Exercises

Exercise 14.1 The mass of a pulsar is 1.5 Me, radius 10 km, and rotation period 0.033 s. What is the angular momentum of the pulsar Variations of 0.0003 s are observed in the period. If they are due to radial oscillations (starquakes), how large are these oscillations Exercise 14.2 In Dragon's Egg by Robert L. Forward a spaceship orbits a neutron star at a distance of 406 km from the centre of the star. The orbital period is the same as the rotation period of the star, 0.1993 s. a) Find the...

The Earth and the Moon

Moon Temperature With Depth

The third planet from the Sun, the Earth, and its satellite, the Moon, form almost a double planet. The relative size of the Moon is larger than that of any other satellite, excluding the moon of Pluto. Usually satellites are much smaller than their parent planets. The Earth is a unique body, since a considerable amount of free water is found on its surface. This is possible only because the temperature is above the freezing point and below the boiling point of water and the atmosphere is thick...

The Hertzsprung Russell Diagram

Around 1910, Ejnar Hertzsprung and Henry Norris Russell studied the relation between the absolute magnitudes and the spectral types of stars. The diagram showing these two variables is now known as the Hertzsprung-Russell diagram or simply the HR diagram (Fig. 8.8). It has turned out to be an important aid in studies of stellar evolution. In view of the fact that stellar radii, luminosities and surface temperatures vary widely, one might have expected the stars to be uniformly distributed in...

The Hot Corona of the Milky

As early as 1956 Lyman Spitzer showed that the Milky Way has to be surrounded by a large envelope of very hot gas (Fig. 15.26). Almost two decades later the Copernicus satellite, whose scientific program was directed by Spitzer, found evidence for this kind of gas, which began to be called galactic coronal gas, in analogy with the solar corona. The satellite observed emission lines of e. g. five times ionized oxygen (O VI), four times ionized nitrogen (NV) and triply ionized carbon (CIV). The...

Temperatures

Temperatures of astronomical objects range from almost absolute zero to millions of degrees. Temperature can be defined in a variety of ways, and its numerical value depends on the specific definition used. All these different temperatures are needed to describe different where a 2R r is the observed angular diameter of the star. For direct determination of the effective temperature, we have to measure the total flux density and the angular diameter of the star. This is possible only in the few...

And the Interstellar Magnetic Field

Elementary particles and atomic nuclei reaching the Earth from space are called cosmic rays. They occur throughout interstellar space with an energy density of the same order of magnitude as that of the radiation from stars. Cosmic rays are therefore important for the ionization and heating of interstellar gas. Since cosmic rays are charged, their direction of propagation in space is constantly changed by the magnetic field. Their direction of arrival therefore gives no information...

Solution of the Equation of Motion

The equation of motion (6.5) is a second-order (i. e. contains second derivatives) vector valued differential equation. Therefore we need six integration constants or integrals for the complete solution. The solution is an infinite family of orbits with different sizes, shapes and orientations. A particular solution (e. g. the orbit of Jupiter) is selected by fixing the values of the six integrals. The fate of a planet is unambiguously determined by its position and velocity at any given moment...

Keplers Second and Third

The radius vector of a planet in polar coordinates is simply Fig. 6.6. Unit vectors er and e f of the polar coordinate frame. The directions of these change while the planet moves along its orbit By comparing this with the length of k (6.23), we find that and integrate over one complete period Fig. 6.7. The areas of the shaded sectors of the ellipse are equal. According to Kepler's second law, it takes equal times to travel distances AB, CD and EF where a and b are the semimajor and semiminor...

Saturn

Its diameter is about 120, 000 km, ten times the diameter of the Earth, and the mass, 95 Earth masses. The density is only 700 kg m-3, less than the density of water. The rotation axis is tilted about 27 with respect to the orbital plane, so every 15 years, the northern or the southern pole is well observable. Fig. 7.43. Saturn and its rings. Three satellites (Tethys, Dione, and Rhea) are seen to the left of Saturn, and the shadows of Mimas and Tethys are...

Positional Astronomy

The position of a star can be measured either with respect to some reference stars (relative astrometry) or with respect to a fixed coordinate frame (absolute astrometry). Fig. 2.22. Astronomers discussing observations with the transit circle of Helsinki Observatory in 1904 Fig. 2.22. Astronomers discussing observations with the transit circle of Helsinki Observatory in 1904 Absolute coordinates are usually determined using a meridian circle, which is a telescope that can be turned only in the...

