to the Course
Scale of the Cosmos
Celestial Sphere, the
and the Calendar
Models of the Universe)
including Tycho Brahe
Origin of Modern
in the Sky:
for the seasons)
and the Sky
of light and matter
(a gentle introduction to atomic
of light and the
History and Tools of Astronomy
to the Course
Tour of the Universe
Scale of the Cosmos
-- our Earth, Moon, individual planets,
the solar system, nearby stars, and star clusters, the Milky Way galaxy,
the Local Group, and galaxy clusters and superclusters (know their sizes
in ascending order and their general appearance)
= rate x time
13-20 billion years
age of the universe
relationship between speed, distance, and time
speed of light: c = 3x105
astronomical length scales
Astronomical Unit (A.U.) = avg. distance
between Earth and Sun
Light Year (l.y.) = distance traveled by light
in one year
scientific notation and powers of Ten
= 1,000,000,000,000 trillion tera-
1,000,000 million mega-
1000 thousand kilo-
= 0.0000000001meter Angstrom
(be able to do calculations using scientific
Increasing length scales (x100 each step):
-- 1 mm: size of an ant
-- 10 cm: size of a flower
-- 10 m: size of a boxing ring or
~10 yards on a football field
-- 1 km: several dozen city blocks
or ~5/8 of a mile
-- 100 km: distance from here to
Panama City Beach for spring break
-- 10,000 km: almost the size of
-- 1,000,000 km: just over the diameter
of the Moon's orbit
-- 108 km:
the radius of the Earth's orbit; light takes 5 minutes to cross
-- 1010 km:
essentially the size of the solar system
-- 1012 km:
the Sun a tiny dot, mostly empty space
-- 1014 km:
nearest half dozen stars, 10 light years across
-- 1016 km:
small fragment of the Milky Way galaxy, ~1000 stars
-- 1018 km:
galaxy, 100,000 light years, 100 billion stars
-- 1020 km:
much larger than the Local Group of galaxies
-- 1022 km:
one billion light years, incl. most of the observable universe, Quasars
-- 1024 km:
100 billion light years; no light has yet traveled that far; well beyond
the edge of the visible universe
Celestial Sphere, the Sky, and the Calendar
What you need to know:
Night Sky: Constellations and Motions, & Magnitude Scale
* features of the
* definition of the
celestial equator, poles, meridian, zenith, nadir, etc.
* definition of angular
diameter, ecliptic, zodiac
* the reasons for
* cause and appearance
of eclipses and phases of the moon.
NOTE THE DIFFERENCES:
"local" coordinate system based on the observer's
horizon (altitude, azimuth) and a "global" coordinate system fixed to the
celestial sphere, and thus also to the stars (right ascension, declination)
(horizon) to +90° (straight up)
(north) to 360° (around horizon eastward)
LATITUDE: North-south position on Earth
South Pole -90°;
Equator 0°; North Pole +90°
* In the northern
hemisphere, Latitude is equal to the altitude (height above the horizon)
of the North Celestial Pole (position of Polaris).
LONGITUDE: East-west position on the
-180° (west) to
can be obtained by knowing the precise time that a star of known right
ascension passes across the local meridian line and comparing that withe
the exact time that the same star crossed the Greenwich meridian.
ZENITH: straight up, direct overhead
NADIR: straight down, directly under your
MERIDIAN: curved arc on the celestial
sphere through North Pole, Zenith, and South Pole.
DECLINATION: North-south position
on the celestial sphere (fixed to the sphere)
Celestial South Pole
Celestial North Pole
RIGHT ASCENSION: East-west position
on the celestial sphere (fixed to the sphere)
NOTICE: the relative elevation of the
Sun at noon at different times of the year as it moves along the ecliptic.
ECLIPTIC (ZODIAC): path of the Earth's
orbit around the Sun; apparent path of the Sun across the sky. The
Zodiac is that band of constellations through which the ecliptic passes,
and through which all the planets (with orbits near the ecliptic) move.
