Download Astrophysics by Daniel Yang

ASTROPHYSICS
1. Our understanding of celestial objects depends upon observations made from Earth or
from space near the Earth
- Discuss Galileo’s use of the telescope to identify features of the Moon
Galileo was not the inventor of the telescope, but he built his own that was clear enough to observe the moon. He
identified features on the moon that contradicted the Aristotelian view that the heavens were perfect and
unchanging. He found that the moon was rough like the earth and had vast plains, high mountains and deep valleys.
He calculated the height of mountains by measuring the shadows cast by the mountains on the moon, and found
that some were several kms high. This was significant in providing evidence for the heliocentric model of the
universe by challenging fundamental beliefs about the universe at the time.
- Discuss why some wavebands can be more easily detected from space
The atmosphere and ionosphere filter out most forms of electromagnetic radiation except visible light and radio
waves (high frequency radio waves, more specifically). This means that most types of electromagnetic radiation are
prevented from reaching the Earth’s surface. As a result, ground-based telescopes can only be operated using visible
light or radio waves. All other wavebands are more easily detected outside of the Earth’s atmosphere.
- Define the terms ‘resolution’ and ‘sensitivity’ of telescopes
The resolution of a telescope is the ability of a telescope to clearly distinguish between two close objects. It is
measured as an angle, and the larger the diameter of the primary light gathering lens or mirror, the better the
resolving power of the telescope. A smaller angle indicates a higher resolution.
2.1 × 105 𝜆
𝑅=
𝐷
R = Theoretical resolution (arcsec)
λ = Wavelength (m)
D = Diameter of telescope’s primary mirror/lens (m)
[1 arc second =1/3600 of a degree]
Sensitivity is the light-gathering power of a telescope. A telescope with high sensitivity can collect large amounts of
light, allowing faint objects to be observed. It is largely determined by the size of the mirror/lens – the bigger the
mirror/lens, the more light is collected and focused and the more sensitive it is. Thus sensitivity is directly
proportional to the diameter/radius of the mirror/lens (For circular mirrors/lens, sensitivity∝r2)
- Discuss the problems associated with ground-based astronomy in terms of resolution and absorption of radiation
and atmospheric distortion
Atmospheric distortion and Resolution
The problem with ground-based astronomy is that the atmosphere is in between the source and the telescope – in
between is a constantly changing atmosphere of gases and dusts that lead to atmospheric distortion. The unstable
nature of the atmosphere leads to constant changes in the density of the atmosphere which in turn produces
fluctuations in its refractive index. The light is bent in different directions, blurring the light source. This is known as
seeing – the twinkling and blurring of a distant light source due to atmospheric distortion. The effect causes images
to blur to about 1 arcsec. This lowers the real resolution of a telescope from its theoretical resolution.
Another effect that affects ground-based astronomy is the distortion of the colour of light passing through the
atmosphere. The scattering of blue light (shorter wavelength) leads to a dominance in red when the light has to
travel further (eg at sunset) so ground-based astronomers must take this effect into account.
Absorption of radiation
Gamma rays, X-rays, UV rays, infrared radiation, and long λ radio waves all are filtered out by the Earth’s
atmosphere and ionosphere. Ground-based astronomy using these wavebands is nearly impossible due to the very
low intensity of radiation received. To combat this, telescopes that use these wavebands must be placed either high
in the atmosphere or above the atmosphere (eg Hubble Space Telescope).
- Outline methods by which the resolution and/or sensitivity of ground-based systems can be improved, including:
Active optics
Active optics uses a slow feedback system to correct for changes in the surface shape of the primary mirror of
reflector telescopes. These deformities may occur due to gravity at different inclination angles (sagging) or due to
temperature changes (expansion and contraction). The back of the primary mirror is fitted with many actuators that
can push or pull the mirror into a desired shape. Active optics involves using a wavefront sensor to monitor the
distortion of the reflected light, and correcting the shape of the mirror using the actuators accordingly. By sampling
slowly, the effect of atmospheric distortion is removed and the remaining distortions are due to the shape of the
mirror.
