Chapter 5 Global Parameters of the Sun PDF

Chapter 5 Global Parameters of the Sun 5.1 Astronomical Unit (AU) 5.2 Orbital Motion of the Earth 5.3 ⨀ and the Mass of the Sun 5.4 Power Output of the Sun: The Solar Luminosity 5.5 Radius of the Sun: ⨀ 5.6 Surface Gravity of the Sun 5.7 Escape Speed from the Solar Surface 5.8 Effective Temperatu...

Chapter 5 Global Parameters of the Sun 5.1 Astronomical Unit (AU) 5.2 Orbital Motion of the Earth 5.3 ⨀ and the Mass of the Sun 5.4 Power Output of the Sun: The Solar Luminosity 5.5 Radius of the Sun: ⨀ 5.6 Surface Gravity of the Sun 5.7 Escape Speed from the Solar Surface 5.8 Effective Temperature of the Sun 5.9 Shape of the Sun 5.10 Critical Frequency for Solar Oscillations 5.11 Mean Density of the Sun  To determine the physical processes occur in the Sun, we need to know certain properties of the Sun, including mass, radius, and other quantities. When it comes to astrophysical measurements, the quantity which can be measured with greatest accuracy is TIME. As a result, we begin our discussion of the determination of solar parameters by referring to measurements of certain intervals of time. 5.1 Orbital Motion of the Earth The single most important property which determines the evolutionary behaviour of a star is its mass; which determines whether its life ends quietly or explosively. To determine the mass of the Sun, ⨀ , we have to know the time that is required for the Earth to orbit the Sun. This is achieved by observing the interval of time required for the Sun to return to a given location relative to the “fixed” stars as seen by an observer on Earth. Actually the “fixed stars” used are a class of galaxies known as quasi- stellar radio sources (“quasars”). Relative to the quasar frame, this defines the unit of one sidereal year: = 365.25636 = 31,558,150. 5.1 Orbital Motion of the Earth For a precise estimates of various parameters of interest to solar system dynamics, visit http://ssd.jpl.nasa.gov/ Now that we know the orbital period of the Earth, we now turn to the equation of motion of the Earth in its orbit. Relative to a zero point which can be arbitrarily chosen, we have the position vector of the Sun is ( ) and the position vector of the Earth is ( ). 5.1 Orbital Motion of the Earth The position vector of Earth relative to the Sun is = ( ) − ( ), and the unit vector, , is directed from the Sun toward the Earth. Applying the Newton’s law of gravitation. The gravitational force causes the Earth to accelerate according to the equation 2 ( ) ⨀ ⨁ ⨁ =− (1.1) 2 2 where is Newton’s gravitational constant, and the negative sign indicates that the force is toward the Sun, i.e., in the negative direction. 5.1 Orbital Motion of the Earth The gravitational force causes the Sun to accelerate according to the equation 2 ( ) ⨀ ⨁ ⨀ =+ (1.2) 2 2 where the positive sign indicates that the force is toward the Earth, i.e., in the positive direction. In terms of the relative position vector r, the above equations can be combined to yield 2 ⨀ + ⨁ =− (1.3) 2 2 The solution of this equation is an ellipse with the centre of mass at one focus. With a semi major axis for the ellipse. 5.1 Orbital Motion of the Earth The period of orbital motion is given by 2 4 2 3 = (1.4) ⨀ + ⨁ This leads to an expression for the mass of the Sun: ⨀ 4 2 = (1.5) 3 2 1+ ⨁ ⨀ The ratio of ⨁ to ⨀ is very small (we will evaluate it shortly). If we were to neglect the ratio ⨁ ⨀ compared to unity, then ⨀ we would get a fairly precise first approximation to. 3 According to this approximation, for each planet in the solar system, the square of the period 2 is proportional to the cube of the mean distance 3. 5.1 Orbital Motion of the Earth This property was first identified empirically by Kepler as his third law of planetary motion. However, with or without the correction for the Earth’s mass, we cannot determine the value of ⨀ unless we first determine the value of D (= 1 astronomical unit).  