P K pn

where K is constant. This relation may be used when the density does not depend on temperature. The index n (the polytropic index) takes the value of 0 for non-compressible objects (the density being constant as a function of P); the value n = 3/2 corresponds to a perfect, monatomic gas (a pure hydrogen plasma). The mass-radius relationship may be written as:

For an object with the mass of Jupiter, the value of n is close to 1. For an object of 10 Jupiter masses, we find that n = 1.3: The effect of compression becomes such that the radius decreases as the mass increases. This effect is shown in Fig. 7.1, which shows the mass-radius relationship for different compositions. The curve

Fig. 7.1 The mass-radius relationship for planets of different chemical compositions. The position of the four planets is indicated by their respective initials. The curve marked x = 0.25 represents a mixture of hydrogen and helium of solar composition (Y = 0.25) (After Marley, 1999)

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