Adaptive optics
Adaptive optics uses a fast feedback system to correct for atmospheric distortion by changing the shape of a
secondary deformable mirror. Similar to active optics, actuators are also fitted on the back of the deformable mirror
and a wavefront sensor is also used. Adaptive optics uses the wavefront sensor to monitor the received light for
atmospheric distortion which is analysed by a computer. The actuators then bend the deformable mirror to correct
for atmospheric distortion and improve resolution. These corrections are made very rapidly, up to 1000 times a
second.
Interferometry
Interferometry is used with radio telescopes to overcome their low resolution (large λ), although the technique can
be used for optical telescopes as well. Interferometry works by superposition of multiple signals from an array of
many telescopes to improve resolution and sensitivity. Signals from each of the telescopes in the array are combined
to form an interference pattern. A computer then analyses this pattern and converted into an image with a
resolution similar to that of a single telescope with a much larger diameter. The Very Large Array in Mexico uses
interferometry – each dish is 25m in diameter, but when an array 36 km across is combined, it provides the
equivalent resolution of a dish with 36km diameter and sensitivity of a dish with 130m diameter.
2. Careful measurement of a celestial object’s position in the sky (astrometry) may be used
to determine its distance
- Define the terms parallax, parsec, light-year
Parallax refers to the apparent change in position of a nearby object against a distant background due to a change in
the position of the observer.
A parsec is defined as the distance from Earth to a point that has an annual parallax of one arcsec. 1 pc = 3.26 ly.
A light-year is the distance travelled by light in one earth-year. 1 ly = 9.4605 x 1015 m.
- Explain how trigonometric parallax can be used to determine the distance to stars
Trigonometric parallax is the technique used to determine the distance to an object by using trigonometry and
parallax.
tan 𝜃 =
𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
The annual parallax, p (arcsec), is half the angle through which a star appears to shift over a 6 month period.
The distance of a star from earth can be calculated if the annual parallax of the star is known.
𝑑=
1
𝑝
d = Distance from Earth (pc)
p = Annual parallax (arcsec)
- Discuss the limitations of trigonometric parallax measurements
Using trigonometric parallax relies heavily on an accurate measurement of the parallax. This places a limitation on
ground-based telescopes which experience atmosphere distortion or seeing, making small parallax measurements
hard to achieve. The smallest parallax that can be measured from the ground is about 0.03’’, leading to a maximum
distance measurable of about 33 pc. Any further away and the parallax becomes too small, not able to be measured
accurately. One way to combat this limitation is to place the telescope outside of the atmosphere to eliminate
atmospheric distortion. For example, the HIPPARCOS telescope could measure parallax to a precision of 0.001’’,
extending the maximum distance measurable to 1000 pc.
4. Photometric measurements can be used for determining distance and comparing objects
- Account for the production of emission and absorption spectra and compare these with a continuous blackbody
spectrum
Continuous spectra are produced by hot bodies, such as in lightbulbs and stars. They emit radiation over all
wavelengths, but at varying intensities.
Wien’s Law:
𝜆𝑚𝑎𝑥 𝑇 = 𝑊
λmax = Dominant wavelength (m)
T = Surface temperature (K)
W = 2.9x10-3
Stefan’s Law:
𝐿 = 4𝜋𝑅 2 𝜎𝑇 4
L = Luminosity (Js-1)
T = Temperature (K)
R = Radius of star (m)
σ = 5.6705x108 Wm2K4
Emission spectra are produced by hot glowing or excited low density gases. When an electron in its ground state
receives energy and elevates to a higher energy level, it soon releases this energy as a photon at a certain frequency
as it returns to ground state. The emitted energy is the difference between the two energy levels.
𝐸2 − 𝐸1 = ℎ𝑓
E2 = Energy of excited state (J)
E1 = Energy of ground state (J)
h = Planck’s constant (6.6x10-34 Js)
f = Frequency (Hz)
Absorption spectra are produced by cool non-luminous gases placed in front of a source of continuous spectra. The
gas absorbs certain frequencies and re-emits them in all directions, leaving the spectrum deficient in those
wavelengths. The frequencies absorbed are identical to the frequencies it would emit when incandescent.
