Download Our galaxy, the Milky Way, has about 3 billion solar masses of HI

Astronomy Assignment #6: Newton's Law of Gravity and Structure of the Milky Way
Your Name______________________________________
Your Class Meeting Time __________________________
This assignment is due on Friday, 16 Oct and Thursday, 15 Oct
Submit this cover sheet with your assignment.
Complete the assigned problems from the text listed below and address the Instructor Assigned Topic. Mathematical
problems may be hand written. Write out the problem, show your work in solving the problem and state your answer in a
complete sentence. Failure to complete all three of these tasks will result in less than full credit awarded.
Read the following sections in Astronomy Notes: Chapter 5 Newton's Law of Gravity
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Introduction
Universal Law of Gravity
o Characteristics of Gravity---why it is the most important force in astronomy.
Inverse Square Law
Gravitational Acceleration
o Measuring the Mass of the Earth
Read the following sections in Astronomy Notes: Chapter14: The Interstellar Medium and the Milky Way
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Introduction
Interstellar Medium (ISM)
o Dust
o Gas
Galactic Structure The Milky Way: our galaxy
o Period-Luminosity Relation for Variable Stars
o Our Location
o Our Motion
o Deriving the Galactic Mass from the Rotation Curve
Answer the following Review Questions from Nick Strobel’s AstronomyNotes: Chapter 5: Newton's Law of Gravity
1. What things does gravity depend on?
Gravity depends on three things. First, the strength of the gravitational attraction between two
objects depends directly on the mass of each object, or equivalently on the product of the masses of
each object. Second, the strength of the gravitational attraction between two objects depends
inversely on the separation of the two objects. In particular, the strength of gravity follows an
inverse square relation so that the strength of the gravitational attraction between two bodies depends
1
on 2 . Finally, the strength of the gravitational attraction depends on the intrinsic strength of
d
gravity as expressed through the Universal Gravitational Constant G. G has a very small value of
6.6710-11 in mks (meters-kilogram-seconds) units and illustrates that gravity is an intrinsically
weak force. So weak, that for gravity to be “felt” between two objects, at least one of the objects
must be astronomical in mass.
2. How does gravity vary with distance between objects and with respect to what do you measure the distances?
The strength of the gravitational attraction between two objects depends inversely on the separation
of the two objects. In particular, the strength of gravity follows an inverse square relation so that the
1
strength of the gravitational attraction between two bodies depends on 2 . The distance d is
d
measured from the center of one object to the center of the other object so long as they are spherical
in shape.
3. What would happen to the Earth's orbit if the Sun suddenly turned into a black hole (of the same mass)?
Why?
If the Sun suddenly turned into a black hole with the same mass as the Sun, the Earth would not feel
any difference in the gravitational attraction to the now-black-hole-Sun. The gravitational attraction
between any two objects depends only on the mass of each object and the separatin distance from
their centers – not on the radius of either object.
The Earth would get quite cold since black holes do not emit any radiation (i.e. heat). However, the
Earth’s orbit would remain unchanged.
4. Why is gravity called an ``inverse square law''?
Gravity called an ``inverse square law'' because its strength depends inversely on the square of the
1
distance between the two objects. In mathematical lingo FGravity  2 . To illustrate, if the
d
separation between two objects is increased by a factor of two, the strength of their mutual
gravitational attraction will decrease by a factor of four. I the separation between two objects is
decreased by a factor of five, the strength of their mutual gravitational attraction will increase by a
factor of 25.
5. What is the difference between a simple inverse relation and an inverse square relation?
The difference between a simple inverse relation and an inverse square relation is that a simple
inverse relation decreases much more slowly than an inverse square relation. For example, in a
simple inverse relation doubling the distance would simply halve the strength. However, with an
inverse square relation doubling the distance would decrease the strength by a factor of four. Inverse
square relations rapidly weaken with distance.
6. If the Earth was 3 A.U. from the Sun (instead of 1 A.U.), would the gravity force between the Earth and the
Sun be less or more than it is now? By how many times?
If the Earth was 3 A.U. from the Sun (instead of 1 A.U.), the gravity force between the Earth and the
Sun be less than it is now by 32 = 9 times?
7. If Mercury was 0.2 A.U. from the Sun (instead of 0.4 A.U.), would the gravity force between Mercury and the
Sun be less or more than it is now? By how many times?
