Observation Opportunities

Abstract

The observations of electromagnetic radiation, particle radiation, and gravitational waves provide information about the physical and chemical properties of celestial bodies and the processes occurring there. One of the most important tasks of astronomical and astrophysical research is the classification of objects in space and time (Kitchin, Astronomical techniques. IoP Publishing, 2003).

1 Classical Methods

In this section, we briefly present the physical foundations for the evaluation of various astronomical and astrophysical observations. Of interest are the methods for determining distance, but also mass and speed of various objects.

We distinguish between terrestrial and extraterrestrial observatories and observation techniques. In the visible and infrared spectral range, terrestrial and extraterrestrial observatories complement each other. Radio astronomy has been predominantly terrestrial in recent decades. In interferometric measurements, several radio telescopes are used in combination. Today, satellites are gaining importance, and there are already the first satellite radio telescopes. Also included in terrestrial observatories are “neutrino telescopes”, detectors for detecting high-energy cosmic radiation and gamma radiation, as well as “antennas” for gravitational waves. Together with space telescopes, the entire electromagnetic spectrum is now accessible for astrophysics. In Sect. 2.2, we will briefly outline some technical implementations.

1.1 Methods of Distance Determination

All direct information about large-scale matter distribution (stars, star clusters, gas clouds, galaxies, quasistellar radio sources or quasars etc.) comes to us in the form of waves or particles. The origin of the cosmic radiation consisting of massive particles is still largely unclear. The gravitational waves resulting from Einstein’s theory of gravitation have just been detected. This refers to the direct detection with the LIGO detector. The indirect detection was awarded the Nobel Prize in 1993 to Joseph H. Taylor and Russell A. Hulse. As early as 1974, they measured the orbital period of the double pulsar PSR 1913 + 16, which decreased by 75 millionths of a second each year, through the change in the actual pulsar period of 59 ms. This corresponds to the prediction for energy loss due to radiation of gravitational waves to within less than 1%.

Considered as a function of celestial coordinates, the received radiation (in the various frequency ranges) consists of a uniform, non-localizable background (e.g., microwave or X-ray radiation) and a portion that comes from discrete sources in very specific directions. The directed-appearing radiation may already be scattered or reflected. Which portion can be interpreted how depends on the model, i.e., the analytical understanding of the system. For this, one must also know whether a radiating object is far away or not. Since initially only angles on the celestial sphere are apparent, one must fundamentally come up with something new for distance determination.

Different methods are required for distance measurement if we want to gather information from the stars closest to our planetary system (i.e., the binary star system $\alpha$-Centauri at a distance of 4.4 light years or Proxima Centauri and Barnard’s Star) to the most distant galaxies and quasistellar sources (so-called quasars) a few hundred megaparsecs (Mpc) away.

The primary methods start with the parallax method known to us from terrestrial procedures. For greater distances, the Cepheid and RR Lyrae stars in the Milky Way are calibrated in the first step. Secondary and tertiary methods, which are based on considerable assumptions, follow the primary methods. This results in a not insignificant source of uncertainty in extragalactic distance determination. Figure 2.1 provides a rough overview of the most commonly used possibilities with which one can “climb up” to the largest scales. It dictates the structure of this section.

Fig. 2.1

Possibilities of distance determination on different scales

Through the overlap of different methods, one gains a “ladder” that can be used very successfully. Before we introduce the peculiarities of the individual “rungs” separately, a basic note on the expected accuracies. The distances between the galaxies of the Local Group reach up to 1 Mpc. The uncertainties in the measured values are estimated to be less than 10% today. Then the galaxies and small galaxy clusters at medium distances (1-10 Mpc) and finally the distant giant clusters are determined. For example, the average distance to the Virgo cluster is 15.8 ± 1.1 Mpc, and for the Coma cluster, it’s 87 ± 6 Mpc. For even greater distances, the Hubble relationship is used itself. As we will see, the measurement of cosmological distances strongly depends on the cosmological model, e.g., on the value of the Hubble constant $H_{0}$. Now, however, the systematic remarks on the individual “rungs”.

Laser and Radar Techniques

In the near range of the Earth, laser and radar techniques can be used to determine distances. We will not go into this here.

Trigonometric Parallax

In the adjacent area, the trigonometric parallax shift is the classic method, where – as already mentioned – the Earth’s orbit is taken as the basis. The measurement of the star parallax in 1838 by Bessel on the star 61 Cyngi serves as a historical example.

Nomenclature of Stars

The designation of stars typically adheres to the following guidelines: The brightest stars are designated with a Greek letter and a three-letter abbreviation of the genitive of the Latin constellation name. Instead of the abbreviation, the full name can also appear. Slightly weaker stars are designated within a constellation with a number instead of a Greek letter, e.g., 61 Cyg = 61 Cygni. Weak stars are only designated with their number in a star catalog (or with their coordinates). Thus, HD 48916 means the star with the running number 48916 in the Henry Draper Catalog. Variable stars often have large Latin letters before the constellation abbreviation, e.g., T Tau or T Tauri.

Back to the parallax method. For the distance d applies

$$\begin{aligned} \boxed { d = \frac{1 \, \text {AU}}{\text {tan} \Pi } \approx \frac{1}{\Pi } \, \text {AU} , } \end{aligned}$$

(2.1)

when the two measurement points are 1 AU apart (Fig. 2.2). It follows immediately

$$\begin{aligned} d \approx \frac{2.063\times 10^5}{\Pi \; ^{\prime\prime}} \, \text {AU} = \frac{1}{\Pi \; ^{\prime\prime}}\, \text {pc} , \end{aligned}$$

(2.2)

at 1 parsec = 1 pc = 2.063$\times 10^5$  AU. With $\Pi$ one denotes the so-called annual parallax,  for which also

$$\begin{aligned} \sin \Pi = \frac{a}{r} \; \end{aligned}$$

(2.3)

applies, where a is the major semi-axis of the Earth’s orbit (AU) and r is the distance of the star. The distance follows in AU, if you divide 206,265 by the parallax in arcseconds.

Fig. 2.2

Measurement of distances using parallax

The nearest star, Proxima Centauri (no rule of naming without exception!), has a parallax of $0.772 \pm 0.002$ arcseconds and is therefore $1.295 \pm 0.004$  pc or $4.225 \pm 0.013$  ly away. Distances in the Milky Way are usually given in kiloparsecs, in the extragalactic area in megaparsecs.

With “normal” terrestrial telescopes, a resolution up to approx. $0.02^{\prime\prime}$ can be achieved; this then corresponds to 50 pc at the distance. At distances > 100 light years, roughly speaking, the previously described parallax method fails. Improvements were achieved by satellites, e.g. Hipparcos, in the period from 1989 to 1993. With it, 118,218 stars were measured. The resolution was increased to $0.001^{\prime\prime}$. This made it possible to determine distances of stars up to approx. 1 kpc. Increases up to approx. $4 \times 10^{-6^{\prime\prime}}$ seem achievable, so that the distance measurement by parallax can be extended to approx. 250 kpc.

Spectroscopic Parallax

In the case of spectroscopic parallax, the spectral class of a star is measured based on its spectrum – so no angle. Its absolute brightness is assigned from the Hertzsprung-Russell diagram. If the absolute brightness is known, the distance can be calculated from it and the measured apparent brightness. The uncertainty of the method is very large, as stars of the same spectral type can have different absolute brightnesses.

Distance Measurement by Cepheids

For the pulsating variable stars, the luminosity L can be determined without knowledge of the distance. As early as 1595, a pulsating star (Mira) was observed by the pastor David Fabricius with a period of eleven months. Today, we count Mira among the pulsating stars with long periods.

The method of distance determination goes back to the American astronomer Henrietta Swan Leavitt (1868–1921) from the Havard Observatory. She measured many hundreds of Cepheids in the Small Magellanic Cloud. She noticed a correlation between the period and the apparent brightness of variable stars.

Today, more than 20,000 pulsating stars (including the classical Cepheids) are known very precisely. The apparent brightness of the classical Cepheids is linearly linked with the logarithms of the periods. This is called the period-luminosity relationship. A recently measured fluctuation in the brightness of a Cepheid is shown in Fig. 2.3.

Fig. 2.3

(Source: STCI)

Light curve of a variable star RR Lyr. Shown is the apparent brightness in the visible spectrum over time (in days).

The fact that the Cepheids of the Small Magellanic Cloud are all at approximately the same distance is quite plausible. The distance that the Cepheids have among each other within the Small Magellanic Cloud is small compared to the distance of the Small Magellanic Cloud. Henrietta Leavitt noticed that the Cepheids are brighter the longer their period is. The first observations (Fig. 2.4) were published in 1912.

Fig. 2.4

(Source: STCI)

Data from “Miss Leavitt,” as reported by Edward Pickering from the Harvard College Observatory in 1912. Shown are the maximum and minimum brightnesses of 25 stars in the Small Magellanic Cloud. Through a fit, one can already recognize the logarithmic dependence.