- Describe the technology needed to measure astronomical spectra
To measure spectra, light from the source must be collected then dispersed to form a spectrum. Afterwards, the
spectrum can be recorded and analysed. The light can be dispersed by a diffraction grating, glass prism or
spectroscope, and can be captured on photographic film or, preferably, CCDs.
- Identify the general types of spectra produced by stars, emission nebulae, galaxies and quasars
Stars produce absorption spectra. The inner layers of the star which are hot dense gases produce a continuous
spectrum, but the cooler outer atmosphere absorb particular frequencies of light and cause dark spectral lines.
Emission nebulae, hot interstellar gas clouds at low pressures, produce emission spectra. They are heated by
radiation from nearby stars and emit their own light at the specific wavelengths of the gases that make it up, usually
red since they are mostly hydrogen gas.
Galaxies produce continuous spectra, a combination of the emission spectra emitted by nebulae and the absorption
spectra emitted by the billions of stars.
Quasars, extremely distant and more luminous than hundreds of galaxies, produce emission spectra. They are
associated with black holes at the centre of galaxies. It is thought that the matter being accelerated by the black
hole’s gravity causes the emission spectra.
- Describe the key features of stellar spectra and describe how these are used to classify stars
By studying the spectra of stars including the intensity and thickness of spectral lines and the peak wavelength, we
can classify stars into spectral classes based on the spectral lines evident and the surface temperature/colour. O is
hottest and M is coolest, and each class is subdivided into 10 sub-classes (O0, O1… O9 etc.)
- Describe how spectra can provide information on surface temperature, rotational and translational velocity,
density and chemical composition of stars
Surface temperature
The surface temperature can be found from the spectrum of a star in two ways; its surface temperature can be
found by determining its spectral class, or the intensity versus wavelength graph can be plotted and the peak
wavelength can be used to determine the surface temperature using Wien’s Law.
Rotational velocity
A rotating star will exhibit red shifting from the receding side and blue
shifting from the approaching side. The spectral lines are therefore
shifted in both directions and are broadened. The faster a star is rotating,
the more broadening is observed and this can be used to determine the
rotational velocity of the star.
Translational velocity
The velocity at which a star is moving towards or away can be determined from the blue or red shift apparent in the
star’s spectrum. By measuring the extent of red or blue shift, the radial velocity can be found.
Density
High density stars have greater broadening of spectral lines compared to lower density stars. The particles in low
density gases travel further before colliding with other particles, producing sharper spectral lines. The particles in
higher density gases produce broader spectral lines.
Chemical composition
By observing the absorption spectrum of a star and matching the deficient wavelengths with the known emission
spectra of the elements, we can determine the elements in the stars cooler outer layers. Spectra of molecules such
as TiO can be observed in cooler stars such as M class stars, but at higher temperatures only stronger molecules such
as CH and CN may be observed. In addition, neutral atoms can usually only be found in lower temperature stars.
These atoms with ionise in hotter stars; easily ionised atoms such as Na and Ca will exist in ionised states in G stars
and hotter, while ionised helium will only exist in O stars.
4. Photometric measurements can be used for determining distance and comparing objects
- Define absolute and apparent magnitude
Absolute magnitude (M) is how bright a star would be if it were 10 pc away.
Apparent magnitude (m) is how bright a star is viewed from Earth.
Both absolute and apparent magnitude are placed on the same scale. The lower the number, the brighter the star.
The brightness ratio of two stars can be calculated using the following:
𝑚𝐵 −𝑚𝐴
𝐼𝐴
= 100 5
𝐼𝐵
𝐼𝐴
𝐼𝐵
= Brightness ratio of A:B
mB = Apparent magnitude of B
mA = Apparent magnitude of A
- Explain how the concept of magnitude can be used to determine the distance to a celestial object
Using trigonometric parallax, we can measure the distance of close stars. However, the distance to a star that is
further away than the parallax measurement allows can be found by using both its apparent and absolute
magnitude.
𝑀 = 𝑚 − 5 log
𝑑
10
Where M = absolute magnitude
m = apparent magnitude
d = distance (pc)
- Outline spectroscopic parallax
Spectroscopic parallax is a method of using the Hertzsprung-Russell diagram to determine the absolute magnitude of
a star knowing its spectral class and luminosity class. With the apparent magnitude measured, the distance to the
stat can be determined using the distance modulus formula.