Reasons for the Seasons
PRECESSION: 25,800 year cycle of the wobble
of the Earth's rotation axis; NCP now pointed at Polaris, but in 3000 B.C.
it was pointed at Thuban (a-Draconis), and will
continue to sweep across the sky.
NOTICE: the relative positioning of
the Sun, Earth, and Moon and the planets as they move through these motions.
ECLIPSES: Solar, Lunar, Umbra, Penumbra
PLANETARY MOTION: Direct and Retrograde
and Ancient Sky Viewers
The Ancients were keen observers of the sky, and
guided their lives by the cycles they saw.
the Copernican Revolution (Models of the Universe)
* cycles of day and night
-- rotation of the earth.
* cycles of the tides and
fishing and commerce -- relative position of the Earth, Moon and Sun, and
the cycles of days and months.
* cycles of the months and
women's monthly cycles.
* cycles of the seasons and
the planting and harvesting of crops.
* etc., . . .
. . . and with good reason.
(Life throughout history has nearly always been
at or near the subsistence level -- right at the boundary between life
and death, between survival and catastrophe.)
* work when it's light, rest
when it's not.
* pragmatism; as a boatman,
you could only get in and out of most harbors safely at high tide,
and wanted to be where the fish were when they were running. further it
was always important not to run ships aground on shallow shoals.
All these depended on knowing the cycles of the tides.
* plant and harvest crops
in favorable seasons; if you plant too soon or too late, your crops will
fail and you will all die.
. . . but superstition grew up around all these
activities. These were life-and-death matters to them! This lead
to. . .
Egyptian planting cycles set to the co-rising of Sirius with the Sun.
Stonehenge, Mayans, American Indian stone circles, etc. -- the ancients
were exceptionally keen observers of the sky, set very precise geometrical
observatories where they set their calendars by aligning with the stars,
and motions of the planets and the Sun.
-- Many careful scientific
studies have shown no correlation between astrological birth sign and events
or patterns in people's life.
Geocentric Model of the Universe
astronomy up to and including Tycho Brahe
Claudius Ptolemius, Greek scholar and
librarian living in Alexandria Egypt 140 AD
* based on the Aristotelian
world view that the world followed patterns of symmetry and truth and beauty
* tied very closely to the
Christian view of the centrality and importance of Man in the universe
* perfectly circular orbits
* cycles upon cycles upon
cycles (epicycles and deferents)
* required great complexity
Heliocentric Model of the Universe
Nicholas Copernicus, Polish astronomer
* radical new view of the heavens.
* strictly against Christian
(Roman Catholic) doctrine, heretical.
* still circular orbits
* explained retrograde
motionmuch more simply and elegantly.
* was in fact much less precise
in finding the exact positions of the planets than earlier models.
Ancient (pre-telescope) Observers:
Origin of Modern Astronomy:
-- The ancients were exceptionally keen
observers of the sky, set very precise geometrical observatories where
they set their calendars by aligning with the stars, with motions of the
planets, and with the Sun. (e.g. Egyptian planting cycles set to the co-rising
of Sirius with the Sun; Stonehenge; Mayans, American Indian stone circles,
-- Erastosthenes: (Greek
living in Egypt, 273 B.C.) measured quite precisely the size (circumference)
of the Earth using gnomon, angles, and the paced distance between
Alexandria and Syene.
-- Tycho Brahe: greatest pre-telescope
observational astronomer; Danish nobleman; measured very precisely the
positions of the planets for over a period of 30 years; Johannes Kepler
was his assistant and used Tycho's data to generate his (Kepler's) three
Kepler, Galileo & Newton
What you need to know:
-- contributions of Tycho, Kepler, Copernicus,
Galileo, and Newton, and about when they lived
-- Kepler's laws and their use.
-- how Galileo's observations could be used
to prove Copernican model of the universe.
-- why Newton should be considered the
greatest thinker of all time.
Kepler, Galileo, Newton, and Edmund Halley (who was
Astronomer Royal to the British crown, and Newton's good friend) were all
overlapping contemporaries with Tycho Brahe and many of the historical
"movers and shakers" of the 17th century.
-- "wandering mathematician;"
assistant to Tycho Brahe.