Since measurements of about 2400 classical Cepheids in the Small Magellanic Cloud are practically available at the same distance (60 kpc), one can equate the apparent brightness m with the absolute brightness M up to a fixed factor. Through observations (by Henrietta Leavit), a linear correlation between the apparent brightness m and the logarithm of the period was found (see Fig. 2.5).

Fig. 2.5

(Source: Original data from Henrietta Leavitt)

Linear relationship between the apparent brightness m and the logarithm of the period of about 2400 classical Cepheids in the Small Magellanic Cloud.

Absolute Brightness of Cepheids

If we take the Cepheids of the Magellanic Cloud, which are all approximately the same distance away, the absolute brightnesses are linearly linked with the logarithms of the periods. After a specific Cepheid distance was determined with parallax, the law

$$\begin{aligned} \log \left( \frac{L}{L_{\odot }} \right) = 1.15 \log (\Pi ^d) +2.47 \end{aligned}$$

(2.4)

could be calibrated, i.e., the constants determined. Here, $\Pi ^d$ is the period in days. Together with (1.211) the absolute brightness follows

$$\begin{aligned} { M= -2.80 \log (\Pi ^d)- 1.43 . } \end{aligned}$$

(2.5)

So, if a pulsating star is located in an arrangement of stars, we can use it as a “standard candle” for determining the distance of the arrangement. Of course, it immediately makes sense to seek a deeper understanding of the relationship (2.5). This already brings us into the realm of star models.

To determine distances using the period-luminosity relationship, it was necessary to establish a zero point. First, the distance had to be directly determined for some Cepheids. This was achieved by Harlow Shapley (1885–1972) in 1918. The parallaxes of eleven Cepheids could be measured. W. Baade discovered in 1952 that different zero points must be set for the period-luminosity relationship for the classical Cepheids (of the so-called Population I) and the 1.5 mag (magnitude) fainter W Virginis (of the so-called Population II).

With an empirically established period-luminosity relationship, it becomes possible to determine distances to any area where a Cepheid is located. The absolute brightness of the Cepheid is determined by its period. With this and the measured apparent brightness, the distance follows. The more Cepheids that can be used from an area, the more accurate the distance determination becomes.

The period-luminosity relationship was confirmed early on for far more than the original 25 Cepheids. Figure 2.5 impressively reflects the “lawfulness” [15].

Cepheids

To summarize: Cepheids are variable Population I or II giant stars. Their effective (surface) temperatures range from 6000–8000 $^\circ$C. Their luminosity is a hundred to ten thousand times the luminosity of the sun. The observed periods range from a few to a hundred days. Since the period-luminosity relationship is one of the most used bases for distance determination, we present simple models in Sect. 2.1.2 that underpin the “lawfulness”.

Bright Stars in Galaxies

When measuring the distance of galaxies and clusters of galaxies, one can also resort to more indirect methods. As long as individual stars or star clusters can still be resolved, it is utilized that the average absolute brightness of the three brightest red supergiant stars seems to be the same in all galaxies.

Globular Clusters

Another indicator is the brightness distribution of globular clusters, which is well described by a Gaussian function. If one again assumes that the absolute brightness of all globular clusters is the same at the maximum, one has another distance indicator.

Tully-Fisher Relationship

To determine the distance of galaxies, it is also possible to use the Tully-Fisher relationship. Tully and Fisher established a correlation in 1977 between the total luminosity of spiral galaxies and their rotation speed. The empirically found relationship is best for the luminosity in the infrared range. The rotation speed of the spiral galaxies is measured by the Doppler broadening of the 21-cm line of the neutral hydrogen present in the galaxy. Greater rotation speeds are associated with greater luminosity and mass of the galaxy.

Tully-Fisher Relationship

The Tully-Fisher “law” provides a correlation between the absolute magnitude of spiral galaxies and their maximum rotation speed $W_{R}$ in the form

$$\begin{aligned} \boxed {M = - a \log W_{R} - b \;} . \end{aligned}$$

(2.6)

Here, M is the absolute (photometric) brightness, and $W_{R}$ is determined by the Doppler width of an observed spectral line. The constants to be determined a, b depend on the frequency range in which the brightness is measured.

Example 2.1

The Tully-Fisher relationship can be made plausible through a very rough consideration. For this, we use the virial theorem in the form

$$\begin{aligned} U \approx -2 K \; \end{aligned}$$

(2.7)

with the average potential energy

$$\begin{aligned} U \sim \frac{G M^2}{R} , \quad M = \text {mass} , \end{aligned}$$

(2.8)

and the average kinetic energy

$$\begin{aligned} K \sim M v^2_{\max } . \end{aligned}$$

(2.9)

It follows

$$\begin{aligned} M \sim \frac{v^2_{\max } R}{G} . \end{aligned}$$

(2.10)

We now assume for the entire galaxy that the luminosity is proportional to the mass,

$$\begin{aligned} L \sim C_{ML} M , \end{aligned}$$

(2.11)

and to the surface,

$$\begin{aligned} L \sim C_{SB} R^2 , \end{aligned}$$

(2.12)

with corresponding constants. We thus find

$$\begin{aligned} \frac{L}{C_{ML}} \sim \frac{v^2_{\max } \sqrt{L}}{G \sqrt{C_{SB}}} \end{aligned}$$

(2.13)

or

$$\begin{aligned} L \sim v_{\max }^4 . \end{aligned}$$

(2.14)

If we use

$$\begin{aligned} M = M_\odot - 2.5 \log _{10} \frac{L}{L_\odot } , \quad M = \text {magnitude} , \end{aligned}$$

(2.15)

we obtain a relationship as stated in the Tully-Fisher “law”.    $\blacksquare$

Faber-Jackson Relationship

The Faber-Jackson relationship exists between the absolute luminosity and the velocity dispersion for elliptical galaxies.

Faber-Jackson Relationship

For the Faber-Jackson relationship, we can proceed similarly to the Tully-Fisher relationship, only that we make the substitution

$$\begin{aligned} v_{\max }^2 \rightarrow \sigma ^2 \end{aligned}$$

(2.16)

with the velocity dispersion $\sigma ^2$. Then, from the virial theorem, we obtain

$$\begin{aligned} { L \sim \sigma ^4 . } \end{aligned}$$

(2.17)

This is a form of the Faber-Jackson relationship.

Angular Diameter Distance

From the measurement of gas temperature and density distribution (e.g., determined from X-ray radiation), the linear dimension of hot gas clouds can be inferred. If the expansion of an object can be assumed to be known, the distance can be determined by comparing it with the measured angular diameter.

There are a number of astrophysical methods for determining the so-called angular diameter distance. For example, one observes hot gas in galaxy clusters that emits X-rays. The so-called Sunyaev-Zeldovich effect  utilizes the attenuation of the amplitude of cosmic microwave rays. The following should be noted: The Sunyaev-Zeldovich effect describes the attenuation of the number of low-energy photons compared to an increased number of high-energy photons relative to the Planck spectrum of cosmic background radiation, caused by interaction with electrons in hot gases in galaxy clusters. The change is based on the scattering of photons on electrons and the transfer of energy through the inverse Compton effect. If one now measures this effect in the cosmic background radiation in the direction of a galaxy cluster and at the same time the X-ray emission of the galaxy cluster, an interesting conclusion can be drawn about the expansion of the galaxy cluster by comparison. The Sunyaev-Zeldovich effect is proportional to the density of the electrons, the thickness L of the cluster along the line of sight, and the electron temperature. The Kompaneets parameter $y= \tau (k_B T_e/m_e c^2)$, with $\tau$ as the optical depth of the scattered photons, results from the measured change. On the other hand, the X-ray emissivity $I_X$ is proportional to the square of the electron density, the thickness L of the cluster along the line of sight, and the square root of the electron temperature. Since $y^2/I_X \propto L f(T_e)$ with a known function f, $y^2/I_X$ can be inferred to L. If we now identify the diameter D of the galaxy cluster with L, we have the opportunity to measure the angular diameter distance.

Novae and Supernovae

Nova (plural Novae) stands for “new star”, but this designation is historically conditioned and astrophysically incorrect. In a nova, no new star is created. Rather, an original star experiences a brightness outbreak.

A supernova (plural Supernovae) refers to a particularly strong brightness outbreak of a star. The brightness of this star sometimes increases by more than 20 mag. This rapid increase is accompanied by an explosive ejection of most of its mass. In contrast to supernovae, novae usually experience multiple brightness outbreaks with a temporal distance of a few years to about 80 years.

A well-known example of a supernova is Supernova 2002er, which was discovered on August 23, 2002, in the spiral galaxy UGC 10743. It belongs to the class of SN Type Ia and can be used as a “standard candle” for distance measurement.

Supernovae were initially named with the year and a large Latin letter according to the order of their discovery, e.g., SN 1993J. Due to the now many discovered supernovae, they have started to be marked with small double letters, e.g., SN 2002bu. Exceptions regarding the naming are some historically significant supernovae. These are named after their discoverers, e.g., Kepler’s Supernova.