- Explain how two-colour values (eg colour index, B-V) are obtained and why they are useful
𝐶𝑜𝑙𝑜𝑢𝑟 𝑖𝑛𝑑𝑒𝑥 = 𝐵 − 𝑉
B = Photographic magnitude
V = Visual magnitude
Photographic magnitude is the magnitude of a star as measured by photographic film, or a photometer fitted with a
blue filter of centre wavelength 440 nm. Visual magnitude refers to magnitude as judged by the eye, or a
photometer fitted with a yellow-green filter of centre wavelength 550 nm.
A red star will have a low V magnitude but a high B magnitude. Thus, its colour index will be positive.
A blue star will have a low B magnitude but a high V magnitude. Thus, its colour index will be negative.
By definition, stars of spectral class A0 have a colour index of 0.
Once the colour index is known, the spectral class can be determined accordingly and spectroscopic parallax applied
to find the distance to the star.
- Describe the advantages of photoelectric technologies over photographic methods for photometry
Photometry is the measurement of the brightness of a light source such as a star. Photoelectric techniques are now
more common than photographic techniques for a number of reasons.
Photographic photometry involves the use of photographic emulsions to record a photograph of a portion of the sky.
Once the photograph has developed, the size and area of each star is measured. Brighter stars appear as a larger,
denser spot, and the stars’ magnitude is determined in this manner.
Photoelectric photometry use a combination of a filter and an electronic sensor such as a charge-coupled device
(CCD) or photomultiplier tube to convert the input light into an electronic signal that can be multiplied, digitised,
analysed and stored electronically.
Advantages of photoelectric technologies:
- Sensitive to a much wider range of wavelengths compared to photographic emulsions.
- More sensitive to faint light sources compared to photographic film.
- Photoelectric techniques are much faster than photographic.
- Information can be collected remotely.
- Easier to transmit and store data because it is digital.
5. The study of binary and variable stars reveals vital information about stars
- Describe binary stars in terms of the means of their detection: visual, eclipsing, spectroscopic and astrometric
A binary star system consists of two stars orbiting a common centre of mass.
Visual binary
A visual binary can be resolved by a telescope with a small enough resolution. To
ascertain if they are actually in motion around each other, they may need to be
observed for many years. The more massive star orbits in a smaller ellipse around
the centre of mass.
Eclipsing binary
An eclipsing binary is a binary system whose orbital plane is parallel to our observation. Periodically, each star
eclipses the other star causing a decrease in brightness. The time between successive primary minima or secondary
minima indicates the period of motion. The duration of the eclipse can also tell us the relative diameter of the stars;
stars that are similar in size will have a sharp V minimum while stars of different sizes have a broader U minimum.
Spectroscopic binary
A spectroscopic binary is one that cannot be resolved and do not orbit in such as plane as to be seen as an eclipsing
binary from earth. However, examination of the spectrum reveals simultaneous red and blue shifting of spectral
lines as the two stars move in different directions.
Regular observation of the spectrum of spectroscopic binaries can reveal its period.
Astrometric binary
In an astrometric binary, one on the stars is too faint to be observed, but a wobble can be detected in the visible
star’s motion. It can be inferred that there must be an unseen partner to the star to cause such a deviation in
motion. Measurement of the wobble can tell us the period and the separating distance, leading to an estimation of
mass.
- Explain the importance of binary stars in determining stellar masses
We can find the mass of binary stars as opposed to an isolated star, which we have no direct way of measuring the
mass of since we cannot observe its gravitational effect on another mass.
4𝜋 2 𝑟 3
𝑚1 + 𝑚2 =
𝐺𝑇 2
m1 = Mass of star 1 (kg)
m2 = Mass of star 2 (kg)
r = Distance of separation of the stars (m)
T = Orbital period of the binary star system (s)
𝑚1 =
𝑀(𝑟 − 𝑟1 )
𝑟
m1 = Mass of star 1 (kg)
M = Combined mass (kg)
r = Distance of separation of the stars (m)
r1 = Distance of star 1 from the common centre of mass (m)
By plotting the luminosity of main sequence binary stars versus mass, the mass-luminosity relationship was found.