The Horizontal System

The most natural coordinate frame from the observer's point of view is the horizontal frame (Fig. 2.9). Its reference plane is the tangent plane of the Earth passing through the observer this horizontal plane intersects the celestial sphere along the horizon. The point just above the observer is called the zenith and the antipodal point below the observer is the nadir. (These two points are the poles corresponding to the horizon.) Great circles through the zenith are called verticals. All...

Stellar Energy Sources

When the equations of stellar structure were derived, the character of the source of stellar energy was left unspecified. Knowing a typical stellar luminosity, one can calculate how long different energy sources would last. For instance, normal chemical burning could produce energy for only a few thousand years. The energy released by the contraction of a star would last slightly longer, but after a few million years this energy source would also run out. Terrestrial biological and geological...

The Classification of Galaxies

A useful first step towards an understanding of galaxies is a classification based on their various forms. Although such a morphological classification must always be to some extent subjective, it provides a framework within which the quantitative properties of galaxies can be discussed in a systematic fashion. However, it should always be remembered that the picture thus obtained will be limited to those galaxies that are large and bright enough to be easily visible in the sky. An idea of the...

The Origin of the Elements

There are just under a hundred naturally occurring elements, and about 300 isotopes in the solar system (Fig. 11.13). In Sect. 11.4, we have seen how the elements up to iron are produced when hydrogen burns to helium and helium further to carbon, oxygen and heavier elements. Almost all nuclei heavier than helium were produced in nuclear reactions in stellar interiors. In the oldest stars, the mass fraction of heavy elements is only about 0.02 , whereas in the youngest stars it is a few per...

Position in the Orbit

Although we already know everything about the geometry of the orbit, we still cannot find the planet at a given time, since we do not know the radius vector r as a function of time. The variable in the equation of the orbit is an angle, the true anomaly f, measured from the perihelion. From Kepler's second law it follows that f cannot increase at a constant rate with time. Therefore we need some preparations before we can find the radius vector at a given instant. The radius vector can be...

The Evolution of Close Binary Stars

If the components of a binary star are well separated, they do not significantly perturb one another. When studying their evolution, one can regard them as two single stars evolving independently, as described above. Fig. 11.9a-c. The types of close binary systems (a) detached, Fig. 11.9a-c. The types of close binary systems (a) detached, However, in close binary pairs, this will no longer be the case. Close binary stars are divided into three classes, as shown in Fig. 11.9 detached,...

Pulsating Variables

The wavelengths of the spectral lines of the pulsating variables change along with the brightness variations (Table 13.1). These changes are due to the Doppler effect, showing that the outer layers of the star are indeed Table 13.1. The main properties of pulsating variables (N, number of stars of the given type in Kukarkin's catalogue, P, pulsation period in days, Am, pulsation amplitude in magnitudes) Table 13.1. The main properties of pulsating variables (N, number of stars of the given type...

Structural Components of the Milky

We have seen how the structure of the Milky Way can be globally described by means of an almost spherical halo of old stars and and a disc of gas and young and middle-aged stars. In a more detailed picture more small-scale features can also be distinguished. The Thick Disc. In the traditional scheme dividing the stars of the Milky Way into a series of populations it was left undecided whether the populations should be considered as qualitatively different classes or merely steps along a...

Interstellar Dust

The first clear evidence for the existence of interstellar dust was obtained around 1930. Before that, it had been generally thought that space is completely transparent and that light can propagate indefinitely without extinction. In 1930 Robert Trumpler published his study of the space distribution of the open clusters. The absolute magnitudes M of the brightest stars could be estimated on the basis of the spectral type. Thus the distance r to the clusters could be calculated from the...

Star Catalogues and Maps

The first actual star catalogue was published by Ptolemy in the second century this catalogue appeared in the book to be known later as Almagest (which is a Latin corruption of the name of the Arabic translation, Al-mijisti). It had 1025 entries the positions of these bright stars had been measured by Hipparchos 250 years earlier. Ptolemy's catalogue was the only widely used one prior to the 17th century. The first catalogues still being used by astronomers were prepared under the direction of...

