## Info

Scalar and vector products satisfy the laws of distributivity Here the cross and dot can be interchanged and the factors permuted cyclically without affecting the value of the product. For example A x B C B x C A B C x A, but A x B C -B x A C. A vector triple product is a vector, which can be evaluated using one of the expansions A x (B x C) B(A C) - C(A B) , (A x B) x C B(A C) - A(B C) . In all these products, scalar factors can be moved around without affecting the product The position vector...

## Exercises

Exercise 9.1 The components of a binary move along circular orbits. The mutual distance is 1 AU, and the mass of each component is 1 Me. An observer in the plane of the orbit will see periodic splitting of the spectral lines. What is the maximum separation of the components of the HY line. Exercise 9.2 A planet (mass m) is orbiting a star (mass M) at a distance a. The distance of the star from the centre of gravity of the system is a'. Show that where P is period in years, distances are in AU's...

## Wi I

In the 19th century, n Carinae was one of the bright- has been called Homunculus. The Hubble photograph has est stars in the sky since then it has dimmed considerably. In been printed negative, to show finer details in the bipolar an outburst in 1843 the star ejected an expanding nebula, which outflow. (Photograph NASA HST University of Minnesota) Fig. 13.10. In the 19th century, n Carinae was one of the bright- has been called Homunculus. The Hubble photograph has est stars in the...

## Q

Surface features of Venus. (Top left) A Magellan image of a 50 km peak-ring crater Barton at 27.4 N and 337.5 E. (Top right) A Magellan radar image of a region 300 km across, located in a vast plain to the south of Aphrodite Terra. The large circular structure near the centre of the image is a corona, approximately 200 km in diameter. North of the corona is a 35 km flat-topped volcanic construct known as a pancake dome. Complex fracture patterns like in the upper right of the image...

## A

In case the required accuracy is higher than about one minute of arc, we have to do the following further corrections. 3. The full nutation correction is rather complicated. The nutation used in astronomical almanacs involves series expansions containing over a hundred terms. Very often, though, the following simple form is sufficient. We begin by finding the mean obliquity of the ecliptic at the observation time 0 23 26' 21.448 - 46.8150T - 0.00059 T2 + 0.001813 T The mean obliquity means that...

## Proton Flux Density In Astronomy

At the very beginning of the radiation era within a few hundred seconds helium was produced. Just before the epoch of helium synthesis, the number ratio of free protons and neutrons was changing because of the decay of the free neutrons. After about 100 s the temperature had dropped to about 109 K, which is low enough for deuterons to be formed. All remaining neutrons were then incorporated in deuterons these, in turn, were almost entirely consumed to produce helium...

## Stellar Evolution

N the preceding chapter we have seen how one can compute the evolution of a star by starting from a homogeneous model representing a newly formed system. When the chemical composition ofthe star changes with time, a new model is computed each time. In this chap ter we shall consider the theoretical evolutionary paths of systems with various masses and see how the computed evolution explains the observational data. The following discussion is rather qualitative, since the details of the...

## 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...

## Conic Section Astronomy

The orbit of an object in the gravitational field of another object is a conic section ellipse, parabola or hyperbola. Vector e points to the direction of the pericentre, where the orbiting object is closest to central body. If the central body is the Sun, this direction is called the perihelion if some other star, periastron if the Earth, perigee, etc. The true anomaly f is measured from the pericentre Fig. 6.4. The orbit of an object in the gravitational field of another object is a...

## 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...

## Photometric Concepts and Magnitudes

Most astronomical observations utilize electromagnetic radiation in one way or another. We can obtain information on the physical nature of a radiation source by studying the energy distribution of its radiation. We shall now introduce some basic concepts that characterize electromagnetic radiation. 4.1 Intensity, Flux Density and Luminosity 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...

## The Milky

On clear, moonless nights a nebulous band of light can be seen stretching across the sky. This is the Milky Way (Fig. 17.1). The name is used both for the phenomenon in the sky and for the large stellar system causing it. The Milky Way system is also called the Galaxy with a capital letter. The general term galaxy is used to refer to the countless stellar systems more or less like our Milky Way. The band of the Milky Way extends round the whole celestial sphere. It is a huge system consisting...

## The Ecliptic System

The orbital plane of the Earth, the ecliptic, is the reference plane of another important coordinate frame. The ecliptic can also be defined as the great circle on the celestial sphere described by the Sun in the course of one year. This frame is used mainly for planets and other bodies of the solar system. The orientation of the Earth's equatorial plane remains invariant, unaffected by annual motion. In spring, the Sun appears to move from the southern hemisphere to the northern one (Fig....

## Black Holes

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...

## Vj

Diffraction and resolving power. The image of a single star (a) consists of concentric diffraction rings, which can be displayed as a mountain diagram (b). Wide pairs of stars can be easily resolved (c). For resolving close bi naries, different criteria can be used. One is the Rayleigh limit 1.22 X D (d). In practice, the resolution can be written X D, which is near the Dawes limit (e). (Photo (a) Sky and Telescope) Fig. 3.6a-e. Diffraction and resolving power. The image of a...

## R

The redshift is z (X0 X) X, and hence i. e. the redshift of a galaxy expresses how much the scale factor has changed since the light was emitted. For example, the light from a quasar with z 1 was emitted at a time when all distances were half their present values. For small values of the redshift, (19.7) approaches the usual form of Hubble's law. This can be seen as follows. When z is small, the change in R during the propagation of a light signal will also be small and proportional to the...

