different regions are easily discernible: I - flat slope with gradient of 1, for the region [-1, 2] in and [-12, 6] in ; II - flat slope with gradient of 0.5, roughly encompassing [4-6, 8] in , and all ; III - "base of the island'', [-8, -1] in and [-12, -6] in ; IV - "island'', [-8, -1] in and [-6, 4-8] in (cf. Fig. 2).

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In the text

	Figure 2: Manifold regions: I - compact non-self-gravitating envelopes, II - compact self-gravitating envelopes, III - uniform non-self-gravitating envelopes, IV - uniform self-gravitating envelopes. The border of the region IV somewhat depends on the choice of the surrounding nebula (cf. Fig. 12); we use here a value from the Hayashi (1985) minimum mass solar nebula model.
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In the text

Figure 3: Demonstration of the self-gravitating effect for sub- and super-critical cores: comparison of cuts through two manifolds, with- (M=M(r) in Eq. (3)) and without- ( ) the envelope's gravitating effect, each for two core masses. Cuts are for a=30 AU and T=51.1 K. Circles and squares represent the envelope mass of the subcritical core, calculated for M=M(r) and in Eq. (3), respectively. White and black triangles have the same meaning but for the supercritical core. Labels without arrows correspond to manifold regions from Fig. 2, while labels with arrows mark interfaces between regions. D corresponds to the "divergent wall'' which surrounds region IV (cf. Fig. 1). Self-gravitating envelopes with M=M(r) in Eq. (3) have a larger envelope mass than the corresponding envelopes with in Eq. (3) (cf. Fig. 6).

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In the text

	Figure 4: Envelope mass as a function of the nebula density . Labels are the same as in Fig. 3. Lines connect states with increasing . Note the strong dependence of on the envelope mass, and a non-trivial behavior of the for region IV (enlargement in Fig. 5).
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In the text

Figure 5: Enlargement of the boxed region of Fig. 4, isothermal curl regularized with the finite-density core; "-1.25'': black squares represent protoplanets with first subcritical line on the mesh of Fig. 1 and the arrow points at the black square with the highest , DS: two protoplanetary states with the largest envelope mass in the manifold, but with typically very different (cf. Sect. 3.9.2); in and out curves are the consequence of the core. The smaller the core, the closer the inand out curves are. The figure corresponds to a V-U plane for the protoplanets (see Sect. 3.5 for further discussion).

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In the text

	Figure 6: Uniform, compact and self-gravitating profiles. The uniform self- gravitating profile resembles the non-self-gravitating one until the envelope mass becomes comparable to the core mass. Then the density profile changes to .
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In the text

	Figure 7: Pressure as a function of density, for T=123 K. Black circles represent the ideal gas, squares are for the Carnahan-Starling EOS, and triangles are for the Saumon-Chabrier EOS. This figure also shows that a completely degenerate electron gas (stars) is not a good assumption for this ( ) parameter range.
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In the text

	Figure 8: Cut through the envelope mass manifold, for a core, a=5.2 AU, and T=123 K. Black circles represent the ideal gas, and squares are for the Carnahan-Starling EOS.
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In the text

	Figure 9: Specific temperature scale-height as a function of the density at the core surface, for different subcritical core masses. Protoplanetary models with cores of -8 (black circles), -5 (stars), and -3 (crosses) in logarithmic  units have much larger than unity. This justifies the isothermal assumption for the manifold regions III and IV.
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In the text

	Figure 10: Envelope mass solutions as a function of gas density at the core surface, for gas temperatures of 100, 500, 1000, 5000, and 10 000 K. A change of T has no influence on the envelope mass of the non-self-gravitating regions, while the same change of T will produce a significant effect for protoplanets in self-gravitating regions.
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In the text

	Figure 11: Envelope mass solutions as a function of gas density at the core surface, for orbital distances of 0.05, 0.1, 1, 5.2, and 30 AU. Enlargement: the transition from uniform to compact envelope solutions is more abrupt for protoplanets at large orbital radii. This is a consequence of the larger Hill-sphere of outer protoplanets.
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In the text

	Figure 12: Solution branches - isobars for - for (a=5.2 AU and T=123 K) manifold: the standard solar nebula solution branch is represented by the innermost solid line; an enhanced nebula with (dashed lines) has multiple solution branches; each solution branch has its own maximum core mass, and hence local critical mass.
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In the text

	Figure 13: For nebula density enhanced relative to a minimum-mass solar nebula, even more than two hydrostatic equilibria could exist; M: protoplanetary solutions with that fit into nebula; DS: double solutions, a special case of multiple solutions, cf. Figs. 5 and 15; S: protoplanetary solutions with the same core, whose envelope fits into the minimum-mass solar nebula.
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In the text

	Figure 14: Density profiles for the solutions which fit into the same (10-6 kg m-3) nebula. These solutions are labelled with M in Fig. 13.
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In the text

	Figure 15: Mass and density radial structure of the special case of multiple solutions, where two protoplanets have the same core, almost the same envelope mass, connect to the same nebula, but have different radial structure. These solutions are labelled DS in Fig. 13.
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In the text