These last chapters may change more than the rest of the book in the future. These changes mean that the numbering of formulas and figures has changed quite a lot since the previous version of the book.
Introduction
- Celestial Objects
- The Role of Astronomy
- Astronomical Objects of ResearchResearch
- The Scale of the Universe
Thus, many of the galaxies in the image are also among the oldest known objects. The solar system scale was established along with the heliocentric system in the 17th century.
Spherical Astronomy
Spherical Trigonometry
The sum of the angles of a spherical triangle is always greater than 180 degrees; the surplus. The anglesψ,θ,ψ,θ andχ can be expressed in terms of the angles and sides of the spherical triangle:. cosBsina= −cosAsinbcosc+cosbsinc, cosa=cosAsinbsinc+cosbcosc.
The Earth
All of these variations on the sine formula can be written in an easy-to-remember form:. Due to the flattening, the perpendicular does not point to the center of the earth except to the poles and at the equator.
The Celestial Sphere
The easiest way to do this is to use rectangular coordinates of the object and the observer (Example 2.6). Since the radius of curvature varies with latitude and increases towards the poles, the length of the nautical mile defined in this way will depend on the latitude (1843 m at the equator, 1862 m at the poles).
The Horizontal System
Since the coordinates of celestial objects in astronomical almanacs are given relative to the center of the Earth, if high accuracy is required, the coordinates of nearby objects must be corrected for the difference in the position of the observer. Therefore, one nautical mile is defined as equal to one minute of arc at φ=45◦, hence 1 nautical mile=1852 m.
The Equatorial System
But the zero point of the true ascension appears to move in the sky, due to the daily rotation of the Earth. The angle φin the last equation is the altitude of the celestial pole, or the latitude of the observer.
Rising and Setting Times
From the same observations, we can also determine the direction of the celestial pole as well as the latitude of the observer. The zero point for the second coordinate (right ascension) can then be defined as the point where the Sun appears to cross the equator from south to north.
The Ecliptic System
For very distant objects the differences are negligible, but not for solar system bodies. To transform heliocentric coordinates to geocentric coordinates or vice versa, we also need to know the distance of the object.
The Galactic Coordinates
The angle ε that appears in these equations is the obliquity of the ecliptic, or the angle between the equatorial and ecliptic planes. This transformation is most easily accomplished by calculating the object's rectangular coordinates and the new origin, then changing the origin, and finally estimating the new latitude and longitude from the rectangular coordinates (see Examples 2.5 and 2.6).
Perturbations of Coordinates
The difference of the observed directions (as well as of the entire phenomenon) is called parallax. By diurnal parallax we mean the change in direction due to the daily rotation of the Earth.
Positional Astronomy
This effect is due to the relative motion of the Sun and the stars through space; is called proper motion. By the Doppler effect we understand the change in the frequency and wavelength of radiation due to the radial velocity of the radiation source.
Constellations
Star Catalogues and Maps
In the zone catalogs the main aim is to give the positions of stars as precisely as possible. The catalog is accompanied by a star chart containing all the stars in the catalog.
Sidereal and Solar Time
As a result, the hour angle of the Sun (which determines solar time) also grows unevenly. Both true solar time and mean time are local times depending on the hour angle of the Sun, real or artificial.
Astronomical Time Systems
The difference between TAI and TT is always the same, but UTC will lag slightly due to leap seconds. The culminations and times of rising and setting of celestial bodies are related to rotation.
Calendars
From. If J is the Julian date at noon, the remainder of J /7 indicates the day of the week as follows:
Examples
Example 2.5 Find the altitude and azimuth of the Moon at Helsinki at midnight in early 1996. Example 2.6 Find the topocentric place of the Moon in the case of the previous example.
Exercises
Until the end of the Middle Ages, the human eye was the most important means of observation in astronomy. Astronomical photography was introduced at the end of the 19th century, and during the last few decades many types of electronic detectors have been used for the study of electromagnetic radiation from space.
Observing Through the AtmosphereAtmosphere
At even longer wavelengths, the ionosphere in the upper parts of the atmosphere reflects all radiation (Fig.3.2). Scattering caused by the molecules in the atmosphere is inversely proportional to the fourth power of the wavelength.