Globular Star Clusters

Globular star clusters usually contain about 105 stars. The distribution of the stars is spherically symmetric, and the central densities are about ten times larger than in open clusters. Stars in globular clusters are among the oldest in the Milky Way, and therefore they are of great importance for studies of stellar evolution. There are about 150-200 globular clusters in the Milky Way. The colour-magnitude diagram of a typical globular cluster is shown in Fig. 16.7. The main sequence only...

The Celestial Sphere

The ancient universe was confined within a finite spherical shell. The stars were fixed to this shell and thus were all equidistant from the Earth, which was at the centre of the spherical universe. This simple model is still in many ways as useful as it was in antiquity it helps us to easily understand the diurnal and annual motions of stars, and, more important, to predict these motions in a relatively simple way. Therefore we will assume for the time being that all the stars are located on...

Solar Activity

The clearest visible sign of solar activity are the sunspots. The existence of sunspots has been known for long (Fig. 12.9), since the largest ones can be seen with the naked eye by looking at the Sun through a suitably dense layer of fog. More precise observations became available beginning in the 17th century, when Galilei started to use the telescope for astronomical observations. A sunspot looks like a ragged hole in the solar surface. In the interior of the spot there is a dark...

Interstellar

The mass of gas in interstellar space is a hundred times larger than that of dust. Although there is more gas, it is less easily observed, since the gas does not cause a general extinction of light. In the optical region it can only be observed on the basis of a small number of spectral lines. The existence of interstellar gas began to be suspected in the first decade of the 20th century, when in 1904 Johannes Hartmann observed that some absorption lines in the spectra of certain binary stars...

Spherical Trigonometry

Spherical Trigonometry

For the coordinate transformations of spherical astronomy, we need some mathematical tools, which we present now. If a plane passes through the centre of a sphere, it will split the sphere into two identical hemispheres along a circle called a great circle (Fig. 2.1). A line perpendicular to the plane and passing through the centre of the sphere intersects the sphere at the poles P and P'. If a sphere is intersected by a plane not containing the centre, the intersection curve is a small circle....

Origin of the Solar System

Cosmogony is a branch of astronomy which studies the origin of the solar system. The first steps of the planetary formation processes are closely connected to star formation. The solar system has some very distinct features which have to be explained by any serious cosmogonical theory. These include - planetary orbits are almost coplanar and also parallel to the solar equator - planets orbit the Sun counterclockwise, which is also the direction of solar rotation - planets also rotate around...

Constellations

In astronomy the Doppler effect can be seen in stellar spectra, in which the spectral lines are often displaced towards the blue (shorter wavelengths) or red (longer wavelengths) end of the spectrum. A blueshift means that the star is approaching, while a redshift indicates that it is receding. The displacements due to the Doppler effect are usually very small. In order to measure them, a reference spectrum is exposed on the plate next to the stellar spectrum. Now that CCD-cameras have replaced...

The Rotation of the Milky

Oort's Formulas. The flatness of the Milky Way is already suggestive of a general rotation about an axis normal to the galactic plane. Observations of the motions both of stars and of interstellar gas have confirmed this rotation and shown it to be differential. This means that the angular velocity of rotation depends on the distance from the galactic centre (Fig. 17.13). Thus the Milky Way does not rotate like a rigid body. Near the Sun, the rotational velocity decreases...

Quantum Numbers Selection Rules Population Numbers

The Bohr model needs only one quantum number, n, to describe all the energy levels of the electron. This can explain only the coarse features of an atom with a single electron. Quantum mechanics describes the electron as a three dimensional wave, which only gives the probability of finding the electron in a certain place. Quantum mechanics has accurately predicted all the energy levels of hydrogen atoms. The energy levels of heavier atoms and molecules can also be computed...

Radio Telescopes

Radio astronomy represents a relatively new branch of astronomy. It covers a frequency range from a few megahertz (100 m) up to frequencies of about 300 GHz (1 mm), thereby extending the observable electromagnetic spectrum by many orders of magnitude. The low-frequency limit of the radio band is determined by the opacity of the ionosphere, while the high-frequency limit is due to the strong absorption from oxygen and water bands in the lower atmosphere. Neither of these limits is very strict,...