If Mercury was 0.2 A.U. from the Sun (instead of 0.4 A.U.), the gravity force between Mercury and
the Sun be more than it is now by 4 times.
8. Why do astronauts in orbit around the Earth feel ``weightless'' even though the Earth's gravity is still very
much present?
Astronauts in orbit around the Earth are in free fall as they orbit the Earth. They feel weightless
because they are falling towards the Earth. They don’t hit the Earth because their horizontal velocity
(i.e. orbital velocity) is so large that as they fall they move over the “edge” of the Earth.
Answer the following Review Questions from Nick Strobel’s AstronomyNotes: Chapter 14: The Interstellar Medium and
the Milky Way.
1. What is the interstellar medium composed of?
The interstellar medium is composed of gas and dust. The gas is primarily hydrogen and helium, with trace
amounts of other gases. The dust is composed of microscopic (mostly) grains of ices (H2O, CO2, NH3, CH4) ,
silicate grains (SiO4), Carbon grains, and metals (Fe, Ni)
2. How does dust make stars appear redder than they actually are?
Dust makes stars appear redder than they are through a process called interstellar reddening. Interstellar
reddening is the process whereby light passing through an interstellar dust cloud is separated into two
components. The long wavelength red light component is allowed to pass through the interstellar cloud
because the wavelength of red light is longer than the typical diameter of the dust grains, while the bluelight
component is scattered (i.e. reflected) in all directions because its wavelength is about the same size as the
diameter of the dust grains. Thus much of the blue light from stars behind an interstellar dust cloud is
removed from the beam causing the star to look redder than it really is.
The front surface of the
cloud will look blue
from the scattered blue
star light and
astronomers call this a
reflection nebula.
The star appears
redder than it really
is because the blue
component of the
light is scattered out
of the beam.
3. How does dust cause the extinction of starlight?
Dust causes the extinction of star light by two processes. If the dust grains are very small, on the order of the
wavelength of light, then the light is scattered out of the beam reducing the beams intensity. If the dust grains
are larger than the wavelength of light, then the dust grains absorb the star light directly and heat up. The
heated dust grains will then re-emit the light energy at infrared wavelengths in all directions. A similar effect
can be seen in terrestrial clouds. If the cloud particles are small, then they reflect the sunlight and appear very
bright. Rain clouds, however, have much larger cloud particles and these larger ccloud particles absorb the
sunlight and appear dark gray in color.
4. Where is the dust thought to come from?
Dust grains form from the material in the outflows of stellar winds. Typically giant stars, planetary nebula and
supernovas create dust particles from their outflows.
5. What are the characteristics of the gaseous part of the ISM? Is the gas all at the same temperature and
density? How do you know?
The presence of interstellar gas can be seen when you look at the spectral lines of a binary star system and
you see narrow spectral lines that do not move among the broad stellar spectral lines that shift as the two stars
orbit each other. The narrow lines are from much colder gas in the interstellar medium between us and the
binary system.
The gaseous part of the ISM is composed primarily of hydrogen (90%), Helium (10%) and trace amounts of
other gasses (e.g. CO, H2O, NH3…). The hydrogen appears in three forms in the ISM; neutral atomic
hydrogen (HI), ionized hydrogen atoms (HII) and molecular hydrogen (H2). The ionized hydrogen emits light
in the visible band as the electrons recombine with the protons and the
neutral atomic and molecular hydrogen emits light in the radio band of the
electromagnetic spectrum.
Our galaxy, the Milky Way, has about 3 billion solar masses of H I gas with
about 70% of it further out in the Galaxy than the Sun. Most of the H I gas
is in disk component of our galaxy and is located within 720 light years
from the mid-plane of the disk.
Distribution of HI in the Galaxy
In our galaxy the H II regions are distributed in a spiral pattern associated
with the very luminous new O, B and A stars that define the spiral arms.
Molecular hydrogen H2 does not produce radio emission. Fortunately,
there is evidence of a correlation between the amount of CO and H2, so
the easily detected CO radio emission lines (at 2.6 and 1.3 mm) are used
to infer the amount of H2. The CO emission is caused by H2 molecules
colliding with the CO molecules. An increase in the density of the H2 gas
results in more collisions with the CO molecules and an increase in the
HII & GMC’s in the plane of the Milky Way
CO emission. There is some evidence that indicates that 90% of the H2 is
locked up in 5000 giant molecular clouds with masses greater than 105
solar masses and diameters greater than 65 light years. The largest ones, with diameters greater than 160 light
years, have more than a million solar masses and make up 50% of the total molecular mass.