## Through the Atmosphere

With satellites and spacecraft, observations can be made outside the atmosphere. Yet, the great majority of astronomical observations are carried out from the surface of the Earth. In the preceding chapter, we discussed refraction, which changes the apparent altitudes of objects. The atmosphere affects observations in many other ways as well. The air is never quite steady, and there are layers with different temperatures and densities this causes convection and turbulence. When the light from a...

## S V

At night, stars seem to revolve around the celestial pole. The altitude of the pole from the horizon equals the latitude of the observer. (Photo Pekka Parviainen) The angular separation of a star from the equatorial plane is not affected by the rotation of the Earth. This angle is called the declination 8. Stars seem to revolve around the pole once every day (Fig. 2.10). To define the second coordinate, we must again agree on a fixed direction, unaffected by the Earth's rotation....

## Example

Example 12.1 Assume that the Sun converts 0.8 of its mass into energy. Find an upper limit for the age of the Sun, assuming that its luminosity has remained constant. The total amount of energy released is E mc2 0.008 Mec2 0.008 x 2 x 1030 kg x (3 x 108 ms-1)2 1.4 x 1045J . The time needed to radiate this energy is _ E _ 1.4 x 1045J t L 3.9 x 1026W 3.6 x 1018s - 1011 years .

## Answers to Exercises

2.1 The distance is 7640 km, the northernmost point is 79 N, 45 W, in North Greenland, 1250 km from the North Pole. 2.2 The star can culminate south or north of zenith. In the former case we get 8 65 , 70 , and in the latter 8 70 , 65 . 2.3 a) > 58 7'. If refraction is taken into account, the limit is 57 24'. b) 8 7 24'. c) 59 10' < < 0 50'. 2.5 70 22', pQ 0 0', X 250 22', 0 0'. 2.8 a 6h 45 min 9 s, 8 16 43'. 2.9 vt 16.7kms 1, v 18.5kms 1, after 61,000 years. x 1.62'' per year, parallax...

## White Dwarfs

As was mentioned in Sect. 10.2, in ordinary stars the pressure of the gas obeys the equation of state of an ideal gas. In stellar interiors the gas is fully ionized, i. e. it is plasma consisting of ions and free electrons. The Fig. 14.1. Two views of the best-known white dwarf Sirius B, the small companion to Sirius. On the left, a picture in visible light by the Hubble Space Telescope. Sirius B is the tiny white dot on lower left from the overexposed image of Sirius. On the Fig. 14.1. Two...

## Sidereal Hour Angle

The vernal equinox measured along the equator. This angle is the right ascension a (or R.A.) of the object, measured counterclockwise from . Since declination and right ascension are independent of the position of the observer and the motions of the Earth, they can be used in star maps and catalogues. As will be explained later, in many telescopes one of the axes (the hour axis) is parallel to the rotation axis of the Earth. The other axis (declination axis) is perpendicular to the hour axis....

## 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....

## 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...

## Xvd

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...

## 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...

## Examples

Example 13.1 The observed period of a cepheid is 20 days and its mean apparent magnitude m 20. From Fig. 13.4, its absolute magnitude is M 5. According to (4.11), the distance of the cepheid is r 10 x 10(m - M) 5 10 x 10(2 + 5) 5 Example 13.2 The brightness of a cepheid varies 2 mag. If the effective temperature is 6000 K at the maximum and 5000 K at the minimum, how much does the radius change

## Planetary Nebulae

Bright regions of ionized gas do not occur only in connection with newly formed stars, but also around stars in late stages of their evolution. The planetary nebulae are gas shells around small hot blue stars. As we Fig. 15.23. The Crab nebula (M1, NGC 1952) in Taurus is a strong radio source. Its energy source is the central rapidly the remnant of a supernova explosion observed in 1054. The rotating neutron star, pulsar, which is the collapsed core of the photograph was taken at red...

## P1 P2

The angle Sun-planet-Earth is called the phase angle, often denoted by the Greek letter a. The phase angle is between 0 and 180 in the case of Mercury and Venus. This means that we can see full Venus, half Venus, and so on, exactly as in the phases of the Moon. The phase angle range for the superior planets is more limited. For Mars the maximum phase is 41 , for Jupiter 11 , and for Neptune only 2 . The sidereal year is the real orbital period of the Earth around the Sun. After one sidereal...

## 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...

## B1 Basic Concepts

Everyone knows the famous Pythagorean theorem where As is the length of the hypotenuse of a right-angle triangle, and Ax and Ay are the lengths of the other two sides. (For brevity, we have denoted As2 (As)2.) This is easily generalized to three dimensions This equation describes the metric of an ordinary rectangular frame in a Euclidean space, i. e. tells us how to measure distances. Generally the expression for the distance between two points depends on the exact location of the points. In...

## 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...

## 1

Where D 2R is the diameter of the hole. In mirror telescopes diffraction is caused also by the support structure of the secondary mirror. If the aperture is more complex and only elementary mathematics is used calculations may become rather cumbersome. However, it can be shown that the diffraction pattern can be obtained as the Fourier transform of the aperture. Example 3.2 A telescope has an objective with a diameter of 90 mm and focal length of 1200 mm. a What is the focal length of an...