The Sun The Sun is a very typical main sequence star. It contains 1000 9mes more mass than the combined mass of all other solar system bodies. By mass it is composed of 74% H, 25% He and 1% heavier elements ( metals ). The mean density is 1400 kg/m 3. Surface temperature is 5800 K, central temperature 15.5 x 10 6 K. Energy is generated through nuclear fusion reac9ons in which 4 H nuclei fuse to form a He nucleus. The set of reac9ons involved is called the proton- proton chain. The combined mass of 4 individual hydrogen nuclei is 6.693 x 10-27 kg, and one He nucleus has mass 6.645 x 10-27 kg, showing that the fusion reac9ons convert mass into energy. Using Einstein s E=mc 2 we see that each complete reac9on leads to 4.3 x 10-12 Joules of energy being released. The luminosity of the Sun is 3.9 x 10 26 W, corresponding to 9 x 10 37 reac9ons per second. The Sun converts 6 x 10 11 kg of mass into energy every second.
The proton- proton chain is represented below. More precisely, the pp- I chain is shown, as there are addi9onal reac9ons (pp- II, pp- III and pp- IV) that also occur leading to more modest energy produc9on. There is also a set of reac9ons called the CNO cycle that contributes only 1.7% of the energy produc9on in the Sun, but is the dominant energy source in more massive, ho_er stars. See on- line supplementary notes for more details.
Two other possible sources of energy are chemical reac9ons (e.g. combus9on) or gravita9onal contrac9on (as during a protostar s Kelvin- Helmholtz contrac9on stage). Chemical reac9ons involved in burning on the Earth release approximately 10-19 Joules of energy per atom. The number of atoms required to generate the Sun s luminosity is approximately 3.9 x 10 45 per second. We know the Sun contains about 10 57 atoms, so the length of 9me required for all atoms to be involved in combus9on reac9ons is 3 x 10 11 seconds. This corresponds to 10 4 years. Clearly chemical energy is not the source of the Sun s luminosity. A similar calcula9on demonstrates that gravita9onal contrac9on can also not be the source of the Sun s luminosity, given our knowledge of the age of the Earth and Sun. In that case we find the life- 9me of the Sun would be approximately 10 7 years, which is also the length of 9me required for the Sun to join the zero- age main sequence ader it contracts during the protostar stage of its evolu9on. See on- line supplementary lecture notes.
Solar neutrinos The energy generated by nuclear reac9ons at the centre of the Sun does not reach us directly. The high energy photons diffuse toward the surface, experiencing many sca_erings and absorp9ons/re- emissions before finally being emi_ed from the surface. Direct evidence of the nuclear reac9ons instead comes from detec9on of neutrinos. Because these par9cles interact only via the weak force most are able to stream out of the Sun s core without being absorbed. Experiments designed to detect solar neutrinos discovered the phenomenon of neutrino oscilla9ons where neutrinos of one flavour (electron, muon or tau) are able to change into a different flavour during their flight from the centre of the Sun to its surface. Thermal equilibrium The Sun is in a state of thermal equilibrium the rate of energy genera9on in the core is equal to the rate of radia9on loss from the surface. Hydrosta9c equilibrium The Sun is in a state of hydrosta9c equilibrium this means that at each radius in the Sun gas experiences an inward accelera9on due to gravity that is balanced by an outward accelera9on due to pressure (or more accurately due to the pressure gradient since pressure acts on both the upper and lower surface of an imaginary slab of gas, but the upward pressure force is greater than the downward pressure force).
Energy transport Heat may be transported by radia9on, convec9on or conduc9on. In the Sun only radia9on and convec9on are important. In the deep interior of the Sun, where the gas is fully ionised because of the high temperatures, energy flows of out the core by a process of radia9ve diffusion. Here, the high energy photons generated by nuclear reac9ons are sca_ered, absorbed and re- emi_ed by the atoms and free electrons, which causes them to degrade in energy. The fact that heat is diffusing tells us that the photons are undergoing a random walk from the solar interior to its surface. This random walk means that energy generated in the Sun s core takes approximately 10 6 years to reach the surface. Between approximately 71% of the Sun s radius up to its surface, heat transport occurs via convec9on where fluid mo9ons carry heat from the hot inner regions to the cool outer regions. Convec9on occurs because the gas becomes increasingly opaque to radia9on as the temperature decreases and atomic nuclei begin to recombine with electrons. A simple picture of convec9on imagines a local element of gas that becomes ho_er than its surroundings, such that it expands slightly. The lower density of the gas element compared to its surroundings causes it to experience a buoyancy force that causes it to move upward. As it moves upward the blob slowly radiates its excess internal energy, cools, and then descends back down again. The process is repeated leading to a net outward flow of heat caused by the convec9ve fluid mo9ons.
