THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE?

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1 THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? The stars are spheres of hot gas. Most of them shine beause they are fusing hydrogen into helium in their entral parts. In this problem we use onepts of both lassial and quantum mehanis, as well as of eletrostatis and thermodynamis, to understand why stars have to be big enough to ahieve this fusion proess and also derive what would be the mass and radius of the smallest star that an fuse hydrogen. Figure 1. Our Sun, as most stars, shines as a result of thermonulear fusion of hydrogen into helium in its entral parts. USEFUL CONSTANTS Gravitational onstant = 11 G = m 3 kg -1 s 3 Boltzmann s onstant = k = J K Plank s onstant = h = m kg s -1 Mass of the proton = Mass of the eletron = mp me = = Unit of eletri harge = q = Eletri onstant (vauum permittivity) = ε 0 = C N -1 m - 8 Radius of the Sun = = m Mass of the Sun = RS M S = kg 19 kg kg C

2 1. A lassial estimate of the temperature at the enter of the stars. Assume that the gas that forms the star is pure ionized hydrogen (eletrons and protons in equal amounts), and that it behaves like an ideal gas. From the lassial point of view, 15 to fuse two protons, they need to get as lose as 10 m for the short range strong nulear fore, whih is attrative, to beome dominant. However, to bring them together they have to overome first the repulsive ation of Coulomb s fore. Assume lassially that the two protons (taken to be point soures) are moving in an antiparallel way, eah with veloity v rms, the root-mean-square (rms) veloity of the protons, in a onedimensional frontal ollision. 1a What has to be the temperature of the gas, T, so that the distane of 15 losest approah of the protons, d, equals 10 m? Give this and all numerial values in this problem up to two signifiant figures Finding that the previous temperature estimate is wrong. To hek if the previous temperature estimate is reasonable, one needs an independent way of estimating the entral temperature of a star. The struture of the stars is very ompliated, but we an gain signifiant understanding making some assumptions. Stars are in equilibrium, that is, they do not expand or ontrat beause the inward fore of gravity is balaned by the outward fore of pressure (see Figure ). For a slab of gas the equation of hydrostati equilibrium at a given distane r from the enter of the star, is given by P G M = r r r ρr, where P is the pressure of the gas, G the gravitational onstant, M r the mass of the star within a sphere of radius r, and ρ is the density of the gas in the slab. r

3 Figure. The stars are in hydrostati equilibrium, with the pressure differene balaning gravity. An order of magnitude estimate of the entral temperature of the star an be obtained with values of the parameters at the enter and at the surfae of the star, making the following approximations: P P o P, where P and P o are the pressures at the enter and surfae of the star, respetively. Sine P >> P, we an assume that P P. o Within the same approximation, we an write r R, where R is the total radius of the star, and M M M, r R = with M the total mass of the star. The density may be approximated by its value at the enter, ρr ρ. You an assume that the pressure is that of an ideal gas. a Find an equation for the temperature at the enter of the star, T, in terms of the radius and mass of the star and of physial onstants only.

4 We an use now the following predition of this model as a riterion for its validity: b Using the equation found in (a) write down the ratio a star in terms of physial onstants and T only. M / R expeted for Use the value of T derived in setion (1a) and find the numerial value of the ratio M / R expeted for a star. d Now, alulate the ratio M ( Sun) / R( Sun), and verify that this value is muh smaller than the one found in (). 3. A quantum mehanial estimate of the temperature at the enter of the stars The large disrepany found in (d) suggests that the lassial estimate for T obtained in (1a) is not orret. The solution to this disrepany is found when we onsider quantum mehanial effets, that tell us that the protons behave as waves and that a single proton is smeared on a size of the order of λ, the de Broglie wavelength. This implies that if d, the distane of losest approah of the protons is of the order of λ p, the protons in a quantum mehanial sense overlap and an fuse. p 3a λp Assuming that d = is the ondition that allows fusion, for a proton 1/ with veloity v rms, find an equation fort in terms of physial onstants only b Evaluate numerially the value of T obtained in (3a). 3 Use the value of T derived in (3b) to find the numerial value of the ratio M / R expeted for a star, using the formula derived in (b). Verify that this value is quite similar to the ratio M ( Sun) / R( Sun) observed. Indeed, stars in the so-alled main sequene (fusing hydrogen) approximately do follow this ratio for a large range of masses.

5 4. The mass/radius ratio of the stars. The previous agreement suggests that the quantum mehanial approah for estimating the temperature at the enter of the Sun is orret. 4a Use the previous results to demonstrate that for any star fusing hydrogen, the ratio of mass M to radius R is the same and depends only on physial onstants. Find the equation for the ratio M / R for stars fusing hydrogen. 5. The mass and radius of the smallest star. The result found in (4a) suggests that there ould be stars of any mass as long as suh a relationship is fulfilled; however, this is not true. The gas inside normal stars fusing hydrogen is known to behave approximately as an ideal gas. This means that d e, the typial separation between eletrons is on the average larger that λ e, their typial de Broglie wavelength. If loser, the eletrons would be in a so-alled degenerate state and the stars would behave differently. Note the distintion in the ways we treat protons and eletrons inside the star. For protons, their de Broglie waves should overlap losely as they ollide in order to fuse, whereas for eletrons their de Broglie waves should not overlap in order to remain as an ideal gas. The density in the stars inreases with dereasing radius. Nevertheless, for this order-ofmagnitude estimate assume they are of uniform density. You may further use that m >>. p m e 5a Find an equation for n e, the average eletron number density inside the star. 5b Find an equation for d e, the typial separation between eletrons inside the star. 5 λe Use the d e ondition to write down an equation for the radius of 1/ the smallest normal star possible. Take the temperature at the enter of the star as typial for all the stellar interior. 1.5

6 5d Find the numerial value of the radius of the smallest normal star possible, both in meters and in units of solar radius. 5e Find the numerial value of the mass of the smallest normal star possible, both in kg and in units of solar masses. 6. Fusing helium nulei in older stars. As stars get older they will have fused most of the hydrogen in their ores into helium (He), so they are fored to start fusing helium into heavier elements in order to ontinue shining. A helium nuleus has two protons and two neutrons, so it has twie the harge λp and approximately four times the mass of a proton. We saw before that d = is the 1/ ondition for the protons to fuse. 6a Set the equivalent ondition for helium nulei and find v rms (He), the rms veloity of the helium nulei and T (He), the temperature needed for helium fusion.