This allows us to estimate the mass of single stars knowing their luminosity.
𝐿 ∝ 𝑚𝑎𝑠𝑠 4
- Classify variable stars as either intrinsic or extrinsic and periodic or non-periodic
Variables are stars that vary in brightness over time. They can be classified as intrinsic or extrinsic and then further
classified if intrinsic.
Extrinsic variables
Extrinsic variables do not vary in luminosity but only in brightness; the observed variation in brightness is due to an
external process.
Intrinsic Variables
Intrinsic variables are stars which vary in brightness due to changes within the star itself, and are further classified
into periodic and non-periodic.
Periodic variables
Periodic variables are intrinsic variables which display a regular periodic variation in brightness. They are also known
as pulsating variables, and the variation in brightness is caused by disequilibrium between the gravitation force and
radiation pressure of the star.
Non-periodic variables
Non-periodic variables are intrinsic variables whose variations in luminosity are not regular.
- Explain the importance of the period-luminosity relationship for determining the distance of cepheids
Cepheid and RR Lyrae stars are of particular importance to
astronomers because they can be used to measure distance.
Cepheid variables have a characteristic light curve, exhibiting a sharp
increase followed by a slow decrease in brightness. It was found that
Cepheids with longer periods were also more luminous. This is
known as the period-luminosity relationship.
By finding the period of a Cepheid variable
and comparing it to the period-luminosity
graph, its absolute magnitude can be found.
Knowing its absolute magnitude and by
finding its apparent magnitude from direct
observation, its distance can be found via the
distance modulus formula. RR Lyrae stars are
also useful for measuring distance; they
always have an absolute magnitude of about
+0.6. Cepheids and RR Lyrae stars are used as
standard distance measurement references
in space.
6. Stars evolve and eventually ‘die’
- Describe the processes involved in stellar formation
Within vast clouds of dust and gasses (mostly hydrogen), “clumps” of matter which are sufficiently cool and massive
will begin to contract under their own gravity. It begins slowly but as it draws itself in becomes faster, as the
gravitational force increases. This causes the density at the centre to increase quickly, forming a rapidly contracting
core and a more slowly contracting surrounding. The gravitational energy lost as gravitational collapse occurs is
converted into heat, creating an outward heat pressure that opposes the gravitational collapse. As the core gets
hotter and hotter, this pressure eventually stops the collapse and stabilises the size of the core, forming a protostar.
As more and more gravitational collapse occurs in the protostar, it may reach a high enough temperature (about
8000K) to trigger the nuclear fusion of hydrogen in its core. It is now a zero-age main sequence star.
- Outline the key stages in a star’s life in terms of the physical processes involved
Main sequence star – H fusion into He in the core, displacing H into the shell around the He core
Red giant – He fusion into C, O, and heavier elements (up to Fe), H fusion in the shell around the core
- Describe the types of nuclear reactions involved in Main-Sequence and post-Main Sequence stars
Main sequence stars
The conversion of H to He takes place by one of either two thermonuclear pathways. The p-p chain is dominant in
smaller, cooler main sequence stars of core temperature less than 1.6x107 K, while the CNO cycle dominates in
larger, hotter main sequence stars of core temperature more than 1.6x107 K.
THE PROTON-PROTON CHAIN
4 11𝐻 → 42𝐻𝑒 + 2𝑒 + + 2𝜈 + 2𝛾
1
1𝐻
+ 11𝐻 → 21𝐻 + 01𝑒 + 𝜈
1
1𝐻
3
2𝐻𝑒
+
2
1𝐻
3
1𝐻𝑒
→
+𝛾
]
x2
+ 32𝐻𝑒 → 42𝐻𝑒 + 2 11𝐻
The two chargeless and massless neutrinos (ν) are carried away at the speed of light. The net energy release of the
p-p chain is 4.278x10-12 J, in the form of gamma rays and neutrinos.