-- published Rudolphian tables of planetary
-- also both Tycho and Kepler observed
and studied separate supernovas (which were the last supernovae seen in
this galaxy, and the last seen up close until SN1987A).
Kepler's three Laws of Planetary Motion:
Kepler's First Law
"The orbits of the planets are ellipses with
the sun at one focus."
eccentricity -- elongation of orbit
Kepler's Second Law
"A line from the planet to the sun sweeps over
equal areas in equal intervals of time."
Kepler's Third Law
"A planet's orbital period squared is proportional
to its average distance from the sun cubed."
These laws are entirely observational -
they tell 'how' the planets move in their orbits, but not 'why'. It wasn't
until Newton's laws that the 'why's started to be answered.
Sir Isaac Newton:
-- He lived and worked in Italy (for a long
time was a professor at the University of Pisa) and because he was so close
to the seat of the Catholic church at Rome, was persecuted as a heretic
for his views.
| -- "Father of Modern Astronomy,"
and in fact also the father of much of the modern scientific method.
-- born in Pisa, Italy in 1564, and active
and devout Catholic.
-- Very important as an observational
He was the first man to use telescope extensively to observe
* observed the mountains
and craters on the Moon.
-- He was strong advocate of Copernicus' heliocentric
model of the universe, and probably the single person whose observations
'proved' that model more strongly than any other.
* first to observe sunspots.
* first to the rings of Saturn.
* observed four largest moons of
Jupiter, which was strong proof that orbital motion can be centered
on some other object than the Earth, and that planets can have satellite
*observed the phases and range
of sizes of Venus, which is only explainable if Venus is orbiting the Sun
rather than orbitting the Earth.
actual drawings from
of the moons of Jupiter
-- forced by Catholic church to recant
his support for his Copernican views; died after over ten years under church-imposed
house arrest on Jan. 8, 1642.
Cycles in the Sky:
Life and Works:
-- born December 25, 1642 in rural English
village of Woolsthorpe.
-- very well educated; studied physics
and mathematics at Trinity College, Cambridge University.
-- spent much of his adult life living
almost in isolation as a country gentleman, specifically to avoid the Black
Plague that was ravaging London and many of the cities of Europe at the
-- probably the single greatest thinker
of all time. Single-handedly did more to advance a broader range of
widely divergent areas of science than anyone else who has ever lived.
-- advanced studies in:
* Optics: developed ways
of describing the path and interactions of light, developed an entirely
new type of telescope (which now carries his name).
-- greatest written work was Principia
Mathematica, published in 1687 (in Latin) which included his law of
gravitation, his laws of motion, and much else.
* Physics: developed three laws
* Mathematics: invented differential
* Astronomy: developed the Law
of Universal Gravitation; used it to predict the return of Halley's comet.
-- placed science on firm analytical basis
-- mathematically showed "why" the planets move in their orbits and how
those motions are related to the motions of al other objects, even those
as simple as an apple falling from a tree.
Gravity and Motion
Newton's Three laws of Motion:
1. inertia: a body at rest
tends to remain at rest, or if in motion tends to remain in uniform linear
motion unless acted upon by a force.
2. F = ma: acceleration (change
in motion) proportional to force applied.
3. action-reaction: when
two bodies interact, they create equal and opposite forces on each other
[e.g. rocket moves forward because it pushed something else (hot gases)
Newton's Law of Universal Gravitation:
-- all orbits are caused by
the falling inward of the planet or satellite as it simultaneously moves
sideways, leading to a curved path around central mass.
-- several other equations fall directly from
the law of gravity:
F = G x m1
where G = 6.67x10-11m3/kg.sec2
-- one can measure a planet's or satellite's
mass by using orbital motion:
-- surface gravity: g
-- escape velocity: Vesc
= square root of (2GM/R)
The following figure is the solar system as visualized
after the work of Isaac Newton, and published in Newton's Principia
Mathematica. It includes the orbits of all the planets visible to the
naked eye in nearly circular orbits and the highly elongated elliptical
orbits of many comets, including Halley's Comet named after Newton's friend,
Sir Edmund Halley. Other than the fact that three more planets, many
more comets, and many asteroids have been found since then, the model we
have of the solar system is almost unchanged from this image.