The top image displays a schema9c representa9on of the Sun s interior showing the radia9ve and convec9ve zones. The lower diagram shows an image of the solar surface showing the effect of granula9on. This granula9on is caused by warmer convec9on cells rising to the surface (the bright regions) and regions that have cooled (the darker regions) descending into the solar interior. The granules observed in the image below have a typical size of 1000 km. A video showing convec9on currents at the Sun s surface can be observed at: h_p://www.youtube.com/watch?v=et6torwqq74
Modelling the Sun Astronomers encapsulate the effects of energy genera9on, heat transport, and hydrosta9c equilibrium in a set of differen9al equa9ons that are solved on a computer in order to determine how the Sun s temperature, density and pressure vary with radius from the core to the surface. The computer models are constrained by the observa9on that the Sun s temperature is 5800 K at the surface, and that the density and pressure go to zero at the surface (these are the bounday condi9ons ). See the on- line supplementary notes for more detail. Results from solar models are shown below. All energy genera9on through nuclear reac9ons occurs within the inner 20 % of the radius, where the temperature is high enough (T ~ 8 x 10 6 K) that hydrogen nuclei are able to overcome the Coulomb barrier caused by the electrosta9c repulsion between protons. This region contains approximately 25% of the Sun s mass.
The solar photosphere The solar photosphere is the thin layer of gas near the surface from which essen9ally all the visible light that we see is emi_ed. We can observe into the photosphere a distance of about 400 km, which is a 9ny frac9on of the Sun s radius, and this explains why the Sun appears to have a sharp rather than fuzzy surface (photons emi_ed from a depth greater than 400 km do not reach us directly because they are absorbed and re- emi_ed before reaching the base of the photosphere). The photosphere is heated from below, and is therefore ho_er at its base than its surface. This leads to the phenomenon of limb darkening, where the brightness of the solar disc as observed from Earth appears to decrease toward the edge (limb) and increase toward the centre. The diagram below explains this phenomenon.
Solar chromosphere Above the photosphere we have the chromosphere which can only be seen during a solar eclipse. The chromosphere is a low density hot region (T ~ 25,000K) whose spectrum is dominated by emission lines (par9cularly the H- alpha line at 656.3 nm that gives the chromosphere it s pink colour). Unlike in the photosphere, the temperature increases with height in the chromosphere, a phenomenon that is caused by hea9ng effects induced by the solar magne9c field. The ver9cal jets of gas known as spicules are also caused by the magne9c field.
Solar corona The outermost layer of the Sun s atmosphere is called the corona. Again this can only be seen during a total solar eclipse when light from the photosphere is blocked out. The temperature in the corona reaches 2 x 10 6 K, but in spite of this the corona has only a modest luminosity because of its low density (10 11 atoms per m 3 compared with 10 23 per m 3 in the photosphere). The high temperature in the corona means that the plasma (a gas consis9ng en9rely of ions and electrons that is charge neutral overall) can escape the Sun s gravity this is the solar wind. Approximately 10 9 kg are lost into the solar wind every second. Hea9ng of the corona, the erup9on of solar flares and coronal mass ejec9ons (that launch up to 10 12 kg of high temperature plasma from the surface of the Sun into interplanetary space), are caused by magne9c reconnec9on where field lines of opposite polarity cancel one another and the lost magne9c energy heats the plasma.
One impact of coronal mass ejec9ons is the genera9on of intense aurora on Earth as the charged par9cles are funneled toward the poles by the Earth s magne9c field.