5.2 Astronomical Unit (AU) Once the orbital periods of planets are known, the application of Kepler’s third law provides a scale model of the solar system. The scale model provides knowledge, at any given instant of time, of the distances of planets and other solar system objects in terms of AU, the semi major axis of the Earth’s orbit. Also, at any given instant of time, we can know distance between any solar system object and the Earth in terms of AU. In favourable conditions, radar reflection can be used to determine the linear distance to the object at that instant.  5.2 Astronomical Unit (AU) Radar reflection measurements were first made 1960 using Venus. Not only the intensity of the signal which was measured, but also the Doppler shift. When Venus is closest to Earth, the round-trip time for radar reflections is close to 5 minutes, and this interval can be measured with a precision of many significant figures. The International Astronomical Union currently defines the AU as follows: 1 ≡ = 149,597,870.691 We note that at a distance equal to D, the linear diameter of any object which has an angular diameter of 1 arc sec is 728.8 km.  5.2 Astronomical Unit (AU) By inserting the value of D in Equation 1.5, a first approximation to the quantity ⨀ can be determined. Kepler’s 3rd law becomes an equality: 2 = 3 provided that is expressed in years and is expressed in AU. To obtain a more precise estimate of ⨀ , we need to evaluate the ratio ⨁ / ⨀. We do that by comparing the motions of two objects, one in orbit around the Sun, the other in orbit around the Earth. For both objects, we need to determine two quantities: a period and a distance.  5.2 Astronomical Unit (AU) For the object (the Earth) that is in orbit around the Sun, with period ( ) and semi-major axis ( ), we know that ( )3 ( )2 ∼ (1.6) ⨀ + ⨁ For the object (an artificial satellite) that is in orbit around the Earth, with period ( ) and semi-major axis ( ), we have that ( )3 ( )2 ∼ (1.6) ⨁ where we have made the reasonable assumption that the mass of the artificial satellite is entirely negligible (by 20 orders of magnitude or more) compared to the mass of the Earth. Combining the above equations, we have that ⨀ ( ) 3 ( ) 2 +1= (1.8) ⨁ ( ) ( )  5.2 Astronomical Unit (AU) There are a large number or choices which we can make for an artificial satellite in orbit around the Earth. Visit the website http://www.heavens-above.com/, where information is available about many satellites in orbit around the Earth. For example, the satellite RADCAT has: –the perigee lies at an altitude of 491 km above the Earth’s surface, –the apogee at 495 km. –The mean altitude above the Earth’s surface is h = 493 km. Note that only three significant digits are provided for these distances: this will limit the precision of our evaluation of ⨁ ⨀. With an mean altitude of ℎ = 493 , the semi-major axis of the orbit is ( ) = ⨁ + ℎ, where is the radius of the Earth.  5.2 Astronomical Unit (AU) The equatorial radius of the earth has been accurately measured, by the International Union of Geodesy and Geophysics, to be ⨁ = 6378.137 km. This leads to ( ) = 6871.137 for RADCAT. Comparing this with the value of ( ) = 1 , we see that ( ) ( ) = 21,771.924 for RADCAT. Thus, the first factor on the right-hand side of Equation 1.8 is 1.0320255 × 1013. Turning now to the period, information on heavens-above.com indicates that the RADCAT satellite orbits 15.24243084 times per day, corresponding to a period ( ) = 5668.387. Compared to the value of ( ) (= 1 ), we find that the second factor on the right-hand side of Equation 1.8 has the value 3.2262344 × 10−8.  5.2 Astronomical Unit (AU) Combining the terms in Equation 1.8, we find that ⨁ ⨀ = 1/332,955 This is the mass ratio which we obtain when we use the orbital data for a single satellite (RADCAT), for which we know the altitude to only three significant digits. When multiple satellites are used, the currently accepted value of ⨁ ⨀ is found to be 1/332,946. Thus, our use of RADCAT data alone leads to an error in the mass ratio of about 1 part in 30,000. Our calculations would have led to the currently accepted value of ⨁ ⨀ if we were to use a value of 493.08 km (rather than 493 km) for the mean altitude of RADCAT. I t s h o u l d b e k e p t i n m i n d t h a t e v e n s m a l l e r ro r s i n a measurement may translate to significant