Interstellar Molecules

The first interstellar molecules were discovered in 1937-1938, when molecular absorption lines were found in the spectra of some stars. Three simple diatomic molecules were detected methylidyne CH, its positive ion CH+ and cyanogen CN. A few other molecules were later discovered by the same method in the ultraviolet. Thus molecular hydrogen H2 was discovered in the early 1970's, and carbon monoxide, which had been discovered by radio observations, was also detected in the ultraviolet. Molecular...

Other Wavelength Regions

All wavelengths of the electromagnetic spectrum enter the Earth from the sky. However, as mentioned in Sect. 3.1, not all radiation reaches the ground. The wavelength regions absorbed by the atmosphere have been studied more extensively since the 1970's, using Earth-orbiting satellites. Besides the optical and radio regions, there are only some narrow wavelength ranges in the infrared that can be observed from high mountain tops. The first observations in each new wavelength region were usually...

Mercury

Mercury is the innermost planet of the solar system. Its diameter is 4800 km and its mean distance from the Sun 0.39 AU. The eccentricity of the orbit is 0.21, which means that the distance varies between 0.31 and 0.47 AU. Because of the high eccentricity, the surface temperature of the subsolar point varies substantially at the perihelion, the temperature is about 700 K at the aphelion, it is 100 K lower. Temperature variations on Mercury are the most extreme in the solar system Table 7.3....

Astrometric Binary Stars

In astrometric binaries, only the orbit of the brighter component about the centre of mass can be observed. If the mass of the visible component is estimated, e. g. from its luminosity, the mass of the invisible companion can also be estimated. The first astrometric binary was Sirius, which in the 1830's was observed to have an undulating proper motion. It was concluded that it had a small companion, which was visually discovered a few decades later (Figs. 9.4 and 14.1). The companion, Sirius...

Minor Bodies of the Solar System

So far we have considered only planets and planetary satellites. There is a great number of other bodies in the solar system, like dwarfplanets, asteroids, comets, meteoroids and interplanetary dust. However, there are no distinct borders between different types of objects. Some asteroids have similar features or origin as the comets, and some near-Earth asteroids are possibly cometary remnants where all volatile elements have disappeared. Thus our classification has been based more on the...

Venus

Venus is the brightest object in the sky, after the Sun and the Moon. Like Mercury, Venus can be seen only in the morning or in the evening sky. (It is sometimes possible to see Venus even if the Sun is above the horizon, if its exact position is known.) In antiquity, Venus was thought to be two different planets, Hesperos and Phosphorus, evening star and morning star. The maximum elongation of Venus is about 47 . Venus is a remarkable object when shining in the dark sky at its brightest, 35...

Planetary Configurations

The apparent motions of the planets are quite complicated, partly because they reflect the motion of the Earth around the Sun (Fig. 7.3). Normally the planets move eastward (direct motion, counterclockwise as seen from the Northern hemisphere) when compared with the stars. Sometimes the motion reverses to the opposite or retrograde direction. After a few weeks of retrograde motion, the direction is changed again, and the planet continues in the original direction. It is quite understandable...

Julian Date

There are several methods for finding the Julian date. The following one, developed by Fliegel and Van Flandern in 1968, is well adapted for computer programs. Let y be the year (with all four digits), m the month and d the day. The Julian date J at noon is then - (3 y + (m - 9) 7 100 +1 ) 4 + 275m 9 + d +1721029 . The division here means an integer division, the decimal part being truncated e. g. 7 3 2 and -7 3 -2. Example. Find the Julian date on January 1, 1990. Now y 1990, m 1 and d 1. J...

Astronomical Time Systems

Time can be defined using several different phenomena 1. The solar and sidereal times are based on the rotation of the Earth. 2. The standard unit of time in the current SI system, the second, is based on quantum mechanical atom-ary phenomena. 3. Equations of physics like the ones describing the motions of celestial bodies involve a time variable Fig. 2.31. The time zones. The map gives the difference of the local zonal time from the Greenwich mean time (UT). During daylight saving time, one...

Intensity Flux Density and Luminosity

Luminous Flux Physics

Let us assume we have some radiation passing through a surface element dA (Fig. 4.1). Some of the radiation will leave d A within a solid angle dw the angle between dw and the normal to the surface is denoted by 0. The amount of energy with frequency in the range v, v + d v entering this solid angle in time dt is d Ev Iv cos 0 d A dv dw dt. (4.1) Here, the coefficient Iv is the specific intensity of the radiation at the frequency v in the direction of the solid angle dw. Its dimension is Wm-2...