The solar cycle Observa9ons of sunspots in the photosphere indicates that their number varies with a cycle of approximately 11 years (see diagram at bo_om of page). Sunspots are dark cooler regions associated with regions of intense magne9c fields at the solar surface. The sunspot cycle is related to a 22 year cycle in which the Sun s global magne9c field direc9on changes polarity (North to South and back to North again). As with the magne9c fields associated with the planets, the solar magne9c field is generated by a dynamo that requires an electrically conduc9ng fluid (gas), rota9on and convec9on. The solar dynamo evolves in a way that causes the Sun s magne9c field direc9on to reverse every 11 years. Sunspots are regions where intense columns of magne9c flux rise into the solar atmosphere (the atmosphere is cooler in the sunspots because the magne9c field causes the gas to expand locally). Hence the rela9on between sunspot ac9vity and the reversal of the magne9c field direc9ons.
Stars Under dark skies the naked eye can see a few thousand stars. A small telescope can reveal the existence of about 2 million stars. We now know that the Milky Way galaxy contains around 100 billion (10 11 ) stars. We have previously discussed how parallax measurements can determine the distances to nearby stars. The nearest star to the Sun has the largest parallax angle of 0.772 arcsec, corresponding to a distance of 1/(0.772) =1.30 pc, where 1 pc = 3.26 light years. The Hipparchos satellite could measure parallax angles with an accuracy of 0.001 arcsec, and determined the distance to 118,000 stars (i.e. those within a distance 1000 pc). Determining the distance to those stars that lie further away requires other techniques. The flux of radia9on from a star (energy per metre 2 per second) is related to its intrinsic luminosity, L, and its distance to an observer, d, by the expression F = L/(4πd 2 ). If we wish to compare the luminosity of a star whose distance has been determined by parallax with the luminosity of the Sun we can write L/L sun = (d/d sun ) 2 x (F/F sun ) where L sun, F sun and d sun are the luminosity of the Sun, and flux from the Sun, and the Sun- Earth distance. Hence we can measure the intrinsic luminosi9es of stars with accurate distance determina9ons. In our following discussion we will use the terms flux and apparent brightness interchangeably.
The magnitude scale Astronomers use the magnitude scale to denote the brightness of stars, a scale that was introduced originally by the Greek astronomer Hipparchus in the 2 nd century B.C. We use the terms apparent magnitude and absolute magnitude, and their rela9on will be defined below. Apparent magnitude measures how bright a star appears to an observer on Earth. Absolute magnitude is used to define the absolute brightness of a star. The apparent magnitude scale of ancient Greece was based on defining the brightest stars as having an apparent magnitude m=+1 and the dimmest stars (just visible to the naked eye) with a magnitude m=+6. Magnitude m=+2 stars were perceived to be about half as bright as magnitude m=+1 stars, and magnitude m=+3 stars were perceived to be about half as bright as magnitude m=+2 stars, and so on. A magnitude m=+1 star was then approximately 2 5 (=64) 9mes as bright as a magnitude m=+6 star. A key point to note is that stars with lower apparent magnitudes are brighter than stars with high apparent magnitudes! This system was formalised in 1856 by defining a m=+1 star as being 100 9mes as bright as a m=+6 star. Therefore stars that differ in apparent magnitude by +1 have a brightness ra9o of 2.512 = (100) 1/5 under this system. Because each step on the magnitude scale corresponds to a factor 2.512 in perceived brightness, the magnitude scale corresponds to a logarithmic scale.
The rela9on between apparent magnitudes and fluxes received from two stars is given by m 2 m 1 = 2.5 log 10 (F 1 /F 2 ) because log 10 (100) = 2, so magnitude +6 and +1 stars differ in their received fluxes by a factor of 100. Although the Greek magnitude scale only went between +1 and +6, the modern scale has nega9ve values to represent the brightest objects in the sky (e.g. the Sun has an absolute magnitude of - 26.74, Venus at maximum brightness has m=- 4.4). The star Vega was originally chosen as the zero point on the scale, but in fact has an apparent magnitude m=+0.03.
Defini9on: The absolute magnitude of a star is defined to be the apparent magnitude that it would have if placed at a distance of 10 parsecs from the Earth. Absolute magnitude is denoted with an upper case M. Apparent magnitude is denoted with a lower case m. If the flux of a star measured at its current distance from the Earth is F *, and the flux measured at 10 parsecs is F 10, then M and m are related by m M = 2.5log 10 (F 10 /F * ). We note that F 10 /F * = L/(4π10 2 ) L/(4πd 2 ) = (10 / d) 2 if we measure d in parsecs. Thus we can write m M = 5log 10 d 5. As an exercise you should prove this result!