uncertainties in some of the quantities which are of interest to us when we study the  5.3 ⨀ and the Mass of the Sun The product of the gravitational constant and the mass of the Sun: ⨀ = 1.327124 × 1026 3 −2 (1.9) The precision of ⨀ has increased over the course of the space age, as more and more spacecraft have travelled throughout the solar system, always subject to a sunward acceleration which is proportional to the above product. Currently, the numerical value of ⨀ is known to 11−12 significant digits, but we do not need all those digits here, because is not that well known. To extract a value for the mass of the Sun, we need to divide the above product by.  5.3 ⨀ and the Mass of the Sun The numerical value of is among the most poorly measured constants of nature, known only to 1 part in 104 : = 6.67428(±0.00067) × 10−8 2 −2 (National Institute of Standards and Technology: http://physics.nist.gov/cuu/Constants/). Using this, we obtain the following estimate of the mass of the Sun, reliable to 1 part in 104 : ⨀ = 1.9884 × 1033 (1.10) 5.4 Power Output of the Sun: The Solar Luminosity Radiometers on spacecraft can measure the flux of radiant energy coming from the Sun: this flux, known as the solar irradiance, , From SOHO the mean value of is about 1366 −2 , i.e., 1.366 × 106 −2 −1 The magnitude of is observed to vary slightly in the course of a sunspot cycle (see Figure 1.1): the variations are at the 0.1% level, i.e., about one part in 1000. 5.4 Power Output of the Sun: The Solar Luminosity FIGURE 1.1: The solar irradiance, normalized to a solar distance of 1 AU, measured over almost 30 years. (Courtesy of SOHO/VIRGO consortium. SOHO is a project of international cooperation between ESA and NASA.) 5.4 Power Output of the Sun: The Solar Luminosity Given D, the mean transforms to an output power from the Sun of ⨀ = 4 2 , i.e., ⨀ = 3.8416 × 1033 −1 (1.11) This is also referred to as the “solar luminosity” varies by roughly ±1 part in 1000 on 10−12 year cycles. During each cycle, the surface of the Sun is occupied by a greater or smaller number of “sunspots.” Comparing Equations 1.11 and 1.10, we note that the ratio of ⨀ / ⨀ has a numerical value close to 2 −1 −1. Its useful to use this ratio when we calculate the internal structure of the radiative interior of the Sun. 5.4 Power Output of the Sun: The Solar Luminosity Also, we note that the power output from the Sun relies on the conversion of (nuclear) mass into energy in the deep inner core of the Sun. Using the conversion formula = 2 , we note that the value of the Sun’s power output requires the conversion of mass to energy at a rate ( ) = 4.274 × 1012 −1 In the course of the Sun’s lifetime, which is estimated to be about 4.6 , the mass of the Sun has been reduced by nuclear processing by a few parts in 104. 5.5 Radius of the Sun: ⨀ Now, the mean distance to the Sun is known, We can obtain the linear radius (diameter) of the Sun by measuring the angular radius (diameter). However, measuring the angular radius of the Sun from the ground is difficult to do with precision on account of the phenomenon of “seeing.” Turbulent eddies in the Earth’s atmosphere make the image of the Sun unsteady, smearing out the edge of the solar disc on angular scales of order 1 arc sec. This leads to an uncertainty in the solar radius of order 700 km derived from ground-based measurements. 5.5 Radius of the Sun: ⨀ Empirically, the existence of two distinct classes of eclipses of the Sun (total and annular) indicates that the Sun has an angular diameter which is comparable to the Moon’s: the latter is close to 32 arc min, i.e., 1920 arc sec. So the Sun’s angular radius is of order 960 arc sec. But from the ground, this cannot be measured to better than about 1 arc sec, i.e., to one part in 1000. In order to obtain more reliable measurements, observations from space are required. The SOHO spacecraft, launched in 1996, has made the most careful measurements in this regard. 