Supernovae

The following types of supernovae are distinguished: The spectra of Type I supernovae have no or only very weak hydrogen lines, while those of Type II have clearly recognizable hydrogen lines. Type I is further divided into three groups: The spectrum of Type Ia contains silicon lines, but only very weak to no helium lines. Type Ib has no silicon lines, but helium lines appear in the spectrum. The spectrum of Type Ic shows (until shortly after the maximum) neither silicon nor helium lines. The spectra of Type II show a much greater individual variety.

The light curves of Supernovae I are very similar. Within a few days, the brightness reaches its maximum. Over the next 20 to 30 days, it decreases by 2 to 3 mag, then it decreases less rapidly. It apparently only takes a few hours for the light curves of Supernovae II to reach their maximum brightness. However, supernovae are only discovered shortly before reaching the maximum brightness. After the maximum brightness, the light curves of Type II can be divided despite the large differences. Most supernovae belong to Type II-P, where the light curves show a brightness plateau with minor brightness changes after 30 to 80 days. This lasts about two to three weeks. After that, the magnitude, just like with the Supernova Type II-L, decreases linearly with time.

To determine the distance of the supernova, it is necessary to set the value of its apparent brightness at the maximum brightness in relation to the absolute brightness. The absolute brightness at the maximum of Supernovae Ia is about 1.5 mag brighter than that of Type Ib and Ic. They have a relatively constant value at approx. $-19$ mag. Therefore, and for reasons of the quite clear history of origin, SN Ia serve as “standard candles” for distance measurement.

In a Type Ia supernova, the maximum of its absolute brightness varies only slightly around the above-mentioned value. From the measured apparent brightness, the distance can be calculated. However, this is not possible with SN II. Their absolute brightness varies greatly around the average value, which is about $-17$ mag. The deviations can be more than 2 mag.

Due to the lack of hydrogen lines in the spectra, it is assumed that the progenitor stars of type I supernovae have already lost their hydrogen-rich shell before the outbreak. Type Ia supernovae occur in all galaxies. It is assumed that their progenitor stars are old objects. The assumption is that they are white dwarfs. These have a fairly well-defined mass between 1.1 and 1.39 solar masses. Through gravitational attraction, a white dwarf can exceed the critical Chandrasekhar limit mass. The star becomes unstable, and an explosive carbon burning begins in the star center. The released energy triggers further nuclear processes, also in the outer regions of the star. The released energy is much greater than the gravitational binding energy. The star is torn apart and completely destroyed, leaving no stellar remnant.

It should be emphasized again that the more or less identical initial conditions explain why the light curves and the absolute brightness of the type Ia supernovae are so similar and therefore they can be used as standard candles for distance measurement.

The energy released in type I and II supernova explosions is approximately $10^{45}$  J. A supernova can shine as brightly as an entire galaxy at its peak. It is estimated that a supernova lights up in a galaxy on average every 25 years. Due to the interstellar extinction in the disk of our galaxy, it is only possible for us to observe about a tenth of the supernova outbreaks.

Example 2.2

In the literature, for the absolute brightness, for example, $M_{\max } \approx - 19.2$ or $M_{\max } \approx - 19.49$ are given. The uncertainty is already in the first place after the decimal point. For the SN 2002er, the apparent brightness $m_{\max } \approx 14.95$ was measured. From this, the distance – without extinction corrections – can be determined to $d \approx 38$ Mpc.   $\blacksquare$

It must be emphasized that the assumption of knowing something about the initial state strongly influences the conclusion of a universal behavior of type Ia supernovae.

Redshift and Hubble Expansion

For even greater distances the Hubble relationship itself is used. As we will see, the measurement of cosmological distances strongly depends on the cosmological model, i.e., on the Hubble constant $H_{0}$.

Example 2.3

We compare the results from various distance determinations of the galaxy UGC 10743. From data of the Supernova 2002er, the maximum magnitude

$$\begin{aligned} M_{\max }^{Supernova Ia} \approx - 19.35 \pm 0.75 \; \text {mag} \; \end{aligned}$$

(2.18)

was obtained and thus the distance to

$$\begin{aligned} d \approx 70 \, \text {Mpc} \end{aligned}$$

(2.19)

determined. If one conversely starts from an escape velocity

$$\begin{aligned} v_r \approx 2652 \, \text {km/s} \end{aligned}$$

(2.20)

and the Hubble constant $H_0 \approx 71$ km/s/Mpc, then follows

$$\begin{aligned} d \approx \frac{v_r}{H_0} \approx 40 \, \text {Mpc} . \end{aligned}$$

(2.21)

At least the orders of magnitude are the same.   $\blacksquare$

1.2 Highly Simplified Models for Cepheids

Since distance determination using Cepheids is very frequently used in astrophysics, we go into a bit more detail about the physical backgrounds here. 

With the Cepheid stars, the color, brightness, and radial velocity of the stellar atmosphere change within a period. The periodic pulsations form a precise clock. In Fig. 2.6, it is indicated in the top left how exactly one finds the sawtooth shape of the brightness curve in observations. The rest of the figure sketches the principal temporal courses of brightness, surface temperature, radius, and radial velocity [16].

Fig. 2.6

Observable parameters (brightness m, temperature T, radius R, radial velocity v) of a pulsating variable star $\delta$ Cephei [16]. In the top left, the many measurement points for brightness are indicated again, which lead to the sawtooth shape of the brightness curve

The physical mechanism behind the pulsations of absolute brightness can be evaluated in detail. The compression of a gas leads to increased energy release. If the energy of the gas is near the ionization limit of a frequently occurring element in it, it can absorb heat and only release it after reaching the highest density. If this happens at a resonance radius inside the star, pulsations occur. It is believed that the He$^{+}$ $\rightleftharpoons$ He$^{++}$ -ionization zone provides the drive for the pulsation, along with an influence from H $\rightleftharpoons$ H$^{+}$. If all Cepheids of the same period also have the same absolute luminosity M, then for a Cepheid in our vicinity (in the Milky Way), whose distance d is known, the absolute brightness can be determined from the observed apparent brightness m and d. Then this absolute brightness of a Cepheid of the same period at an unknown distance is assigned, and the distance d is determined from the apparent magnitude measured for it. Cepheids from the open star cluster of the Hyades (approx. 40–45 pc away) or the Pleiades (approx. 130 pc away) are used for calibration. The period-brightness relationship for Cepheids in the Large Magellanic Cloud has been measured in various wavelength ranges.

Estimating Pulse Duration with a Sound Wave Model

Based on ideas from Eddington it is interesting to create a simple model for the relationship between the period and the brightness of a pulsating star. The model assumes a radial “breathing” of the star. However, it does not take into account the previously mentioned ionization processes.

This model assumes that the radial oscillations arise from the resonance of sound waves inside the stars. The adiabatic speed of sound is given by

$$\begin{aligned} \text {v}_{s} = \sqrt{\frac{\gamma \; P}{\rho }} \end{aligned}$$

(2.22)

where P is the pressure and $\rho$ is the mass density. For an ideal monoatomic gas, we can use $\gamma =\frac{5}{3}$.

A first qualitative consideration leads to the period length

$$\begin{aligned} \Pi \sim \frac{R}{v_s}\sim \frac{R}{\sqrt{P/\rho }} \sim \frac{R}{\sqrt{\rho R^2}}\sim \frac{1}{\sqrt{\rho }}\sim R^{3/2} \sim L^{3/4} . \end{aligned}$$

(2.23)

Here we have $\rho \sim R^{-3}$, $L \sim R^2$ and $P \sim \rho ^2 R^2$ used. We justify the last proportionality in the following example.

Example 2.4

The estimation for the pressure dependence is made under the assumption of a (not necessarily realistic) hydrostatic equilibrium with constant density. We balance the “pressure force”  on a volume element at a distance r from the center of the star with the gravitational attraction.   Here, $M_r$ is the mass of the star enclosed up to a radius r:

$$\begin{aligned} \frac{d P}{d r} = - \frac{G M_{r} \rho }{r^{2}} = - \frac{4}{3} \pi G \rho ^{2} r \; \end{aligned}$$

(2.24)

This equation, which we will encounter often, arises as a static approximation of the momentum balance.  Themovement of a fluid element is described in the momentum balance by

$$\begin{aligned} \rho \frac{d \mathbf {\text {v}}}{d t} = - \nabla P - \rho \nabla \phi \end{aligned}$$

(2.25)

where the gravitational potential follows from

$$\begin{aligned} \nabla ^{2} \phi = 4 \pi G \rho . \end{aligned}$$

(2.26)

The last equation provides under the assumption of radial symmetry

$$\begin{aligned} \frac{1}{r^{2}} \frac{d}{dr} r^{2} \frac{d \phi }{d r} = 4 \pi G \rho , \end{aligned}$$

(2.27)

i.e., after integration

$$\begin{aligned} \left. \frac{d \phi }{d r} \right| _{r} = \frac{G M_{r}}{r^{2}} , \end{aligned}$$

(2.28)

since

$$\begin{aligned} \int _{0}^{r} \rho dV = M_{r} . \end{aligned}$$

(2.29)

So we get for the radial component of the momentum balance

$$\begin{aligned} \rho \frac{d \text {v}_r}{d t} = - \frac{dP}{dr} - \rho \frac{G M_{r}}{r^{2}} . \end{aligned}$$

(2.30)

If we first calculate statically, (2.24) follows. The solution with constant $\rho$ is obtained directly by integration:

$$\begin{aligned} P (r) = \frac{2}{3} \pi G \rho ^{2} (R^{2} - r^{2}) \; \end{aligned}$$

(2.31)

   $\blacksquare$

The time for sound to traverse the distance 2R (R being the radius of the star) is then

$$\begin{aligned} \Pi \approx 2 \int _{0}^{R} \frac{d r}{\text {v}_{s}} = \sqrt{\frac{3\pi }{2 \gamma G \rho }} . \end{aligned}$$

(2.32)

It is interpreted as the period of the star. Interestingly, this time has the same order of magnitude as the observed pulsation period! This has been particularly confirmed in the dependence on $\rho$.