Model Atmospheres

The stellar atmosphere consists of those layers of the star where the radiation that is transmitted directly to the observer originates. Thus in order to interpret stellar spectra, one needs to be able to compute the structure of the atmosphere and the emerging radiation. In actual stars there are many factors, such as rotation and magnetic fields, that complicate the problem of computing the structure of the atmosphere. We shall only consider the classical problem of finding the structure, i....

The Harvard Spectral Classification

Harvard Classification Spectral Type

The spectral classification scheme in present use was developed at Harvard Observatory in the United States in the early 20th century. The work was begun by Henry Draper who in 1872 took the first photograph of the spectrum of Vega. Later Draper's widow donated the observing equipment and a sum of money to Harvard Observatory to continue the work of classification. The main part of the classification was done by Annie Jump Cannon using objective prism spectra. The Henry Draper Catalogue (HD)...

Examples

Example 2.1 Trigonometric Functions in a Rectangular Spherical Triangle Let the angle A be a right angle. When the figure is a plane triangle, the trigonometric functions of the angle B would be sin B b a , cos B c a , tan B b c . For the spherical triangle we have to use the equations in (2.7), which are now simply cos b sin c , cos a cos b cos c . The first equation gives the sine of B Dividing the second equation by the third one, we get the cosine of B And the tangent is obtained by...

Photometry Polarimetry and Spectroscopy

Asteroid Phase Curve

Having defined the phase function and albedos we are ready to derive a formula for planetary magnitudes. The flux density of the reflected light is At the phase angle zero 0(a 0 ) 1 and the reflected flux density is If we replace the object with a Lambertian surface of the same size, we get The ratio of these flux densities is and the constant factor expressed in terms of the geometric albedo The observed solar flux density at a distance of a 1 AU from the Sun is L0 Now we have found a physical...

Astronomical Objects of Research

Modern astronomy explores the whole Universe and its different forms of matter and energy. Astronomers study the contents of the Universe from the level of elementary particles and molecules (with masses of 10-30 kg) to the largest superclusters of galaxies (with masses of 1050 kg). Astronomy can be divided into different branches in several ways. The division can be made according to either the methods or the objects of research. The Earth (Fig. 1.3) is of interest to astronomy for many...

Y n T T

Equivalent Width

It can be shown that the corresponding line profile is 2n (v - vo)2 + y2 4 ' where v0 is the frequency at the centre of the line and I0 the total intensity of the line. At the centre of the line the intensity per frequency unit is 2 Thus the width Y is the width of the line profile at a depth where the intensity is half of the maximum. This is called the full width at half maximum (FWHM). Doppler Broadening. Atoms of a gas are moving the faster the higher the temperature of the gas. Thus...

Black Holes

Ergosphere Radius

If the mass of a star exceeds MOV, and if it does not lose mass during its evolution it can no longer reach any stable final state. The force of gravity will dominate over all other forces, and the star will collapse to a black hole. A black hole is black because not even light can escape from it. Already at the end of the 18th century Laplace showed that a sufficiently massive body would prevent the escape of light from its surface. According to classical mechanics, the escape velocity from a...

Perturbations of Coordinates

Even if a star remains fixed with respect to the Sun, its coordinates can change, due to several disturbing effects. Naturally its altitude and azimuth change constantly because of the rotation of the Earth, but even its right ascension and declination are not quite free from perturbations. Precession. Since most of the members of the solar system orbit close to the ecliptic, they tend to pull the equatorial bulge of the Earth towards it. Most of this flattening torque is caused by the Moon and...

Methods of Distance Measurement

Measuring The Distance Stars

In order to study the structure of the Milky Way, one needs to know how various kinds of objects, such as stars, star clusters and interstellar matter, are distributed in space. The most important ways of measuring the distances will first be considered. Trigonometric Parallaxes. The method of trigonometric parallaxes is based on the apparent yearly back-and-forth movement of stars in the sky, caused by the orbital motion of the Earth. From Earth-based observations the trigonometric parallaxes...

Stellar Statistics

By systematically observing all stars in the solar neighbourhood, one can find the distribution of their absolute magnitudes. This is given by the luminosity function M), which gives the relative number of main sequence stars with absolute magnitudes in the range M 1 2, M +1 2 . No stars appear to be forming at present in the region of space where the luminosity function has been determined. The age of the Milky Way is 10-15 Ga, which means that all stars less...