The Milky Way has about 2.5 billion solar masses of molecular gas with about 70% of it in a ring extending
from 13,000 to 26,000 light years from the center. Not much molecular gas is located at 4,900 to 9,800 light
years from center but about 15% of the total molecular gas mass is located close to galactic center within
4,900 light years from the center. Most of the molecular clouds are clumped in the spiral arms of the disk and
stay within 390 light years of the disk mid-plane.
6. How is the 21-cm line radiation produced?
21-cm line radiation is produced when an electron in excited atomic hydrogen radiatively decays to the
ground state by emitting a photon of 21-cm wavelength in the radios portion of the electromagnetic spectrum.
Solitary hydrogen has only two allowed spins states (for reasons that only nature knows). Spin is analogous
to the spinning of a child’s top although it is quantum mechanical interpretation is less intuitive. An electron
and proton process a property a physicist calls “spin”. The electron and proton of the hydrogen atom can
spin in opposite directions – called anti-parallel spins – or the electron and proton can spin in the same
direction – called parallel spins. The state of parallel spins is just ever so slightly higher in energy – thus
parallel spins represent an excited state of the hydrogen atom. The lower energy antiparallel spins represent
the lower ground state. The energy difference between these two states is so slight that a hydrogen atom can
be excited simply by a collision with another atom. These collisions are common in interstellar clouds of gas
and dust. So there are a significant fraction of hydrogen atoms in an interstellar cloud that are in this parallel
spin excited state. The excited state, however, is unstable. Given a short time, the excited hydrogen atom
will decay back to the ground state by radiating a low energy 21-cm wavelength photon - so called radiative
decay.. These 21-cm photons reside in the radio portion of the electromagnetic spectrum. Although it is
impossible to detect one 21-cm photon from a cloud of interstellar gas, it is quite simple with a radio
telescope to detect the many many billions of 21-cm photons that a single cloud can emit. The detection of
21-cm radiation by radio telescopes along with a measurement of its Doppler-shifted velocity is a cornerstone
of modern astronomy and has been used to determine our place in the galaxy and properties of the galaxy.
7. Why is the 21-cm line radiation so important for determining galactic structure and mass?
The 21-cm line radiation is so important for determining galactic structure and mass because the 21-cm
photons can travel vast distances even through tick interstellar clouds of gas and dust that visual photons
cannot pass through. Radio telescopes are extraordinarily sensitive due to their very large diameters and very
sensitive electron detectors and amplifiers. 21-cm radio photons from neutral hydrogen clouds can be
detected even if they originate from the very opposite side of the galaxy.
8. How is the 21-cm line radiation used to determine galactic structure and mass?
The distance of individual neutral hydrogen clouds in the galaxy’s disk can be determined using their radial
velocities (that are determined from their Doppler shifts). Thus, astronomers can determine both the distance
to these clouds of hydrogen and their velocity. With the hydrogen cloud’s velocity and distance an
astronomer can assemble a galactic rotation curve from which the orbital velocity and orbital radius of
hydrogen clouds around the center of the galaxy can be deduced. With these two pieces of information the
orbital velocity equation can be used to determine the mass of the galaxy within each clouds orbital radius.
9. What is the name for our galaxy and what kind of galaxy is it?
Our galaxy is called the Milky Way Galaxy. It is a barred spiral galaxy that consists of a large (100,000 ly
diameter) disk of stars and gas and dust around a central bulge and bar ( galactic bars are still pretty
mysterious) all embedded in a larger (diameter about 150,000 ly) oblate spheroidal halo of dark matter,
globular clusters, and halo stars (very rare).
10. How big is our galaxy? How many stars are in it and how do we know?
It is a barred spiral galaxy that consists of a large (100,000 ly diameter) disk of stars and gas and dust around
a central bulge and bar ( galactic bars are still pretty mysterious) all embedded in a larger (diameter about
150,000 ly) oblate spheroidal halo of dark matter, globular clusters, and halo stars (very rare).