Stellar spectra We know already that for a black- body emi_er the colour of the object is related to its temperature. Wien s displacement law tells us that the peak wavelength of emission is directly related to the temperature. A quick look at an image containing numerous stars shows that they have different colours. Blue stars are ho_er than yellow stars which are ho_er than red stars. To gain further insight into the proper9es of stellar atmospheres, it is necessary to analyse their spectra, which differ for different stars. Some stars show absorp9on features due to hydrogen Balmer lines. The Sun displays prominent lines due to calcium, iron and sodium. Astronomers have developed a classifica9on system for describing the spectra of different stars. The classifica9on system used now was developed originally in the 1890 s. At that 9me a star was assigned a classifica9on running from A to O depending on the appearance of Balmer absorp9on lines in the stars spectrum. The scheme has been reordered, and some classes have been dropped, leaving only the classes OBAFGKM.
As shown by the diagram below, these spectral classes correspond to the temperature of the star s photosphere. For example, O stars have surface temperatures greater than 25,000 K, so the Balmer lines (formed by raising an electron occupying the n=2 energy level) are weak in this star because a large frac9on of the hydrogen is ionised and therefore unable to produce Balmer absorp9on features. Similarly, if the star is much cooler than 10,000 K most of the hydrogen is in the n=1 (ground) state, and few H atoms have electrons in the n=2 orbit, making the Balmer lines weak. The op9mal temperature for strong Balmer features is approximately 9000 K, so that most hydrogen atoms have electron in the n=2 level and very few are ionised. We see that stars with spectral classifica9on M are cool enough even to have molecular lines present in their spectra. Each spectral class is divided into 10 subclasses running from 0 to 9. For example, the Sun is a G2 star. We say that the Sun is of spectral type G2.
The sensi9vity of the ionisa9on state of an element to temperature drives changes in the appearance of spectra as one moves from cool stars to hot stars. For example, helium becomes singly ionised at temperatures above 30,000 K, so prominent He II lines are visible in the spectra of O stars. Note that neutral helium is denoted He I, and singly ionised helium is denoted He II. Neutral silicon is denoted Si I and doubly ionised silicon is denoted Si III. Further details about the rela9on between spectral class, temperature, and the visible lines are given in the table.
Determining stellar radii Measuring the flux, F, from a star and measuring its distance, d, using parallax allows us to determine the intrinsic luminosity, L, through the expression: F=L/(4πd 2 ) Measuring the temperature by determining the star s spectral type then allows the radius of the star, R, to be calculated using the following rela9on: L = 4πR 2 σt 4. The range of stellar radii measured for nearby stars whose parallax can be measured is enormous. We have white dwarf stars with surface temperatures T=25,000 K and radii ~ 6000 km (essen9ally the radius of the Earth). We also have supergiants with radii 10 3 9mes larger than the Sun s and luminosi9es that are 10 5 9mes larger. All this informa9on can be obtained by making just three measurements: parallax (distance), flux (or apparent brightness), spectral type (temperature). An important point to note is that spectral type is a direct measure of the surface temperature of a star.
The Hertzsprung- Russell diagram This is a plot of log (luminosity) versus log (temperature) for observed stars, and provides a visual representa9on of the different types of stars that exist. Note that temperature increases from right to led on the x- axis. About 90% of stars in the night sky lie on the red curve called the main sequence. Stars which fall on the line when plo_ed in a H- R diagram are called main- sequence stars, and these are all steadily burning hydrogen into helium in their cores.