One-Zone Model

All this suggests that the pulsations are standing sound waves. We therefore derive a one-zone model from (2.30). The geometry is outlined in Fig. 2.7. From (2.30) directly follows

$$\begin{aligned} \boxed { \rho \frac{d^{2}r}{dt^{2}} = - G \frac{M_{r} \rho }{r^{2}} - \frac{d P}{d r} . } \end{aligned}$$

(2.33)

The model neglects many effects. Computer programs that, for example, take ionization into account, exist. However, we do not want to discuss possible improvements at this point.

Fig. 2.7

Sketch of the radial geometry of a one-zone model for pulsating stars

In the so-called one-zone model, the movement of a shell that is removed from the center by R is examined. For simplicity’s sake, we assume that effectively the entire (inner) mass is concentrated in the center. In the “intermediate space”, a pressure is to be maintained that prevents gravitational collapse. If we integrate (2.30) over the thin shell, we get

$$\begin{aligned} m \frac{d^{2} R}{dt^{2}} \approx 4 \pi R^{2} P - \frac{G M m}{R^{2}} , \end{aligned}$$

(2.34)

if the pressure in the outer area disappears. Here, m is the shell mass.

The equilibrium is given by

$$\begin{aligned} \frac{G M m}{R_{0}^{2}} = 4 \pi R_{0}^{2} P_{0}. \end{aligned}$$

(2.35)

Linearizing around the equilibrium leads to

$$\begin{aligned} m \frac{d^{2} \delta R}{dt^{2}} = \frac{2\,G M m}{R_{0}^{3}} \delta R + 8 \pi R_{0} P_{0} \delta R + 4 \pi R_{0}^{2} \delta P . \end{aligned}$$

(2.36)

An adiabatic relationship $PV^{\gamma } = \text {const}$ provides via $\delta V = 4 \pi R_{0}^{2} \delta R$

$$\begin{aligned} \frac{\delta P}{P_{0}} = - 3 \gamma \frac{\delta R}{R_{0}} . \end{aligned}$$

(2.37)

From this, we obtain an oscillation differential equation. After Fourier transformation

$$\begin{aligned} \omega ^{2} = ( 3 \gamma - 4) \frac{G M}{R_{0}^{3}}, \end{aligned}$$

(2.38)

and thus the period $\Pi = 2 \pi / \omega$ leads to

$$\begin{aligned} \boxed { \Pi = \frac{2 \pi }{\sqrt{\frac{4}{3} \pi G \rho _{0} (3 \gamma - 4)}} } \end{aligned}$$

(2.39)

where $\rho _{0} = 3 M / 4 \pi R_{0}^{3}$ is. We will address the case $\gamma \le \frac{4}{3}$ later.

The next idea is to associate the radius changes with a compression or expansion, where the surface temperature (and thus the luminosity) increases or decreases (Fig. 2.6). Quantifying the luminosity fluctuations requires additional calculations, which we will not further execute here. It is possible, using complex physical models, to infer the absolute brightness from the oscillations. Then, together with the measurement of the apparent brightness according to (1.217), the distance d follows.

It should be emphasized again that we have presented a very simple model here, which does not adequately address many effects, but should convey the basic ideas. The nonlinear solution of the equation

$$\begin{aligned} \frac{d^2R}{dt^2} = -\frac{\alpha }{R^2} + \frac{\beta }{R^3} \end{aligned}$$

(2.40)

for $\alpha = 1.4\times 10^{-10}$ and $\beta = 1.2 \times 10^{-10}$ is shown in Fig. 2.8. Subsequently, the pressure is calculated using

$$\begin{aligned} P \approx \frac{\kappa }{R^5} \end{aligned}$$

(2.41)

with $\kappa = 5.6 \times 10^4$.

Fig. 2.8

Oscillations of radius, edge speed, and pressure of a Cepheid, as they result from a simple one-zone model

Example 2.5

The period-luminosity relationship follows from the simple model calculations. First, we have

$$\begin{aligned} \Pi \sim L^{3/4} \end{aligned}$$

(2.42)

found. We combine this with the previously stated proportionality

$$\begin{aligned} M \sim -2.5 \log L \end{aligned}$$

(2.43)

for the absolute brightness M. The result

$$\begin{aligned} M \sim - \frac{10}{3} \log \Pi \end{aligned}$$

(2.44)

agrees surprisingly well in order of magnitude with the more exact relation

$$\begin{aligned} M \sim -2.80 \log \Pi + \dots \end{aligned}$$

(2.45)

$\blacksquare$

1.3 Determination of Mass

In addition to measuring distance, the determination of mass is an important astronomical goal. In the near-Earth area, we can easily imagine that masses can be determined from complex orbits using model calculations based on gravity. In particular, mass calculation for binary stars is relatively simple. Surprisingly, a large part (more than 50%) of the stars appear as binary stars.

For the visible stars, we distinguish between optical (apparent) and physical (real) pairs. The latter can be used for mass determination. There are also spectroscopic binary stars that are adjacent but cannot be resolved in the telescope. Their relative movement is only expressed in the periodic shifts of the spectral lines due to the Doppler effect.

If the orbital period and the semi-major axis are known, the sum of the masses can be calculated using Kepler’s third law:

$$\begin{aligned} \boxed { M_1 + M_2 =\frac{4 \pi ^2 a^3}{G U^2} \; } \end{aligned}$$

(2.46)

Here, a is the semi-major axis and U is the orbital period. Note: M is (now) the mass of a star!

If the movement can be resolved in more detail, i.e. if the individual semi-major axes around the common center of gravity can also be measured individually, then

$$\begin{aligned} \boxed { \frac{M_1}{M_2}=\frac{a_2}{a_1} . } \end{aligned}$$

(2.47)

Then $M_1$ and $M_2$ can be determined individually.

Since Kepler’s laws and the two-body problem are extensively covered in mechanics, we will not go into them further here. However, there are other, much more indirect methods of mass determination. The following example: In the Hertzsprung-Russell diagram, the luminosity of the stars is plotted against their surface temperature. It can be seen that most stars lie on a curve, the so-called main sequence. The total mass in the core of a star of a given chemical composition determines its luminosity. For the stars of the main sequenceof the Hertzsprung-Russell diagram, there is the relationship according to Eddington

$$\begin{aligned} \frac{L}{L_{\odot }} \sim \left\{ \begin{array}{ll} (M/M_{\odot })^4 &{} \text {for} \; L \lesssim L_{\odot } ,\\ \\ (M/M_{\odot })^{2.8} &{} \text {for} \; L \gtrsim L_{\odot } , \\ \end{array} \right. \end{aligned}$$

(2.48)

from which one can also infer the mass. Red giants and white dwarfs do not follow this relationship. In Fig. 2.9, the actual relationship is shown using some examples. Another method is the so-called spectral mass determination, which we will not go into here.

Fig. 2.9

Mass-luminosity relationship for main sequence stars in the Hertzsprung-Russell diagram. The two drawn straight lines have the slopes 3 and 3.43. The points correspond to observed stars

1.4 Radius Determination

From the light curve of a binary star system, in which one star periodically covers the other, information about the size can be obtained. Let D and d be the star diameters, L the orbital length, and P the orbital period, then with the parameters given in Fig. 2.10 ($t_d$ is the time at the moment d etc.)

$$\begin{aligned} \frac{t_d -t_a}{P} = \frac{D+d}{L} , \quad \frac{t_c -t_b}{P} = \frac{D-d}{L} . \end{aligned}$$

(2.49)

From the measurement of the times, D/L and d/L follow. If the orbital speed $v=L/P$ is also known from spectroscopic observations, d and D follow individually.

Fig. 2.10

Binary star system in which the smaller star is hotter. On the right, the light curve is shown during temporary complete coverage

The observation of the light curve of a star during the passage of the lunar orbit through the line of sight (lunar occultation) provides the star radius with an accuracy up to a maximum of $0.01^{\prime\prime}$. 