The Solar System

The solar system consists of a central star, called the Sun, eight planets, several dwarf planets, dozens of moons or satellites, millions of asteroids and Trans-Neptunian Objects (TNOs), and myriads of comets and meteoroids. Borders between the categories are not clear. Discoveries of new Solar System bodies caused that in 2006 the International Astronomical Union (IAU) in its General Assembly defined three distinct categories to clarify the situation (1)Aplanet is a celestial bodythat (a) is...

And Visibility of the

The sidereal year is the real orbital period of the Earth around the Sun. After one sidereal year, the Sun is seen at the same position relative to the stars. The length of the sidereal year is 365.256363051 days of 86,400 SI seconds at the epoch J2000.0 2000 January 1 12 00 00 TT. We noted earlier that, owing to precession, the direction of the vernal equinox moves along the ecliptic at about 50 per year. This means that the Sun returns to the vernal equinox before one complete sidereal year...

The Orbit of the Moon

The Earth's satellite, the Moon, circles the Earth counterclockwise. One revolution, the sidereal month, takes about 27.322 days. In practise, a more important period is the synodic month, the duration of the Lunar phases (e. g. from full moon to full moon). In the course of one sidereal month the Earth has travelled almost 1 12 of its orbit around the Sun. The Moon still has about 1 12 of its orbit to go before the Earth-Moon-Sun configuration is again the same. This takes about 2 days, so the...

Jupiter

The realm of terrestrial planets ends at the asteroid belt. Outside this, the relative abundance of volatile elements is higher and the original composition of the solar nebula is still preserved in the giant planets. The first and largest is Jupiter. Its mass is 2.5 times the total mass of all other planets, almost 1 1000 of the solar mass. The bulk of Jupiter is mainly hydrogen and helium. The relative abundance of these elements are approximately the same as in the Sun, and the density is of...

Supernova Remnants

In Chap. 11 we have seen that massive stars end their evolution in a supernova explosion. The collapse of the stellar core leads to the violent ejection of the outer layers, which then remain as an expanding gas cloud. About 120 supernova remnants (SNR's) have been discovered in the Milky Way. Some of them are optically visible as a ring or an irregular nebula (e. g. the Crab nebula see Fig. 15.23), but most are detectable only Fig. 15.24. The radio spectra of typical HII regions and supernova...

A2 b2 c2 2bc cos A 22 The Earth

Longitude Latitude Formula

A position on the Earth is usually given by two spherical coordinates (although in some calculations rectangular or other coordinates may be more convenient). If neces- sary, also a third coordinate, e. g. the distance from the centre, can be used. The reference plane is the equatorial plane, perpendicular to the rotation axis and intersecting the surface of the Earth along the equator. Small circles parallel to the equator are called parallels of latitude. Semicircles from pole to pole are...

Classification

When a new variable is discovered, it is given a name according to the constellation in which it is located. The name of the first variable in a given constellation is R, followed by the name of the constellation (in the genitive case). The symbol for the second variable is S, and so on, to Z. After these, the two-letter symbols RR, RS, to ZZ are used, and then AA to QZ (omitting I). This is only enough for 334 variables, a number that has long been exceeded in most constellations. The...

Measuring Spectra

The most important methods of forming a spectrum are by means of an objective prism or a slit spectrograph. In the former case one obtains a photograph, where each stellar image has been spread into a spectrum. Up to several hundred spectra can be photographed on a single plate and used for spectral classification. The amount of detail that can be seen in a spectrum depends on its dispersion, the range of wavelengths per millimetre on the Fig. 8.1a-g. Typical stellar spectra. The spectrum of n...

Apparent Magnitudes

In time di, the radiation fills a volume d V c di dA, where d A is the surface element perpendicular to the propagation direction of the radiation Fig. 4.5. In time di, the radiation fills a volume d V c di dA, where d A is the surface element perpendicular to the propagation direction of the radiation As early as the second century B.C., Hipparchos divided the visible stars into six classes according to their apparent brightness. The first class contained the brightest stars and the...