The upper right of the H- R diagram shows cooler higher luminosity stars. For a star to be highly luminous but low temperature we know that it must have a large radius. The expression L = 4πR 2 σt 4 demonstrates that this is the case, and so these stars must be giants. Furthermore, we find that taking the log of both sides of this expression leads to: log (L) = 4 log (T) + log (4πR 2 σ). This shows that if the radius, R, is constant then lines of log (L) versus log (T) should be straight lines in the H- R diagram, as indicated in the diagram on the previous slide. Most giant stars are 100 1000 9mes more luminous than the Sun and have temperatures between 3000 6000 K. Cooler members of this class are called red giants because they appear red- ish in colour. Examples are Aldebaran in the constella9on Taurus and Arcturus in the constella9on Bootes. A few rarer giant stars are larger and brighter than typical red giants, with radii up to 1000 R sun. These stars are called super- giants, and examples include Betelgeuse in Orion and Antares in Scorpius. Giants & super- giants make up about 1% of stars. The remaining 9% of stars are the white dwarfs located in the lower led of the HR diagram. As their name suggests, these are small (approximately the size of the Earth) hot stars that are so dim they can only be seen with a telescope. They are the cooling embers of stars that were once main sequence stars.
Luminosity classes The appearance of a star s spectrum is not only determined by its temperature (and chemical composi9on), but also by the density and pressure in the atmosphere. In a dense medium in which collisions between atoms are frequent and atoms are close together, the energy levels are modified by the proximity of other atoms, and emission is affected if collisions occur during the emission process. These effects broaden spectral lines in a predictable manner. In a low density atmosphere collisions are less frequent and energy levels remain essen9ally unchanged and narrow. Super- giant stars tend to have low density atmospheres because they are large with extended atmospheres. Main- sequence stars have denser atmospheres. The diagram shows two stars that have the same basic spectral classifica9on, B8, and the same temperature - a super- giant and a main - sequence star. The subtle differences in spectral lines were used to define a system of luminosity classes in the 1930s. Theses luminosity classes are shown on the diagram on the next slide.
Luminosity classes Ia and Ib correspond to super- giants. Luminosity class V corresponds to main- sequence stars. The Sun is a G2 V star. The spectral type of a star indicates the star s surface temperature. It s luminosity class indicates its luminosity. Both pieces of informa9on determine its posi9on on a H- R diagram. A G2 V star is a main sequence star with temperature 5800 K. Aldebaran is a K5 III star, which tells us it is a red giant with luminosity around 370 9mes that of the Sun. Spectroscopic parallax Knowing the spectral type and luminosity class allows a star s distance to be es9mated because we can measure the apparent brightness (flux) of the star. Having an es9mate of the actual luminosity through the luminosity class, combined with the inverse- square law that states F = L/(4πd 2 ) allows the distance d to be es9mated. Distances measured in this way are accurate to about 10%.
Determining stellar masses Stellar masses are determined from binary stars pairs of stars that orbit their mutual centre of mass. Visual binaries are binary systems in which the two stars can be resolved because their separa9on is large enough. Their orbit can be measured by tracing the paths of the stars on the sky. By measuring the orbital period, semi- major axis of the rela9ve orbit, and the distance of each star from the common centre of mass, the mass of each star can be determined using Kepler s 3 rd law. Spectroscopic binaries are binary systems whose orbital separa9on is too small for the individual stars to be resolved in an astronomical image. Measurement of their spectra may reveal the orbital mo9on through the periodic doppler shid. If the spectrum of each individual star can be measured then the individual velocity of each star around the common centre of mass can be determined, and their individual masses calculated. Accurate mass determina9on requires the stars to have an orbit plane that is edge- on to the line of sight (as in the case of extrasolar planets). See on- line supplementary lecture notes for a descrip9on of how these methods work.
The images above show the orbit of the visual binary 2MASSW J0746425
Mass- luminosity rela9on for stars We can determine the masses of the individual stars in binary systems. We can determine their luminosi9es from either their luminosity class via their spectra, or more accurately by measuring the flux from each star and using parallax to determine the distance and the intrinsic luminosity of each star. The diagram below shows the mass- luminosity rela9onship that exists for main- sequence stars (not for other types of stars). The plot shows log (L) versus log (M), and the approximate straight line shown by the data suggests that L is propor9onal to M q where q 3.5. Higher mass stars have higher luminosity, and the main reason for this is that, compared to low mass stars, the central pressures and temperatures in the cores of high mass stars need to be large in order to support the weight of the overlying gas. The proton- proton chain and CNO cycle are highly temperature dependent, so higher central temperatures lead to significantly larger luminosi9es.