Optical Telescopes
In night vision (when the eye is perfectly adapted to the darkness), the resolving power of the human eye is about 2. A disadvantage of the Schmidt telescope is the curved focal plane, consisting of part of a sphere.
Detectors and Instruments
The image on a plate is practically permanent - the picture can be studied at any time. Therefore, data calibration is much easier than with photographic plates (Fig. 3.25). the number of electrons per incident photon is high and the CCD camera is much more sensitive than a photographic plate.
Radio Telescopes
The main difference between a radio telescope and an optical telescope is in the recording of the signal. With the VLBI technique, the spacing between antennas is limited only by the size of the Earth.
Other Wavelength Regions
In the 1990s, the United States built a chain of antennas spanning the entire continent, and the Australians have constructed a similar but north-south antenna chain across their country. The resolution in X-ray telescopes is of the order of a few arcseconds and the field of view about 1 degree.
Other Forms of Energy
Coherent light enters the hole from the direction of the negative z-axis (see figure). The radius of the diffraction disc in angular units can be estimated from the condition.
Examples
In mirror telescopes, diffraction is also caused by the supporting structure of the secondary mirror. However, it can be shown that the diffraction pattern can be obtained as the Fourier transform of the aperture.
Exercises
We can obtain information about the physical nature of a radiation source by studying the energy distribution of its radiation.
Intensity, Flux Density and LuminosityLuminosity
If the flux density of the radiation passing through this surface is F, the total flux is For extended objects (as opposed to objects such as stars that are only visible as points) we can define the surface brightness as the flux density per solid angle unit (fig. 4.4).
Apparent Magnitudes
The response of the human eye to the brightness of light is not linear. The brightness class or magnitude can be accurately defined in terms of the observed flux density F ([F] =W m−2).
Magnitude Systems
The more the radiation distribution differs from that of the Sun, the higher the bolometric correction. One of the multicolor magnitude systems widely used in photoelectric photometry is the UBV system developed in the early 1950s by Harold L.
Absolute Magnitudes
If the UBV system is used, it is usual to give only the V magnitude and the U−BandB−V color indices. The constants F0 in (4.8) for magnitudes U, BandV were chosen so that the color indices B −V and U−B are equal to zero for stars of spectral type A0 (for spectral types, see Chapter 8).
Extinction and Optical ThicknessThickness
The magnitude observed depends on the location of the observer and the object's zenith distance, as these factors determine the distance the light must travel. If the thickness of the atmosphere is used as a unit, the light must travel a distance.
Examples
Let R be the radius of the Sun, F the current density on the surface of the Sun and F the current density at a distance. If the magnitudes of the components of a binary star are 1 and 2, the total magnitude is certainly not 3.
Exercises
In reality, a particle of light is multiscattered and some of the multiscattered photons leave the cloud along the line of sight, reducing total extinction. Extinction in the direction of the star v. visual band isaV=1 mag kpc−1.
Radiation of Atoms and MoleculesMolecules
The odd part of the absorbed energy goes to the kinetic energy of the emitted electron. If polarized radiation passes through a magnetic field, the direction of the polarization will rotate.
The Hydrogen Atom
The polarization direction of linearly polarized light means the plane determined by the electric vector and the direction of the light ray. Scattered light is always polarized, with the degree of polarization highest in the direction perpendicular to the direction of the original radiation.
Line Profiles
In any experiment, only one component of the angular momentum can be measured at a time. The fourth quantum number is the spin which describes the intrinsic angular momentum of the electron.
Molecular Spectra
Continuous Spectra
Blackbody Radiation
This is true if the size of the radiator is very large compared to the dominant wavelengths. We can find the wavelengthλmax corresponding to the maximum intensity by differentiating Planck's functionBλ(T ) with respect toλ and finding zero of the derivative.
Temperatures
The ratio of the flux densities at wavelengths λ1 and λ2 gives the temperature of a blackbody with the same ratio. The color temperature can be determined even if the angular diameter of the source is unknown (Fig. 5.11).
Other Radiation Mechanisms
If the distribution of atoms in different levels is the result of only mutual collisions of atoms, the excitation temperature is equal to the kinetic temperature, Texc=Tk. As seen from the direction of the field, the motion is circular and therefore accelerated.