We cannot count the stars in our galaxy because we cannot see them all due to the dust in the disk of the
galaxy that limits our visibility to within 10,000 light years of the Sun. However, using the galactic rotation
curve derived from 21-cm radiation from interstellar neutral hydrogen clouds astronomers can calculate the
mass of our galaxy. From this mass, astronomers can estimate the number of stars.
For example, the mass of the Milky Way within the Sun’s orbital radius is about 1041 kilograms. If this mass
was solely portioned into sun-like stars, there would be about 1011 sun-like stars – about 100 billion stars.
However, we know that the most common stars in the Milky Way galaxy are not sun-like starsbut K & M
main sequence stars whose masses are a few tenths of the Sun’s mass. So, the number of K & M main
sequence stars equivalent to 100 billion sun-like stars is about 300 billion stars. The number of about 300
billion stars in roughly the number of stars astronomers believe reside in the Milky Way galaxy.
11. How are Cepheids and RR-Lyrae stars considered to be standard candles? How can you find their
luminosity?
Cepheids and RR-Lyrae variable stars are standard candles because their luminosities are either known “a
priori” or can be determined. The luminosity of these objects must first be established by directly measuring
their distance by some means such as stellar parallax or some other method. Once the luminosity a these stars
are measured (by comparing their apparent magnitudes and their distance) and once the luminosity is
established as constant for all objects of that class, then the luminosity of one of these standard candles can be
safely assumed when it is seen in another location.
12. How can you use the period-luminosity relation to find distances?
The luminosity of a Cepheid is found to correlate
very closely to its period of oscillation; the longer
the oscillation period the greater the peak
luminosity of the star. Thus the luminosity of a
Cepheid can be determined by measuring its period
of oscillation. Form the period of oscillation, the
absolute magnitude can be determined. For
example, if the period of oscillation of a Type I
Cepheid variable was 10 days, then its absolute
magnitude would be about 3.9. If you measured
the apparent magnitude of the Cepheid at peak
brightness, you could then find its distance using
the distance modulus (m-M). For example if the
Type I Cepheid with the 10 day period had a peak
apparent magnitude of 9 (you need a telescope to
see it), the distance modulus would be (m-M) = 6.1
and the distance is given by
mM 5
Dpc  10
5
 166 pc.
13. Where are we in the galaxy and how do you know? How can the distribution of globular clusters tell you
about our place in the Galaxy?
Our place in the Milky Way galaxy is defined by comparing our position to the center of the distribution of
globular clusters. Globular clusters are not confined to the disk of our galaxy and therefore are visible beyond
the 10,000 ly visibility limit imposed by dust within the Milky Way’s disk. We can see globular clusters
many tens of thousands of light years away and measure their distances using the standard candles (Cepheid
Variables and RR Lyrae) that they contain in abundance. Thus, we know that the Sun is approximately 9,000
parsecs from the center of the globular cluster distribution. This center astronomers had previously assumed
and now know represents the center of the Milky Way galaxy. Thus globular clusters were used to define
where the center of the galaxy is and our distance from it.
14. What are the four basic components of our galaxy?
The four basic components of our galaxy are the bulge/bar/core, the disk, the halo, and the dark matter
envelope.
Component
Composition
Dimensions
Bulge is about 5 kpc in diameter. The supermassive
Old (Population II) and new
black hole is located at the galactic center within 10
(Population I) stars, a
Bulge/Bar/Core
light days of the center. The bar is a complex structure
supermassive black hole (about 4
that is not well understood and is around 9 kpc long
million solar masses)
inclined relative to our position in the galaxy
36 kpc in diameter and 1 kpc in thickness. The gas
Old (Population II) and new
layer is about 250 pc thick and is centered along the
(Population I) stars with gas and galactic equator. The Sun is about 9kpc from the
Disk
dust concentrated into a thinner
galactic center situated between the two dominant
gas layer.
spiral arms of the galaxy in the smaller Orion Spur
spiral arm fragment.
Globular Clusters, Halo Stars
An oblate spheroid about 50 kpc in diameter along its
Halo
(Both Population II stars)
equatorial plane.
Dark Matter
The composition of dark matter
is as mysterious today as it was
when it was first recognized in
the second half of the 1900’s.
Astronomers have no idea what
dark matter is.
Most models of the galaxy suggest that dark matter is
distributed principally within the galactic halo.