If the luminosity L of a star (and thus its distance) and its effective temperature T are known, the radius follows from

$$\begin{aligned} \boxed { L = 4 \pi R^2 \sigma T^4 . } \end{aligned}$$

(2.50)

1.5 Surface Temperature

In general, one compares a star spectrum with a Planck spectrum, which contains temperature as the only parameter. The surface temperature is actually the temperature of a star’s photosphere. Within the framework of the Eddington approximation for stellar atmospheres, an effective temperature is defined, which we will discuss in the context of star models.

1.6 Velocity Determination

Velocities can be determined from the Doppler shift. When the source approaches, the frequency increases, when it moves away, it decreases. Spectroscopically, for example, a velocity of 100 km/s corresponds to a shift of 1.7 Ångstrom = 0.17 nm at a wavelength of 5000 Ångstrom. The non-relativistic formula (one-dimensional) is

$$\begin{aligned} \boxed { \frac{\Delta \lambda }{\lambda } = \frac{u}{c} } \end{aligned}$$

(2.51)

given a difference velocity u.

The speeds of particles and objects are often relativistic. The one-dimensional Lorentz transformation for a relative speed in the x direction

$$\begin{aligned}&x^{\prime} = \gamma (x-ut) , \end{aligned}$$

(2.52)

$$\begin{aligned}&y^{\prime} = y , \end{aligned}$$

(2.53)

$$\begin{aligned}&z^{\prime} = z , \end{aligned}$$

(2.54)

$$\begin{aligned}&t^{\prime} = \gamma \left( t - \frac{ux}{c^{2}}\right) \end{aligned}$$

(2.55)

with

$$\begin{aligned} \gamma = \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} \end{aligned}$$

(2.56)

determines the transformation behavior. The inverse transformation results from changing the sign of u and swapping the unprimed with the primed variables:

$$\begin{aligned} x= & {} \gamma (x^{\prime}+ut^{\prime}) \; \end{aligned}$$

(2.57)

$$\begin{aligned} y= & {} y^{\prime} \; \end{aligned}$$

(2.58)

$$\begin{aligned} z= & {} z^{\prime} \; \end{aligned}$$

(2.59)

$$\begin{aligned} t= & {} \gamma \left( t^{\prime} + \frac{ux^{\prime}}{c^{2}}\right) \; \end{aligned}$$

(2.60)

Within the framework of Special Relativity, length contraction and time dilation are discussed. An astrophysical example of time dilation is the observation of muons on Earth despite their short lifespan.

Now some remarks on the relativistic Doppler effect. We consider a distant object that emits light. The object should be so far away that the light rays can still be considered (almost) parallel even after small time differences (Fig. 2.11). If a pulse is emitted at the same location $x^{\prime}$ at times $t^{\prime}_{1}$ and $t^{\prime}_{2}$ with $\Delta t^{\prime} = \Delta t_{\text {rest}} = t^{\prime}_{2} - t^{\prime}_{1}$in the rest system of the particle (dashed coordinates), then in the laboratory system (observer with undashed coordinates)

$$\begin{aligned} \Delta t = \gamma \Delta t_{\text {rest}} . \end{aligned}$$

(2.61)

In this time, the particle has traveled the distance

$$\begin{aligned} u \Delta t \cos \theta \end{aligned}$$

(2.62)

as seen by the observer. The angle $\theta$ is explained by Fig. 2.11; it is defined by the relative motion and the current connecting line of the particle to the observer. In addition to $\Delta t$ the runtime difference due to the different locations of the particle at two pulse emissions is added (Fig. 2.11), i.e., the observed time difference is

$$\begin{aligned} \Delta t_{\text {obs}} = \Delta t + \frac{u}{c} \Delta t \cos \theta . \end{aligned}$$

(2.63)

Fig. 2.11

Sketch for the transformation between moving and rest system

Relativistic Doppler Effect

The frequency transformation $\left( \nu _{\mu } = 1 / \Delta t_{\mu } \right)$ within the framework of the relativistic Doppler effect is given by

$$\begin{aligned} { \nu _{\text {obs}} = \nu _{\text {rest}} \frac{\sqrt{1- \frac{u^{2}}{c^{2}}}}{1+\frac{u_{r}}{c}} , } \end{aligned}$$

(2.64)

where we identify $u_{r} = u \cos \theta$ as the radial velocity of the light source. For $\theta = 0^{\circ }$ applies $u_{r} = u$, and the light source is moving away from the observer, while for $\theta = 180^{\circ }$ the light source is moving towards the observer with $u_{r} = -u$. In these special cases, the following applies

$$\begin{aligned} \nu _{\text {obs}} = \nu _{\text {rest}} \sqrt{\frac{1- \frac{u_{r}}{c}}{1+\frac{u_{r}}{c}}} \qquad \text {(radial movement)}. \end{aligned}$$

(2.65)

Non-relativistically, $|u_{r}|/c \ll 1$, follows

$$\begin{aligned} \nu _{\text {obs}} \approx \nu _{\text {rest}} \left( 1 - \frac{u_{r}}{c} \right) . \end{aligned}$$

(2.66)

With $c = \lambda \nu$ we can convert from frequencies to wavelengths. The so-called redshift parameter is defined as

$$\begin{aligned} \boxed { z = \frac{\lambda _{\text {obs}} - \lambda _{\text {rest}}}{\lambda _{\text {rest}}} = \frac{ \Delta \lambda }{\lambda _{\text {rest}}} . } \end{aligned}$$

(2.67)

In the case of radial motion, the following applies

$$\begin{aligned} \boxed { \frac{u_{r}}{c} = \frac{(z+1)^{2}-1}{(z+1)^{2}+1} , } \end{aligned}$$

(2.68)

which in the non-relativistic limit case $(z \ll 1)$ simplifies into

$$\begin{aligned} \frac{u_{r}}{c} \approx z . \end{aligned}$$

(2.69)

With this, we have achieved (2.51).

Using the Doppler effect, the escape velocity of galaxies at a distance D has been determined in particular.

By now, more than $3 \times 10^4$ redshifts of galaxies have been measured. Theoretical considerations by DeSitter, Silberstein, and Weyl led to a linear dependence between the escape velocity v of a galaxy and its distance D from us. But only when the astronomer Hubble succeeded in measuring the distances of galaxies using Cepheids, the cosmological significance of the so-called Hubble-Lemaître law [18] (see also Fig. 2.12) was recognized

$$\begin{aligned} \boxed { {v} \approx H_0 \; D . } \end{aligned}$$

(2.70)

Here, the Hubble constant is of the order of magnitude

$$\begin{aligned} H_0 \approx 71 \;\text {(km/Mpc) s}^{-1} ; \end{aligned}$$

(2.71)

the value is continuously improved.

Fig. 2.12

Various representations of the relationship found by Hubble. a Original data from Hubble from 1936. b Much larger distance range in a preparation by S. Jha from 2002 [17]. A tiny area on the bottom left corresponds to all the data in a

The age of the universe can be estimated – as previously discussed – for example, through Hubble’s law. Often in Hubble’s constant

$$\begin{aligned} H_{0} \approx h \; 100 \; \frac{\text {km}}{\text {s Mpc}} \end{aligned}$$

(2.72)

a factor of uncertainty

$$\begin{aligned} 0.4 \le h \le 1 \end{aligned}$$

(2.73)

is used.

$$\begin{aligned} \boxed { t_{\text {cosmos}} \le \frac{1}{H_{0}} } \end{aligned}$$

(2.74)

From this, we obtain an approximate order of magnitude for the age of the universe.

We will not further discuss age determinations from radioactive decay and the Hertzsprung-Russell diagram here.

2 Observation Instruments

On the topic of “observation instruments” only keywords will be listed in this theoretically oriented book. And that only insofar as they are useful for an understanding of the applicability of theoretical models or the data situation as a starting point for theoretical models. In itself, the range of topics in observation instruments is huge. Reference must be made to the relevant literature. Also, current information on the websites of research institutions is useful, as the commissioning and availability of observation instruments change quickly. In this short section, we focus on a rough overview of the most common detection methods.

Optical Telescopes

Telescopes are used to observe the stars. These are used to distinguish neighboring radiation sources and to bundle electromagnetic radiation. The most well-known are optical telescopes, which are based on refractors or reflectors. The Galilei telescope (Fig. 2.13) was invented by Hans Lipperhey around 1608 in Holland and subsequently developed by Galileo Galilei. The Kepler telescope was built by Johannes Kepler in 1611. Around 1616, Nicolaus Zucchius created the first reflecting telescope. Today, large reflecting telescopes are exclusively used for large projects, as the precisely manufactured mirrors allow much larger entrance openings than lenses.

Fig. 2.13

(With kind permission from Space Telescope Science Institute at copyright@stsci.edu)

Artistic representation of the Galilei telescope.

There are various technical “tricks” to increase the resolution. Above all, air turbulence sets clear limits to the resolution. There are sophisticated evaluation techniques that minimize this effect (speckle analysis, adaptive optics, etc.).

