PHY111 Lecture 7: Stellar Properties and Lifetimes PDF

PHY111 Lecture 7: Stellar properties and lifetimes Lecture aims Have an idea of how common stars of different masses are Know that there is a relationship between the mass of a star and its radius and luminosity while on the main sequence Understand how energy is generated in the core of the S...

PHY111 Lecture 7: Stellar properties and lifetimes Lecture aims Have an idea of how common stars of different masses are Know that there is a relationship between the mass of a star and its radius and luminosity while on the main sequence Understand how energy is generated in the core of the Sun via nuclear fusion Understand how the mass-luminosity relationship helps us estimate the lifetime of stars Physical properties of stars By combining the data from the HR diagram and eclipsing binaries we can see how the physical properties of stars change with their masses and can use these results to understand how stars evolve. For now we will focus on main sequence stars, since these are by far the most common type of star and so the ones we have the most information about Stellar masses When we look at the masses of stars we find that they span a wide range, from around 10-2 M⊙ to about 102 M⊙. Technically ‘stars’ less than about 0.1 M⊙ are ‘brown dwarfs’ not stars as they do not burn H (we’ll see why this is the case soon). The (initial) mass function We find that stars always seem to be born with a similar distribution of masses (shown here is the Pleiades, which is very typical) log10(mass) mass) log10(Number of stars of this By far the most common types of stars are low mass stars (0.1-0.5M⊙) Note: the distribution on the right (specifically The red line) is known as a log-normal distribution. It is what you get when you plot a Gaussian in log space. For every solar mass star made, roughly 100 low mass stars are made The mass-radius relationship On the main sequence we find a relationship between the mass and radius of stars: Where the exponent varies from around 0.6 at the lowest masses to about 0.8 for the highest masses This isn’t a particularly informative relationship, but it can be useful if you want to know the mass of a star but you only know its radius, or vice versa. It also allows you to go straight from a surface gravity measurement to mass and radius The mass-luminosity relationship On the main sequence we find a relationship between the mass and luminosity of stars: This is true for roughly Solar-mass stars – the power-law varies a bit (you might see it as 3 or 4, it depends on the range of masses that are used to fit it). This is an extremely important relationship and the fact that the exponent is much greater than 1 means that stars of different masses will live very different lives The mass-luminosity relationship From the HR diagram we know that on the main sequence the luminosity is a function of the temperature/spectral type Therefore, spectral type is also a measure of the mass of a star (note that this only works on the main sequence) M dwarfs are low mass (0.1-0.5M⊙) G dwarfs are roughly 0.8-1.2M⊙ O dwarfs are 20M⊙ to more than 100M⊙ The mass-luminosity relationship Why are some stars so much more luminous than others? To be more luminous they must be producing more energy, and on the main sequence their luminosity depends strongly on their mass. More massive stars must produce more energy per unit time to be more luminous. A 10 M⊙ star is around 3000 L⊙ but are rare A 0.1 M⊙ star is around 0.0003 L⊙ but are extremely common Energy generation We can age the Earth to about 4.5 Gyr from radiometric dating – so the Sun must be at least this old as well. The only process that could keep the Sun producing 3.8 x 1026 W (the luminosity of the Sun) for 4.5 Gyr (that’s E = power x time = 5x1043 J so far!) is nuclear fusion. We convert mass to energy from E=mc2 – and c is big, so a small amount of mass turns into a very large amount of energy. We saw the Sun is about 75% H and 25% He – the easiest way to make energy with this mixture is fusing H to He. Fusion To fuse H to He we need very high temperatures (T) and pressures (P). High T means the nuclei move very fast, and high P means they meet each other often. The higher the T and P, the more reactions there are and the more energy is generated. More massive stars have higher central T and P because the gravitational forces are larger – so produce more energy per unit time (luminosity). Fusion To fuse two nuclei we need to overcome the the ‘Coulomb barrier’. As both nuclei have +ve charge they will be repelled by the electromagnetic force unless they can get close enough (~10-15m) for the strong nuclear force