5.5 Radius of the Sun: ⨀ In order to provide the best possible calibration of the CCD pixels in the SOHO/MDI detector, a transit of Mercury was observed on May 7 2003. In the course of a 5–6 hour period, Mercury moved along a track which was known from planetary dynamics to a precision of ±0.025 arc sec. Kuhn et al. (2004) report that the angular radius of the Sun (when observed at D =1 AU) is 959.28 ± 0.15 arc sec. When this is converted to linear measure, it corresponds to a (linear) solar radius of ⨀ = 6.9574(±0.0011) × 105 5.5 Radius of the Sun: ⨀ An alternate method of determining the (linear) radius of Sun is provided by helioseismic data. When we study oscillatory modes in the Sun, we shall see that the periods of various modes can be measured to better than 1 part in 30,000. The best determination of solar radius from helioseismic data is ⨀ = 6.9568 ± 0.0003 × 105 (1.12) This result overlaps with the above estimate of angular diameter from SOHO, but is about three times more precise. 5.5 Radius of the Sun: ⨀ The improvement in precision can be attributed to the fact that the analysis depends on measurements of time (frequency) rather than angle. For future reference, when we come to discuss the solar wind, it will be helpful to know how far the Earth is from the Sun in units of the solar radius. Combining with ⨀ , we see that 1 AU is equivalent to 215.04 ⨀. 5.6 Surface Gravity of the Sun Now that we know the mass and radius of the Sun, we can calculate the acceleration due to gravity at the solar surface. ⨀ 1.327124 ×1026 = = = 27,421.6 −2 (1.13) 2⨀ (6.9568 × 1010 )2 For future reference, we note that a convenient way to remember this value is to recall the logarithmic value: log = 4.44 5.7 Escape Speed from the Solar Surface The escape speed from the surface of the Sun is given by 2 ⨀ = = 617.7 −1 (1.14) ⨀ This escape speed is a measure of the depth of the gravitational potential well due to the mass of the material in the entire Sun. It is a measure of how strongly the Sun’s weight crushes the gas in the core of the Sun. It is a law of physics that, if the Sun is to remain in hydrostatic equilibrium, the crushing effects of the weight of the overlying material on the core have to be balanced by outward-directed pressure. 5.7 Escape Speed from the Solar Surface Now, pressure is determined by the momentum flux of the individual gas particles. As a result, the thermal pressure in the core is related to the mean square velocity of the thermal particles there. Thermal particles, each with mass , and in a medium with temperature , have a root-mean-square (rms) velocity = (3 ) where = 1.3806504 × 10−16 −1 is Boltzmann’s constant. 5.7 Escape Speed from the Solar Surface The existence of the two velocities, and , which are both characteristic of the Sun, suggests that in a model of the Sun which is in hydrostatic (i.e., mechanical) equilibrium, and should have comparable magnitudes. We shall check on this expectation when we complete our calculation of a mechanical model of the Sun. For future reference, we note that for a gas consisting of hydrogen atoms, 1 = 1 (where is the mass of a hydrogen atom), and this equals Avogadro’s number , which is the number of molecules in one mole. 5.7 Escape Speed from the Solar Surface The combination is referred to as the gas constant = 8.314472 × 107 −1 −1 For a gas consisting of particles with atomic mass , the rms velocity = 3 5.8 Effective Temperature of the Sun Now that we know the output power of the Sun as well as the radius, we can calculate the effective temperature. This is the temperature of the equivalent black-body which would radiate a flux equal to that emitted by the Sun: ⨀ = 4 2⨀ 4 where the Stefan–Boltzman coefficient = 5.67040 × 10−5 −2 −1 −4 The surface flux of energy at the Sun, ⨀ = ⨀ 4 2⨀ has the numerical value 6.3155 × 1010 −2 −1. Using this, we find that the effective temperature of the Sun is = 5777 (1.15) 5.9 Shape of the Sun To the unaided eye, the Sun appears to be essentially circular in shape. But careful measurements reveal a slight departure from circularity. The difference between the solar radius at the equator and the solar radius at the pole is expressed in terms of the oblateness = ( − )/ First attempts to measure were made using ground-based observations. But the effects of seeing make this very difficult to do. Early results in the 1960s claimed that the Sun was oblate with = 4.2 × 10−5 5.9 Shape of the Sun Such an oblateness would correspond to a linear difference of 30 between the equatorial radius and the polar radius. It would mean that the difference in angular radii would be of order 0.04. This is much smaller than the effects of seeing (typical amplitude ≈ 1 ), and so it is not surprising that it is difficult to make the measurements reliably from the ground. A balloon-borne instrument, the Solar Disk Sextant (SDS), flown in 1992 and 1994, made observations at altitudes which were above most of the atmosphere. 