The Yerkes Spectral Classification

The Harvard classification only takes into account the effect of the temperature on the spectrum. For a more precise classification, one also has to take into account the luminosity of the star, since two stars with the same effective temperature may have widely different luminosities. A two-dimensional system of spectral classification was introduced by William W. Morgan, Philip C. Keenan and Edith Kellman of Yerkes Observatory. This system is known as the MKK or Yerkes classification. (The MK...

Magnitude Systems

The apparent magnitude m, which we have just defined, depends on the instrument we use to measure it. The sensitivity of the detector is different at different wavelengths. Also, different instruments detect different wavelength ranges. Thus the flux measured by the instrument equals not the total flux, but only a fraction of it. Depending on the method of observation, we can define various magnitude systems. Different magnitudes have different zero points, i. e. they have different flux...

Radiative Transfer

Propagation of radiation in a medium, also called radiative transfer, is one of the basic problems of astrophysics. The subject is too complicated to be discussed here in any detail. The fundamental equation of radiative transfer is, however, easily derived. Assume we have a small cylinder, the bottom of which has an area d A and the length of which is dr. Let Iv be the intensity of radiation perpendicular to the bottom surface going into a solid angle dw ( Iv Wm 2 Hz-1 sterad-*). If the...

Internal Structure

The Sun is a typical main sequence star. Its principal properties are On the basis of these data, the solar model shown in Fig. 12.1 has been calculated. The energy is produced by the pp chain in a small central region. 99 of the solar energy is produced within a quarter of the solar radius. The Sun produces energy at the rate of 4 x 1026 W, which is equivalent to changing about four million tonnes of mass into energy every second. The mass of the Sun is so large, about 330,000 times that of...

Blackbody Radiation

A blackbody is defined as an object that does not reflect or scatter radiation shining upon it, but absorbs and re-emits the radiation completely. A blackbody is a kind of an ideal radiator, which cannot exist in the real world. Yet many objects behave very much as if they were blackbodies. The radiation of a blackbody depends only on its temperature, being perfectly independent of its shape, material and internal constitution. The wavelength distribution of the radiation follows Planck's law,...

Sidereal and Solar Time

Time measurements can be based on the rotation of the Earth, orbital motion around the Sun, or on atomic clocks. The last-mentioned will be discussed in the next section. Here we consider the sidereal and solar times related to the rotation of the Earth. We defined the sidereal time as the hour angle of the vernal equinox. A good basic unit is a sidereal day, which is the time between two successive upper culminations of the vernal equinox. After one sidereal day the celestial sphere with all...

Extinction and Optical Thickness

Equation (4.11) shows how the apparent magnitude increases (and brightness decreases ) with increasing distance. If the space between the radiation source and the observer is not completely empty, but contains some interstellar medium, (4.11) no longer holds, because part of the radiation is absorbed by the medium (and usually re-emitted at a different wavelength, which may be outside the band defining the magnitude), or scattered away from the line of sight. All these radiation losses are...

Uranus and Neptune

Uranus Bilder Voyager

The planets from Mercury to Saturn were already known in antiquity. Uranus and Neptune can only be observed with a telescope. Uranus and Neptune are giants, similar to Jupiter and Saturn. Uranus. The famous German-English amateur astronomer William Herschel discovered Uranus in 1781. Herschel himself first thought that the new object was a comet. However, the extremely slow motion revealed that the body was far beyond the orbit of Saturn. Based on the first observations, the Finnish astronomer...

The Jeans Limit

We shall later study the birth of stars and galaxies. The initial stage is, roughly speaking, a gas cloud that begins to collapse due to its own gravitation. If the mass of the cloud is high enough, its potential energy exceeds the kinetic energy and the cloud collapses. From the virial theorem we can deduce that the potential energy must be at least twice the kinetic energy. This provides a criterion for the critical mass necessary for the cloud of collapse. This criterion was first suggested...

Optical Telescopes

Yerkes Observatory Refractor Telescope

The telescope fulfills three major tasks in astronomical observations 1. It collects light from a large area, making it possible to study very faint sources. 2. It increases the apparent angular diameter of the object and thus improves resolution. 3. It is used to measure the positions of objects. The light-collecting surface in a telescope is either a lens or a mirror. Thus, optical telescopes are divided into two types, lens telescopes or refractors and mirror telescopes or reflectors (Fig....