to become dominant. Separation of nuclei Repulsive Force between the nuclei Attractive Fusion If the mass is less than about 0.1M⊙, the central T and P never get high enough to fuse H. This sets the limit of what a star is: it has to be massive enough to fuse H (or will do, or has done). Between 0.013M⊙ and 0.1M⊙ objects can fuse deuterium (2H): this defines a ‘brown dwarf’. Below 0.013M⊙ (13 Jupiter masses) they can’t even fuse deuterium so are classed as ‘planets’. The pp-chain The way most stars fuse H to He is via the pp-chain. The first step is to make deuterium (a proton+neutron): 1 H + 1H -> 2H + e+ + ν + energy where e+ is a positron and ν is a neutrino (the positron then annihilates with an electron and makes more energy). The energy is released in the form of a photon Then another 1H is added to make 3He: 2 H + 1H -> 3He + energy Then two 3He meet to make 4He and 2 protons 3 He + 3He -> 4He + 21H + energy The pp-chain There are slightly different ways of going from 4H to 1He, but this is the most common route in the Sun. In stars >1.3M⊙ there is the CNO cycle that involves C, N and O in the fusion process – but we’ll ignore that in this module. The final 4He nucleus has 0.7% less mass than the 4 1H nuclei – this is released as energy: about 4x10-12 J per He. So to make the ~4x1026 J the Sun releases every second needs about 10 38 reactions per second! Solar neutrinos All of the fusion in the Sun occurs deep within the core, so how can we be confident that this is what is actually happening? We can estimate the number of neutrinos that must be being produced every second, since one neutrino is made per reaction chain (1038-ish). Neutrinos hardly interact with anything – most pass right through the Sun (unlike the photons produced by fusion, which are quickly scattered and absorbed by the Sun) and Earth and we never notice them. But a tiny fraction do interact with large underground detectors in exactly the right numbers. The main sequence lifetime of the Sun Main sequence stars produce energy by fusing H into He. The final 4He nucleus has 0.7% less mass than the 4 1H nuclei – this is released as energy: about 4x10-12 J per He created. How long can a star keep doing this? When do they run out of H? The main sequence lifetime of the Sun We can estimate the length of time the Sun can produce energy this way. The core where it is hot and dense enough for reactions happen is about 10% of the mass of the Sun: 0.1 M⊙ = 0.1 x (2x1030) = 2x1029 kg So this sets the total amount of fuel that the Sun has access to over its lifetime. The main sequence lifetime of the Sun Each 4He nucleus weighs about 0.7% less than the 4 1H nuclei (protons) that made it: so that missing mass must be converted into energy. Therefore the total mass available that is converted into energy during the Sun’s lifetime is 0.7% of the core mass: 0.007 x (2x1029) = 1.4x1027 kg Using e=mc2 we can then calculate the total amount of energy that the Sun can produce via H fusion: (1.4x1027 kg) x (3x108 m s-1)2 = 1.3x1044 J We know that the Sun produces 3.8x1026 W (remember 1 Watt = 1 Joule per second), so at this rate it will take: 1.3x1044 J / 3.8x1026 W = 3.4x1017 seconds Or about 10 billion years, for the Sun to use up all its H fuel. Therefore, we expect the Sun to spend around 10 Gyr on the main sequence. The mass-luminosity relationship We saw that on the main sequence, and there is a mass-luminosity relationship This tells us that more massive stars produce much more energy per unit time (L) than they have extra fuel (M). A star with twice the mass of the Sun (so twice the fuel), produces 2 3.5 = 11 times more energy per second than the Sun Lifetimes of stars Main sequence lifetimes are a very strong function of mass. We can estimate the main sequence lifetime of a star as: More massive stars have more fuel, but burn it much more rapidly. Low-mass stars burn at a very slow rate. A 10 M⊙ star has L~3 000 L⊙ – 10x more fuel, but burnt 3 000x faster 🡪 lifetime of about 50 Myr. A 0.1 M⊙ star has L~3x10-4 L⊙ – 10x less fuel, but burnt 3 000x slower 🡪 lifetime of 30 trillion years (30 000 Gyr). Key points to Remember ‘Stars’ cover a range of masses from about 10-2 M⊙ to about 102 M⊙. Most (>90%) of stars are 0.1-0.5 M⊙. Technically something is only a star if it is >0.1 M⊙, less than this it is a brown dwarf. On the main sequence there is a strong relationship between mass and luminosity where L/L⊙ ~ (M/M⊙)3.5. On the main sequence spectral class is a measure of mass from high (O) to low (M). Key points to Remember Stars on the main sequence produce energy by the fusion of H to He in their cores. Fusion occurs in stars, with masses >0.1M⊙ Stars with M