Diffraction and interference occur in optical observations. When light of wavelength $\lambda$ from an infinitely distant point source falls on a lens with a circular entrance pupil and radius $r_0$, the intensity in the focal plane is given by the Airy distribution

$$\begin{aligned} I_1(\theta )= \left[ \frac{2J_1(2\pi r_0/\lambda \theta )}{2 \pi r_0/\lambda \theta } \right] ^2 . \end{aligned}$$

(2.75)

The angular deviation from the vertical is denoted by $\theta$. For two equally bright incoherent point sources at an angular distance $\theta$ (as seen by the observer), the resolution limit is reached when the maximum of one function coincides with the first zero of the other. This provides the Rayleigh criterion

$$\begin{aligned} \boxed { \theta _{\text {min}} = 0.61 \frac{\lambda }{r_0} . } \end{aligned}$$

(2.76)

The angle is given in radians.

Example 2.6

A large optical telescope with a mirror diameter of $D=2 r_0 =10$ m has a theoretical resolution of $\theta _{\text {min}}=0.01$ arcseconds at $\lambda =0.5 \, \upmu$m. At 100 m mirror diameter and $\lambda =0.2$ m, the resolution is 0.1$^{\circ }$. The main advantage of large telescopes is not in the resolution, but in the light intensity, as more radiation is collected (see below).   $\blacksquare$

Example 2.7

To estimate the light intensity, the linear expansion l of the diffraction disk in the focal plane (at a focal length f) of the telescope is important. It applies

$$\begin{aligned} l=2f \tan \theta . \end{aligned}$$

(2.77)

Together with the Rayleigh criterion, it follows that

$$\begin{aligned} l = 1.22 \frac{\lambda f}{r_0} . \end{aligned}$$

(2.78)

The light intensity indicates how strongly the light flux is concentrated on the receiver surface. Because of

$$\begin{aligned} I \sim \frac{D^2}{l^2} \sim \left( \frac{D}{\lambda } \right) ^2 \; \left( \frac{D}{f} \right) ^2 \; \end{aligned}$$

(2.79)

the light intensity increases with the square of the aperture ratio D/f.   $\blacksquare$

Technical possibilities limit the diameters of lens telescopes (refractors). Diameters of over one meter are only difficult to achieve. Mirror telescopes (reflectors) have significantly more favorable properties. Modern large optical telescopes have mirror diameters of 8 to 10 m. Mega telescopes with mirror diameters of 30 and 100 m are in planning.

In a reflecting telescope, light is focused via a curved mirror and directed into the eyepiece through one or more secondary mirrors. The light is not refracted, and thus the chromatic aberrations of a refractor do not occur. Also, the production of a mirror is significantly cheaper than that of a lens.

On Earth, we are bound by the observation windows (Fig. 1.24). With space telescopes, the problems of absorption are circumvented. Outside of the optical window, the radio window is available, which is used in radio telescopes. In addition, there is also infrared, ultraviolet, and X-ray astronomy. 

Example 2.8

We approximate the solar constant $1367 \; \frac{W}{m^2}$  for simplification of the calculation by $1400 \; \frac{W}{m^2}$. For simplicity, we also assume that all photons coming from the sun have the same energy of 2 eV. This is an average value, which corresponds to the wavelength $\lambda = \frac{hc}{E} \approx 600$ nm. The color is a yellow-orange.

Now we calculate the number N of photons that, under the assumed conditions, hit an area of 1 cm$^2$ per second on Earth:

$$\begin{aligned} N= \frac{0.14 \; J}{2 \;eV} = \frac{0.14 \; J}{2 \times 1.602 \times 10^{-19} \; J} =4 \times 10^{17} \; \end{aligned}$$

(2.80)

   $\blacksquare$

Example 2.9

We assume that a 100 W source emits its energy entirely in the form of photons of wavelength $\lambda =600$ nm. With the energy $E= \frac{hc}{\lambda }$ per photon, the power is $P = N E$, when N photons are emitted per second. We ask at what distance R the source can still be perceived by the human eye. 

Under ideal conditions, the eye just barely detects 1000 photons per second with a pupil opening area $\pi r^2$ with $r=3.5$ mm. The eye receives at a distance R

$$\begin{aligned} n= \frac{N \pi r^2}{4 \pi R^2} \end{aligned}$$

(2.81)

photons per second. If we set $n = 1000$ and solve for R, we get

$$\begin{aligned} R = \sqrt{\frac{N r^2}{4n}} =r \sqrt{\frac{P \lambda }{4 n h c}} \approx 10^6 \, \text {m} . \end{aligned}$$

(2.82)

The result of 1000 km only applies when all absorption and scattering losses are neglected.   $\blacksquare$

Examples of Large Telescopes

• The Daniel K Inouye Solar Telescope is located in Hawaii and is the largest telescope in the world for solar observations.

• The South African Large Telescope could, for example, detect a candle on the moon.

• The Gran Telescopio Canaris was able to detect a galaxy in 2016 that is 500 million light years away from us.

• The Keck Observatory in Hawaii enabled supernova observations with the discovery of the accelerated expansion of the universe. 

• The Hobby-Eberly Telescope in Texas (USA) was used to measure the mass of a supermassive black hole that is 220 million light years away from us.

Infrared Astronomy

Between the optical spectral range and the radio range lie the areas of infrared radiation (1–100 $\upmu$m), the submillimeter (100 $\upmu$m–1 mm) and the millimeter waves (1–4 mm). Absorption by the atmosphere and thermal emission of the atmosphere in these parameter ranges make measurement difficult. To optimize observation conditions, infrared telescopes are installed on high mountains and in satellites. Interesting problems at these wavelengths are, for example, associated with

• objects in the solar system,

• interstellar medium and molecular clouds,

• dust shells of stars,

• star-forming regions,

• 3 K background radiation

Radio Telescopes

The radio window extends in wavelengths from 10 mm to several 10 m. However, many terrestrial sources of interference make measurement difficult. Therefore, for example, the 100-m radio telescope in Effelsberg was built in a protected valley basin. Another example of a radio telescope is the Very Long Baseline Array (VLBA), which consists of ten 25-m telescopes distributed across the Earth. By connecting neighboring telescopes, a much higher angular resolution can be achieved. The Very Long Baseline Interferometry (VLBI) should also be mentioned.  Radar astronomy is also operated from satellites.

Interesting research topics include, for example,

• observation of meteors and their radar signals,

• exploration of the Milky Way,

• evaluation of the 21-cm line of atomic hydrogen,

• pulsars,

• supernova remnants.

Here too, many of the physically relevant applications can only be meaningfully discussed after a further deepening of the physical fundamentals.

Examples of Large Radio Telescopes

• The RATAN 600 in Russia is the largest radio telescope in the world with a main mirror diameter of around 600 m.

• The FAST in China is the second largest with a diameter of 520 m.

• The ALMA in the Atacama Desert consists of an array of 66 antennas.

• The Arecibo Observatory in Puerto Rico became widely known through the James Bond film Golden Eye. Unfortunately, it collapsed on December 1, 2020. 

• The Effelsberg radio telescope in Germany is the second largest movable telescope (after the Green Bank Observatory in West Virginia, USA). 

Observation of Cosmic Radiation

High-energy cosmic radiation generates extensive air showers, which can be detected with large-scale detectors distributed on the ground. Figure 2.14 shows the principle of the development and detection of an extensive air shower.

Fig. 2.14

(Source: Auger collaboration)

a Sketch of the Auger hybrid experiment, b Picture of a water tank.

The particle avalanches triggered by a high-energy primary particle develop along the direction of incidence of the primary particle. Upon entry into the Earth’s atmosphere, the highest particle intensity is detected. A primary proton of energy $10^{15}$ eV (1 PeV) generates, for example, near the Earth’s surface on average $10^6$ secondary particles (80% photons, 18% electrons and positrons, 1.7% muons and 0.3% hadrons). At energies above $10^{17}$ eV, fluorescence light in the wavelength range between 300 and 400 nm can also be efficiently detected. It is produced by the interaction of charged particles with nitrogen molecules of the atmosphere and can be observed at such energies with imaging mirror systems on clear nights at a distance of up to 30 km. The hybrid experiment Auger makes use of this, for example.

The physically interesting quantities such as direction, energy and mass of the primary particles must be derived from the properties of the air showers. Numerical simulations accompany the evaluation of the enormous amounts of data.

Different experiments are conducted to detect cosmic radiation:

• Balloon experiments,

• Satellite experiments, e.g. PAMELA and AMS,

• Air shower experiments, e.g. KASCADE and Auger.

Pierre Auger Observatory

The Pierre Auger Observatory is the largest facility for measuring cosmic radiation of the highest energies. The particle shower triggered by primary particles stimulates air molecules, especially nitrogen, to ultraviolet fluorescence during its flight to Earth. In darkness, the trail of weak luminescence can be recorded with ground telescopes, which is particularly used for the detection of rare events up to 20 km in altitude. The Auger project applies a hybrid technique. Surface detectors measure the secondary particles, and large telescopes register the fluorescence light.