5.9 Shape of the Sun The reported oblateness was = 9 ± 1 × 10−6 considerably smaller than had been suggested by the earlier ground based data. Measurements from space were made by SOHO: the spacecraft was rolled through 360 degrees in small angular increments, each 0.7 degrees in extent, corresponding to 360/0.7 = 514 individual “pie slices” of data around the entire circumference. Each “pie slice” was fitted with a radial profile: taking a numerical radial derivative of each profile, and squaring the derivative, the location of the peak of squared derivative was defined to be the location of the limb. 5.9 Shape of the Sun With more than 500 samples, the rules of statistics suggest that the noise in individual “pie slices” can be reduced from 0.15 arc sec to 0.15 514 = 0.007 = 5. By making multiple observations over several months, the authors claimed that they could achieve a precision of 0.5 km in the solar radius (Kuhn et al., 1998). Observations obtained in 1996–1997 indicated a solar oblateness of = (7.77 ± 0.66) × 10−6 (1.16) This oblateness overlaps with the 1992/1994 results from SDS, although with somewhat improved error bars. 5.9 Shape of the Sun The existence of a finite oblateness in the solar figure is expected because the material in the Sun is subject to forces arising from rotation. If rotation were absent, the Sun’s figure would settle into an equi- potential surface, for which the potential would be spherically symmetric: =− ⨀ / With such a potential, the surface acceleration due to gravity = − / is also symmetric. 5.9 Shape of the Sun In the presence of rotation, however, the (inward) force due to gravity is counter-acted to some extent by the (outward) centrifugal force. With a solar angular velocity , the net gravitational acceleration at colatitude becomes ( ) = − 2 2 (1.17) corresponding to a potential =− 0.5 2 2 2 (1.18) This leads to an equi-potential surface which, in the presence of an equatorial rotational velocity ( ) = , has an oblateness of = 0.5 ( )2 5.9 Shape of the Sun What is the rotational velocity of the Sun? We can answer this question as regards the surface of the Sun by means of direct observations. Rotational periods of material on the surface of the Sun can be measured from the Doppler shifts of spectral lines at east and west limbs. An important finding is that the rotational period is not constant at all latitudes. Instead, the period is found to be shortest at the equator, and the period becomes longer as we observe closer to the poles. This behaviour is called “latitudinal differential rotation.” 5.9 Shape of the Sun An empirical fit to the rotation can be achieved by the following expression for angular velocity as a function of latitude : ( ) = (0) 1 − 2 − 4 (1.19) In a study involving Doppler shift data from many points on the surface, obtained in the course of 14 years, Howard et al. (1983) reported average values for the parameters in this fit: (0) = 2.867 × 10−6 −1 , = 0.121, and = 0.166. At the equator, the measured angular velocity (0) corresponds to a rotational period of ( , ) = 2 (0) = 25.4 days. The equatorial rotational velocity ( ) = (0) ⨀ has a numerical value of 1.99 −1. 5.9 Shape of the Sun At latitudes of 60∘ , the rotational period ( , 60) = 31.3 At the north and south poles, Equation 1.19 indicates that (90) = 0.713 (0) = 2.044 × 10−6 −1 , corresponding to a polar rotational period ( , ) = 2 / (90) = 35.6 Remarkably, the gas in the polar regions of the Sun rotates almost 30% more slowly than the gas near the equator. If we needed any reminder that the Sun is not a solid body (but is composed entirely of gas), differential rotation would provide the evidence. 