THE CNO CYCLE
4 11𝐻 → 42𝐻𝑒 + 2𝑒 + + 2𝜈 + 3𝛾
+ 126𝐶 →
13
7𝑁
+𝛾
13
7𝑁
→
13
6𝐶
+ 01𝑒 + 𝜈
1
1𝐻
+ 136𝐶 →
14
7𝑁
+𝛾
1
1𝐻
+ 147𝑁 →
15
8𝑂
+𝛾
15
8𝑂
15
7𝑁
+ 01𝑒 + 𝜈
1
1𝐻
1
1𝐻
→
+ 157𝑁 →
12
6𝐶
+ 42𝐻𝑒
In the CNO cycle, C-12, N-13, C-13, N-14, O-15 and N-15 are all catalysts. The net energy release of the CNO cycle is
4.289x10-12 J, in the form on gamma rays and neutrinos, higher than that of the p-p chain.
Post-main sequence stars
Once a main sequence star has consumed most of its H fuel, the star begins to collapse into its core. If the star is not
massive enough, the rise in temperature due to the loss of gravitational energy will not be sufficient to trigger He
fusion, and becomes a white dwarf with no nuclear fusion. If the mass of the star is sufficient, He fusion into heavier
elements is triggered called the core helium flash.
He is fused into C and then into O via the triple alpha reaction:
3 42𝐻𝑒 →
12
6𝐶
12
6𝐶
+ 42𝐻𝑒 →
+𝛾
16
8𝑂
+𝛾
Once the He supply has been exhausted, if the star is very massive then further
shell-burning can take place. This converts C to Ne and Mg, O to Si and S and then
Si and S to Fe. Here the energy source ends as further fusion past Fe becomes
endothermic rather than exothermic, providing no outward heat pressure.
- Discuss the synthesis of elements in stars by fusion
Elements are synthesised in stars from the nuclear fusion of H to He via the p-p chain or the CNO cycle, or the postmain sequence fusion of He to C and O, C to Ne and Mg, O to Si and S and finally Si and S to Fe.
- Explain how the age of a globular cluster can be determined from its zero-age main sequence plot for a H-R
diagram
A star cluster consists of many hundreds of thousands of stars that are similarly aged, that formed from the same
cloud at the same time. There are two distinct types of star clusters: globular and open clusters. Globular clusters are
older and have more stars than open clusters.
The age of a cluster can be determined by its turn-off point – the point at which the cluster leaves the main
sequence. An older cluster will have a lower turn-off point, while a younger cluster will have a higher turn-off point.
Past the turn-off point, the larger, more massive stars have depleted their supply of hydrogen and have moved off
the main sequence and have become red giants. Older clusters have a lower turn-off point since most of the stars in
the clusters have already evolved past the main sequence.
- Explain the concept of star death
Planetary nebula
Once a red giant of mass ≤5M⊙ runs out of fuel, it sheds its outer layers into space as a planetary nebula. The core
collapses under its own gravity to form a white dwarf.
Supernovae
Once a red giant of mass >5M⊙ runs out of fuel, it explodes in a supernova; the energy released can be greater than
an entire galaxy for a few days or weeks. What remains depends on the mass of the remaining core: a white dwarf,
neutron star or black hole.
White dwarfs
A white dwarf is a dense, collapsed star corpse, composed of degenerate matter. The pressure of the high-speed
electrons within this matter, known as electron degeneracy pressure, is all that is countering gravitational collapse. A
non-rotating white dwarf must have a mass less than 1.4M⊙, known as the Chandrasekhar limit, otherwise its gravity
will collapse it into a neutron star. A white dwarf will eventually cool into a black dwarf.
From main sequence star: If, once the H fuel runs out, the mass of the star is too small, then the star will collapse
into a white dwarf.
From planetary nebula: The remaining core after the outer layers are shed as a planetary nebula forms a white
dwarf.
From supernova: If the mass of the remaining core is less than 1.4M⊙, the star will collapse into a white dwarf.
Neutron stars/pulsars
If the mass of the remaining core after a supernova is between 1.4 and 3M⊙, the gravity of the star will be sufficient
to collapse electrons into protons forming a neutron star. A rapidly rotating neutron star emitting radiation is
detected as a pulsar.
Black holes
If the mass of the remaining core after a supernova is greater than 3M⊙, the remnants collapse into a black hole.
- Present information by plotting on a H-R diagram the pathways of stars of 1, 5 and 10 solar masses during their
life cycle