The secondary particle population contains a large proportion of (charged) $\pi$-mesons. However, these decay into $\mu$-mesons, which are then detected by 1600 tanks with Cherenkov¹ radiation. The simultaneous detection method significantly reduces the systematic uncertainties of the measurement, especially in the energy determination of the primary particles.²

Observation of X-ray and Gamma Radiation

The designations for high-energy electromagnetic radiation are occasionally confusing, especially when considering the historical development. X-ray radiation is not as short-wavelength as gamma radiation and thus not as energy-rich. One occasionally finds the following different designations:

Electromagnetic radiation with a wavelength

• from $10^{-9}$ m to $10^{-11}$ m is called X-ray radiation,

• from $10^{-11}$ m to $10^{-13}$ m is called gamma radiation,

• from $10^{-13}$ m to $10^{-17}$ m is called cosmic radiation.

Sometimes gamma radiation is also referred to as high-energy X-ray radiation. We will not delve further into this terminology discussion and speak of high-energy electromagnetic radiation.

X-ray Telescopes

The Nobel Prize  2002 for Physics was (in part) awarded to Riccardo Giacconi. In 1959, Giacconi began his work on X-ray telescopes. Initially, he used rockets. Since the observation time is very limited with high-altitude research rockets, he started with satellites (Uhuru, HEAO-2, Chandra) from 1970. Chandra was launched into space in 1999 and became the main instrument of X-ray astronomy. The X-ray satellite ROSAT must also be mentioned in this context today.

Gamma radiation, especially VHE radiation (VHE = very high energy), cannot be of thermal origin. The high temperatures corresponding to a Planck distribution are not reached in the universe. Therefore, it is assumed that charged particles are accelerated in cosmic fields and transfer their energy to electromagnetic radiation in secondary processes. Other possibilities are the decay or annihilation of very heavy particles, e.g., those that could also make up dark matter. An important goal of gamma astronomy is to clarify the origin of the radiation and the underlying acceleration mechanisms.

As already discussed, cosmic particles trigger showers when they hit the atmosphere (Fig. 2.15a). A photon hitting the atmosphere forms an electromagnetic shower with different components at a height of about ten kilometers (Fig. 2.15b).

Fig. 2.15

Two examples of showers: a Particle shower caused by a proton. (Source: Auger Collaboration). b Development of an electromagnetic shower as a sequence of bremsstrahlung and pair formation processes

The electrons and positrons in the electromagnetic shower radiate Cherenkov radiation in a cone that has a diameter of about 120 m on the ground. This light is focused on a grid of photosensitive detectors by the telescope mirror. The photomultiplier tubes (PMT) react to individual photons. The arrangement of several telescopes (e.g., the H.E.S.S. telescopes) allows for particularly accurate direction determination (about 0.1$^{\circ }$ with H.E.S.S.).

H.E.S.S.

H.E.S.S. is a telescope for Cherenkov gamma rays. So, the primary goal of H.E.S.S. is to determine extragalactic gamma-ray sources. However, high-energy gamma radiation can also be a secondary product that is produced when the accelerated particles collide with surrounding matter. A newer image of H.E.S.S. is shown in Fig. 2.16.

Fig. 2.16

(Source: H.E.S.S. Collaboration)

H.E.S.S. arrangement in Namibia.

To register photons between 50 GeV and 50 TeV, the reaction with the Earth’s atmosphere is used.³ So, it is an indirect observation. Predominantly electrons and positrons penetrate into the lower atmospheric layers very quickly (with local superluminal speed in the atmosphere) in the form of so-called air showers and generate the visible Cherenkov light, i.e., ultra-short bluish light flashes with a duration of only a few billionths of a second. The air showers consisting of secondary particles reach their maximum at a height of about ten kilometers and fade out in lower atmospheric layers. Through multiple repetition, tightly bundled Cherenkov radiation is created, which is registered in telescopes as a blue flash of light. A photon of gamma radiation with an energy of one trillion electron volts ultimately only generates about 100 photons of visible light per square meter of ground area, and it is precisely this tiny amount of light that was measured with the initially four telescopes.

Gamma rays with energies from 100 keV to dozens of GeV can also be observed with satellites outside the Earth’s atmosphere. Gamma ray bursts (GRBs) in this energy range are radiation events in which more energy is emitted within ten seconds than our sun emits during its entire lifetime. Since 2008, the Fermi satellite has been in use for the evaluation of such events.⁴  

Models for GRBs assume gigantic supernova explosions (hypernovae) or collisions of two neutron stars as causes. The gamma radiation originates from the emission of charged particles that are accelerated to relativistic energies.

Neutrino Astronomy

The relatively undisturbed propagation over cosmological distances makes high-energy neutrinos interesting messenger particles for astronomical processes and events that lead to the generation of the huge energies observed in the charged and electromagnetic component of cosmic radiation. The low probability of neutrino interactions is helpful for undisturbed observations, but on the other hand, detection requires very large detector volumes. So far, no high-energy (above the TeV scale) neutrinos from astronomical point sources have been observed. A breakthrough is expected with detectors of the latest generation, cubic kilometer-sized detectors in ice and water.

In addition to electromagnetic radiation, particle radiation in the form of neutrinos is another important window into space. Neutrinos were postulated in the explanation of beta decay

$$\begin{aligned} \boxed { n \rightarrow p + e^- + \bar{\nu }_e } . \end{aligned}$$

(2.83)

The interaction with matter is very low, as the cross sections of the reactions

$$\begin{aligned} \nu _e +n\rightarrow & {} p + e^- , \end{aligned}$$

(2.84)

$$\begin{aligned} \bar{\nu }_e +p\rightarrow & {} n + e^+ \end{aligned}$$

(2.85)

are very small. At a typical neutrino energy of 1 MeV, they are less than $10^{-43} \,$ cm$^2.$ This naturally makes the detection of neutrinos difficult, despite their large number. Conversely, the Earth’s atmosphere is not a hindrance.

Neutrinos can, for example,

• originate from cosmic background radiation (signals of nuclear processes after the Big Bang),

• be generated by the interaction of cosmic radiation with the Earth’s atmosphere,

• come from the nuclear reactions inside the sun (should be observed most frequently here),

• come from the radiation of a collapsing supernova 1 kpc away.

The detection of neutrinos is carried out radiochemically. Chlorine, argon, and gallium atoms are suitable for this. However, the corresponding detectors have no directional and only a very coarse time resolution. Things look better with the water detector Kamiokande or Super-Kamiokande.

Nobel Prize 2002

Here comes  again the Nobel Prize 2002 for Physics (in part). For the detection of solar and cosmic neutrinos, Raymond Davis Jr. and Masatoshi Koshiba  were awarded.

About 50 years ago, Davis had a tank with 615 tons of tetrachloroethylene installed in a gold mine in South Dakota. It contained approximately $2 \times 10^{30}$ chlorine atoms, which according to the prediction of a standard solar model should capture 20 neutrinos each month and thus produce 20 argon atoms. In fact, significantly fewer were detected until 1994. Koshiba installed a huge water tank as a neutrino detector in a mine in Japan with the Kamiokande. When neutrinos collide with the atoms of the water, electrons are produced, whose light flashes can be detected with photomultipliers placed all around. The results of Davis were confirmed. With the Kamiokande, it was also possible to determine the reaction time and direction of the neutrinos. It turned out that the neutrinos actually came from the sun. With the Kamiokande, Koshiba also registered twelve neutrinos in February 1987, which had been hurled into space during a supernova explosion in the Large Magellanic Cloud. This was the first detection of neutrinos originating from extragalactic space. With a larger detector, the Super-Kamiokande, which has been in operation since 1996, Koshiba finally solved the mystery of the “missing” solar neutrinos. There are three “oscillating” types of neutrinos that can transform into each other. One cannot conclude the total number of neutrinos from the measurement of one type. 

In the comparison between theory and experiment, the discrepancy arose for a long time that the detectors provided results that were 50–70% below the expected count rate for solar neutrinos (Table 2.1).

Table 2.1 Results of solar neutrino experiments, compared with the theoretical expectations of the standard solar model [19]

The theoretical prediction for solar neutrinos comes from the standard solar model with the pp cycle as shown in Fig. 2.17.

Fig. 2.17

Fusion process in the sun with $\nu _e$ neutrino formation

A crucial step in explaining the solar neutrino problem was achieved in 1998, when it was shown that muon neutrinos transform into tau neutrinos as they pass through the Earth. It is also said that they oscillate from one state to another. The convincing proof of the oscillations of solar neutrinos was presented in mid-2001. It is plausible to assume an oscillation of the electron neutrinos during their passage through the solar matter to explain the deficit of solar neutrinos. In this case, the electron neutrinos from the fusion processes could transform into muon or tau neutrinos, to which the detection reactions are not sensitive.

Nobel Prize 2015

This is where the Nobel Prize for Physics 2015 comes into play, which was awarded to Takaaki Kajita and Arthur McDonald for the proof of neutrino oscillations (and thus a finite mass of neutrinos).