5.9 Shape of the Sun If the entire Sun were to rotate at a period of 25.4 days, then the oblateness due to rotation would have the numerical value ( ) = 10.4 × 10−6 This is several standard deviations larger than the oblateness reported by SOHO. It seems that the entire Sun cannot be rotating with a period that is as short as 25.4 days: some regions must be rotating more slowly than that. The observed oblateness values ( = (8–9) × 10−6 ) would be consistent with rotational effects if the entire Sun were to rotate with a period which is longer than 25.4 days by a factor of (10.4/(8–9)) = 1.07 − 1.14. 5.9 Shape of the Sun The observed oblateness could be due entirely to rotation if the Sun were to rotate as a solid body with angular velocity ( ) = (2.5–2.7) × 10−6 −1 Converting the angular velocities to (temporal) frequency, = 2 , we note that the equatorial rotation (0) corresponds to (0) = 456 ( ), while ( ) corresponds to ( ) = 398–430. As it turns out, the analysis of helioseismological data has revealed that the inner regions of the Sun do not rotate as a solid body. Different regions in the Sun rotate with different periods, depending on latitude and radial location. 5.9 Shape of the Sun The fastest rotation, at equatorial latitudes, and at radial locations close to the surface, is about = 470 , while the slowest (at polar latitudes, and also close to the surface) is about = 320. Thus, the material inside the Sun spans a rather broad range of rotational frequencies: 320–470. The rotational frequencies which are derived when the entire observed oblateness is attributed to rotational effects viz. ( ) = 398–430 , are entirely consistent with the range of rotational frequencies which exist inside the Sun. It appears therefore that most (or all) of the observed oblateness of the Sun can be ascribed without serious contradiction to rotational effects. 5.10 Critical Frequency for Solar Oscillations Now that we know the radius and mass of the Sun, there is a critical frequency which can be constructed from ⨀ , ⨀ , and G which will be relevant when we come to discuss the various modes of oscillations inside the Sun. By analogy with a pendulum, for which the period is given by = 2 √( ) if the length of the pendulum is and the local acceleration due to gravity is , a critical period in the gravity field of the Sun in a global sense can be written down by considering a pendulum with a length that is equal to the natural length of the system: the solar radius. 5.10 Critical Frequency for Solar Oscillations This leads to = 2 ( ). This can be written as 3⨀ = 2 (1.20) ⨀ Substituting ⨀ = 1.327124 × 1026... and ⨀ = 6.9568 × 1010 , we find = 10,008 The associated frequency = 1 has the numerical value 99.92 ( ). Thus, a fundamental frequency which is very close to 100 is expected to provide a significant marker among the oscillation frequencies of the modes in which the Sun oscillates on a global scale. 5.11 Mean Density of the Sun Another quantity which can be calculated once the mass and radius of the Sun are known is the mean density: ⨀ = (1.21) (4 3) 3⨀ Inserting the values of ⨀ and ⨀ from Equations 1.10 and 1.12, we find = 1.410 −3. That is, the mean density of the (gaseous) Sun is somewhat greater than the mean density of (liquid) water. Once we calculate a model for the interior of the Sun, it will be a matter of interest to compare the density at the center of the Sun to the mean density. 5.11 Mean Density of the Sun We shall find that the central density in the Sun is much larger than the density of liquid water: the central density is actually about ten times larger than that of solid lead. Despite these large densities, the material of which the Sun is composed does not behave as a liquid or a solid: instead, we shall find that it obeys the laws which govern the behaviour of a gas. We note that the critical period in Equation 1.20 scales as 1. Now that we have information on the relevant physical parameters on a global scale, we can turn to a study of the internal structure of the Sun.