Radiative Transfer
If Sν(τν) is constant in the cloud and the background radiation is ignored, we get An important area of application of radiative transfer theory is in the study of planetary and stellar atmospheres.
Examples
With these notational conventions, (5.39) is now given. 5.44) This is the form of the radiative transfer equation commonly encountered in the study of stellar and planetary atmospheres. Example 5.4 (a) Find the fraction of radiation emitted by a blackbody in the range [λ1, λ2], where λ1 and λ2λmax. b) How much energy is reflected by a 100 W incandescent lamp in radio wavelengths, λ≥1 cm.
Exercises
If the flux densities at wavelengths λ1 and λ2 are F1 and F2, respectively, the color temperature can be solved from Eq.
Celestial Mechanics
- Equations of Motion
- Solution of the Equation of MotionMotion
- Equation of the Orbit and Kepler’s First LawKepler’s First Law
- Orbital Elements
- Kepler’s Second and Third Law
- Systems of Several Bodies
- Orbit Determination
- Position in the Orbit
- Escape Velocity
- Virial Theorem
- The Jeans Limit
- Examples
- Exercises
A particular solution (eg Jupiter's orbit) is chosen by fixing the values of the six integrals. The variable in the equation of orbit is an angle, the true anomaly, measured from perihelion.
The Solar System
Classification of Objects
For example, in the case of the Earth and Moon, the barycenter of the system is within the Earth, and the Moon is the Earth's satellite. Most of the satellites circle their parent planets in the same direction as the planet moves around the Sun.
Planetary Configurations
When Earth transits a superior planet, the planet's motion reverses or reverses. The synodic period depends on the difference in the sidereal periods of the two bodies.
Orbit of the Earth and Visibility of the Sun
At any given time, the Sun is seen at the zenith from a point on the Earth's surface. The third factor is the length of time the Sun is above the horizon.
The Orbit of the Moon
When the ascending node of the lunar orbit is near the vernal equinox, the Moon may be north or south of the equator. When the descending node is near the vernal equinox, the area where the Moon can be found lies just north or south of the equator.
Eclipses and Occultations
If the Moon is near its apogee, its apparent diameter is smaller than that of the Sun and the eclipse is annular. A lunar eclipse is complete when the Moon is entirely in the Earth's shadow; otherwise, the eclipse is partial.
The Structure and Surfaces of Planets
For Earth, the percentage includes both the outer and inner core. The giant planets' magnetic fields may originate from the layer of metallic hydrogen.
Albedos
If we assume that the reflecting object is a homogeneous sphere, the distribution of the reflected radiation depends only on the phase angle α. The geometric albedo depends on the reflectance of the surface, but also on the phase functionΦ.
Photometry, Polarimetry and SpectroscopySpectroscopy
The shape of the phase curve is very different for objects with or without an atmosphere. The degree of polarization as a function of the phase angle depends on the surface structure and the atmosphere.
Thermal Radiation of the PlanetsPlanets
In practice, the number of spectrum points (or the number of available filters) is often limited to 20-30. Some of the red light is also absorbed, giving rise to the planet's green color.
Origin of the Solar System
The sun lost its angular momentum to the surrounding gas through the magnetic field. Many low-density objects, comets, were flung toward the outer regions of the solar system.
Nice Models
Examples
Example 7.2 (Solar energy flux on the earth) Calculate the daily solar energy flux per unit area at the distance from the earth. However, atmospheric extinction reduces these values and the loss is greatest at the poles, where the height of the Sun is always relatively small.
Exercises
Find the apparent size of the asteroid (a) one week, (b) one day before the collision. What do you think about the chances of finding the asteroid well before the crash.
Mercury
The two-thirds factor meant that the same side of the planet was illuminated during each flyby. Due to the proximity of the Sun, the temperature of the primordial nebula at the distance of Mercury was quite high during planetary formation.
Venus
The North Pole is at the center of the Magellan Synthetic Aperture Radar mosaic image. Above this, there are fog-like layers that form the visible "surface" of the planet.
The Earth and the Moon
The lithosphere is the rigid outer part of the Earth (the crust and the upper part of the upper mantle). Plate tectonics ("continental drift") is driven by the movement of material in the mantle (Fig. 8.7).