Neutrino research has been honored with several Nobel Prizes. In 1988, Leon Max Lederman, Melvin Schwartz, and Jack Steinberger received the prize because they discovered the second generation of neutrinos, the muon neutrinos. Frederick Reins received the Nobel Prize only in 1995, although he was the first to discover a neutrino. In 2002, as already described, Raymond Davis Jr. and Masatoshi Koshiba received the Nobel Prize for the detection of cosmic neutrinos.

We are able to determine mass differences of the three types of neutrinos, but not the mass itself. However, it is doubted that the derived upper limits of the neutrino masses are sufficient to explain dark matter.

Currently, the following mass limits are given:

$$\begin{aligned} m(\nu _e)< 2 \, \text {eV} , \quad m(\nu _\mu )< 190 \, \text {keV} , \quad m(\nu _\tau ) < 18.2 \, \text {MeV} \; \end{aligned}$$

(2.86)

The neutrino masses are likely far below these limits.

Gravitational Astronomy

On a current note, here is a remark on the just beginning gravitational astronomy, i.e., the observations of the cosmos through gravitational waves (more on this in Part II).

Gravitational waves were predicted by Schwarzschild and Einstein from Einstein’s General Theory of Relativity. Only after Einstein’s death (1955) did a serious search for evidence begin, including by Joseph Weber. He used two tons-heavy cylinders more than 1000 km apart to detect excited oscillations in them with highly sensitive amplifiers – in vain, as it turned out despite some interim success reports.

Temporal changes in mass distributions trigger gravitational waves according to the General Theory of Relativity. A binary star system represents the simplest realization of oscillating masses. Figure 2.18 is intended to symbolize the configuration. However, the change in mass distribution as well as the direction and polarization of gravitational waves are not detailed there. Realistic examples of such “binary star systems”, which were used for the indirect detection of gravitational waves, are the pulsars PSR 1913 + 16, the quasar OJ287 (in whose center probably two black holes rotate around each other), J0651 and J0348 + 0432 (in which a neutron star and a white dwarf orbit each other).

Fig. 2.18

Binary star system and gravitational radiation

A small strain h of the Minkowski metric is caused by a change $L_{GW}$ (intensity $\sim$ luminosity $\sim$ energy $E_{GW}$ per time $\tau$). For the flux $F_{GW}$ at distance r, the theory states

$$\begin{aligned} F_{GW} \equiv \frac{L_{GW}}{4 \pi r^2} =\frac{c^3}{16 \ \pi \ G}\left( \frac{dh}{dt}\right) ^2 . \end{aligned}$$

(2.87)

The proportionality factor $\frac{c^3}{G} \approx 4 \times 10^{35}$ kg/s explains why it is so difficult to detect gravitational waves. If we solve for gravitational waves of frequency $f_{GW}$, which are emitted by an energy release in time $\tau$, for h at distance r, we find

$$\begin{aligned} \boxed { h \approx 10^{-22} \left( \frac{E_{GW}}{10^{-4} M_\odot c^2}\right) ^{1/2} \left( \frac{1 \, \text {kHz}}{f_{GW}}\right) \left( \frac{\tau }{1 \, \text {ms}}\right) ^{-1/2} \left( \frac{15 \, \text {Mpc}}{r}\right) . } \end{aligned}$$

(2.88)

Example 2.10

A supernova in the Virgo Cluster (15 Mpc) with an energy release of one thousandth of a solar mass per millisecond results in the kHz range $h \approx 10^{-22}$. Therefore, with a detector length of $l=4$ km, we must measure a length change of

$$\begin{aligned} \boxed { \Delta l \sim h l \sim 4 \times 10^{-17} \, \text {cm} } . \end{aligned}$$

(2.89)

This corresponds to the $10^{-9}$ times the diameter of a hydrogen atom. $\blacksquare$

Indirect Evidence

The indirect evidence of gravitational waves was achieved relatively early. Hulse and Taylor received the 1993 Nobel Prize in Physics for this. They had discovered the double pulsar PSR 1913 + 16 in 1974 and measured the period change due to gravitational wave radiation. We will return to this in Part II. Here, the principle is only indicated by the sketch in Fig. 2.19.

Fig. 2.19

Sketch of the change in pulsar rotation when radiating gravitational waves. The circular frequency $\omega$ changes because the pulsar loses overall energy by radiating gravitational waves

In addition to further indirect evidence (e.g., decrease in the orbital period of orbiting neutron stars due to radiation), there were experiments that utilized the principle of the Michelson interferometer for the detection of the relative change of two test masses. Such experiments were initiated by Heinz Billing. However, the laser arms were initially relatively short (30 m in Munich, 600 m in Hanover). The Laser Interferometer Gravitational-Wave Observatory (LIGO), which was put into operation in 2002, has arm lengths of four kilometers. A space laser interferometer (LISA) consisting of three identical satellites (which orbit the sun in the form of an almost equilateral triangle behind the earth along the earth’s orbit) is being planned as a joint project by NASA and ESA; a decision could, after several delays, be made in 2025. The satellites together form a laser interferometer with almost five million kilometers arm length. LISA could detect gravitational waves from supermassive black holes and perhaps even those waves that originate from the Big Bang.

Direct Evidence

A breakthrough in direct evidence was achieved by LIGO with a successful measurement on September 14, 2015, and the publication in February 2016. The 2017 Nobel Prize was the deserved reward. The end stage GW150914 in the merger of two black holes led to a strong emission of gravitational waves. The sketch in Fig. 2.20 serves to illustrate the principle.

Fig. 2.20

Sketch of the principle of radiation during the merger of two black holes

An impression of the experiment and the measurement results is conveyed by Fig. 2.21.

Fig. 2.21

(Source: LIGO Collaboration)

a LIGO detector in Livingston (Louisiana). b LIGO detector in Hanford (Washington). c The matching measurement series during the last tenths of seconds.

Evidence of Black Holes

Theorists believed in the existence of black holes after the fundamental work on the theory of general relativity. The work of Roger Penrose and colleagues recently provided a solid mathematical basis. In addition to stellar and primordial black holes, the observability of massive and supermassive black holes, which should be located in the centers of galaxies, was particularly discussed.  

Nobel Prize 2020

The Physics Nobel Prize of  the year 2020 went to three researchers who theoretically described (Roger Penrose) and demonstrated (Reinhard Genzel and Andrea Ghez) the existence of black holes. To investigate the structure and dynamics of such objects, Genzel and his colleagues developed a series of new observation techniques and instruments in the field of infrared, submillimeter, and millimeter astronomy. This allowed him and his team to initially prove at the La Silla Observatory and then at the Very Large Telescope (VLT) over long-term observations of the orbits of stars near the center of the Milky Way that there is a supermassive black hole of about 4.3 million solar masses. Independently, this was also achieved by astronomers around Andrea Ghez at the Keck Observatory.

The Event Horizon Telescope is a network of radio telescopes to investigate distant black holes using very long baseline interferometry (VLBI). Radio telescopes around the world record signals caused by the black holes. On April 10, 2019, the collaboration announced:⁵

Scientists have obtained the first image of a black hole, using Event Horizon Telescope observations of the center of the galaxy M87. The image⁶ shows a bright ring formed as light bends in the intense gravity around a black hole that is 6.5 billion times more massive than the Sun. This long-sought image provides the strongest evidence to date for the existence of supermassive black holes and opens a new window onto the study of black holes, their event horizons, and gravity.

Appendix: Satellite Projects

There are numerous space projects. The German Aerospace Center (DLR) alone lists 28 projects in 2020 in which it is involved:

• AMS: Search on the ISS for dark matter and antimatter

• BebiColombo: Mission to Mercury starting in 2025

• Cassini/Huygens: Mission to Saturn from 2004 to 2017

• Cluster II: Satellite quartet for the exploration of the magnetosphere

• CoRoT: Satellite project for the search for extrasolar planets

• Dawn: NASA mission for the exploration of asteroids

• eROSITA: X-ray experiment to explore dark energy

• Fermi: NASA space telescope for gamma ray detection

• Gaia: Astrometry mission to capture our Milky Way

• Herschel: ESA space observatory in the infrared range

• INTEGRAL: European gamma observatory

• LISA Pathfinder: Mission to prepare for the launch of LISA in 2034

• Mars Express: European Mars probe

• MICROSCOPE: French satellite mission to verify the general theory of relativity

• PAMELA: Search for dark matter

• Parker Solar Probe: Space probe to the Sun

• Planck: Satellite for the study of cosmic background radiation

• PLATO: Search for exoplanets (launch 2026)

• Rosetta: Exploration of a comet

• SOFIA: Aircraft-based infrared observatory

• SOHO: Solar observatory operating since 1995

• SolACES: EUV/UV spectrophotometer on the ISS

• Solar Dynamics Observatory: Observation of solar storms

• Solar Orbiter: Satellite for observing the Sun and the heliosphere

• STEREO: Observatory for coronal mass ejections

• Sunrise: Balloon telescope for solar research

• Ulysses: ESA space probe for the exploration of the Sun’s polar regions

• XMM-Newton: X-ray satellite