The Contraction of Stars Towards the Main Sequence

Hayashi Track

The formation and subsequent gravitational collapse of condensations in the interstellar medium will be considered in a later chapter. Here we shall follow the behaviour of such a protostar, when it is already in the process of contraction. When a cloud contracts, gravitational potential energy is released and transformed into thermal energy of the gas and into radiation. Initially the radiation can propagate freely through the material, because the density is low and the opacity small....

Internal Equilibrium Conditions

Mass Loss Rate Sphere Density

Mathematically the conditions for the internal equilibrium of a star can be expressed as four differential equations governing the distribution of mass, gas pressure and energy production and transport in the star. These equations will now be derived. Hydrostatic Equilibrium. The force of gravity pulls the stellar material towards the centre. It is resisted by the pressure force due to the thermal motions of the gas molecules. The first equilibrium condition is that these forces be in...

Variable Stars

Stars with changing magnitudes are called variables (Fig. 13.1). Variations in the brightness of stars were first noted in Europe at the end of the 16th century, when Tycho Brahe's supernova lit up (1572) and the regular light variation of the star o Ceti (Mira) was observed (1596). The number of known variables has grown steadily as observational precision has improved (Fig. 13.2). The most recent catalogues contain about 40,000 stars known or suspected to be variable. Strictly speaking, all...

F

Introducing the observed I(r) into this expression, one obtains the actual luminosity distribution p(R). In the figure the solid curve shows the three-dimensional luminosity distribution obtained from the Vancouleurs' law (the dashed line). Since z2 R2 r2, a change of the variable of integration yields If the galaxy is not spherical, its three-dimensional shape can only be determined if its inclination with respect to the line of sight is known. Since galactic discs are thin and of constant...

Continuous Spectra

We have already mentioned some processes that produce continuous spectra. Continuous emission spectra can originate in recombinations and free-free transitions. In recombination, an atom captures a free Fig. 5.9. Spectrum of carbon monoxide CO from 430 nm to 670 nm. The various bands correspond to different vibrational transitions. Each band is composed of numerous rotational lines. Near the right edge of each band the lines are so closely packed that they overlap and at this resolution, the...

Reduction of Coordinates

Star catalogues give coordinates for some standard epoch. In the following we give the formulas needed to reduce the coordinates to a given date and time. The full reduction is rather laborious, but the following simplified version is sufficient for most practical purposes. We assume that the coordinates are given for the epoch J2000.0. 1. First correct the place for proper motion unless it is negligible. 2. Precess the coordinates to the time of the observation. First we use the coordinates of...

Mars

Mars is the outermost of the terrestrial planets. Its diameter is only half of that of the Earth. Seen through a telescope, Mars seems to be a reddish disk with dark spots and white polar caps. The polar caps wax and wane with the Martian seasons, indicating that they are composed of ice. Darker areas were suspected to be vegetation. At the end of the 19th century, an Italian astronomer, Giovanni Schiaparelli claimed that there are canals on Mars. In the United States, the famous planetary...

Preface to the First Edition

The main purpose of this book is to serve as a university textbook for a first course in astronomy. However, we believe that the audience will also include many serious amateurs, who often find the popular texts too trivial. The lack of a good handbook for amateurs has become a problem lately, as more and more people are buying personal computers and need exact, but comprehensible, mathematical formalism for their programs. The reader of this book is assumed to have only a standard high-school...

Assume All Stars Have Same Absolute Magnitudes

Exercise 4.1 The total magnitude of a triple star is 0.0. Two of its components have magnitudes 1.0 and 2.0. What is the magnitude of the third component Exercise 4.2 The absolute magnitude of a star in the Andromeda galaxy distance 690 kpc is M 5. It explodes as a supernova, becoming one billion 109 times brighter. What is its apparent magnitude Exercise 4.3 Assume that all stars have the same absolute magnitude and stars are evenly distributed in space. Let N m be the number of stars brighter...

The RoleofAstronomy

On a dark, cloudless night, at a distant location far away from the city lights, the starry sky can be seen in all its splendour Fig. 1.1 . It is easy to understand how these thousands of lights in the sky have affected people throughout the ages. After the Sun, necessary to all life, the Moon, governing the night sky and continuously changing its phases, is the most conspicuous object in the sky. The stars seem to stay fixed. Only some rela tively bright objects, the planets, move with respect...