Earth, Moon (phases, tides), and the Sky
Reasons for the Seasons
|The 23.5° tilt of the Earth's
axis with respect to the ecliptic causes the Sun to appear to move
northward or southward with the yearly cycles, shining light more strongly
alternately on the Earth's Northern and then Southern Hemispheres.
The tilt of the Earth's axis causes two secondary
(1) the Sun beats more directly down in the summer
than in the winter --> more direct heating.
(2) the days are longer in the summer than in
the winter --> more accumulated sunshine per day, and thus more accumulated
Both cause the summer days to be hotter than
is summer on the northern hemisphere at the same time it is winter in the
southern hemisphere, and vice versa.
would think that the Earth's elliptical orbit would significantly affect
the seasons, but itseffect is minimal. The earth is nearest the Sun (perihelion)
on ~Jan. 3, and the net effect of the distance variation is to make the
seasonal temperature variation slightly wider in the southern hemisphere
that in the northern hemisphere.
Earth in June
Earth in December
* Notice that above the Arctic Circle, the
Earth is continually in light in June (Summer solstice in the northern
hemisphere) and continually in darkness in December (Winter solstice in
the northern hemisphere). Also notice that the Seasons are exactly reversed
in the southern hemisphere.
-- approximately every seven days
(one week!) the moon goes one quarter of the way through the cycle of phases.
-- new moon, waxing crescent, first
quarter, waxing gibbous, full moon, waning gibbous, third
(or last) quarter, waning crescent, new moon.
| -- the illuminated fraction of the moon,
as seen from the Earth is based on the Earth-Moon-Sun geometry (giving
the phases), but the same side of the Moon always points towards
the earth (with a little bit of wobble, called nutation).
Given two of the three following: (1)the time
of day or night, (2)the position across the arc of the sky where the Moon
will appear, and (3)the phase of the Moon, you should be able to generate
eclipses and the small angle formula:
-- as seem from the surface of the Earth,
both the Moon and the Sun are the same angular size ( about 1/2
degree across), meaning they appear as big as each other.
umbra -- full shadow
L = 2pDA(degrees)/360
L = 2pDA(arcseconds)/360x3600
where L is the actual
size (width) of the object,
D is the distance
to the object,
and A is the angular
size of the object (L and D
must be given in the same units, and the units of A
are listed beside it.)
penumbra -- partial shadow
-- only happen exactly at New Moon
-- happen when the Earth passes
through moon's shadow.
-- eclipses are only visible for
observers who are themselves under the moon's shadow (the area of totality
is a small patch <70km wide speeding across the Earth's surface at 1000km/hr)
* total solar eclipses
only if the angular size of moon is larger than the angular diameter of
the Sun (i.e. at lunar perigee and Earth's aphelion) and
the Sun/Earth/Moon alignment is perfect (exactly at Line of Nodes);
useful for astronomers studying the Sun's atmosphere
* annular solar eclipses -
happen only if the angular size of Moon is smaller than the Sun.
* partial solar eclipses -
happen if the Sun/Earth/Moon alignment is not perfect; Earth will only
pass into the Moon's
-- only happen exactly at Full Moon
-- happen when moon passes through
the Earth's shadow.
-- visible from anywhere on the
Earth in shadow (that is, anywhere that it is nighttime (and thus roughly
half the world's population can see each lunar eclipse.)
Line of Nodes and "eclipse seasons"
-- Eclipses do not happen every month
because the Moon's and EarthÝs orbits are tipped by ~5°; eclipses
can only happen when the New or Full Moon falls near where those two orbital
planes cross (i.e. along the Line of Nodes)
-- Thus eclipses only happen for a short
period (of about a month's duration, called an "eclipse season") twice
each year. The time during the year when this happens changes because of
an 18.3-year wobble in the planes of the orbits called the Saros cycle.)
-- The variation of the Moon's
gravity across the Earth is an example of a differential or
Atoms and Starlight:
-- Deforms Earth' s oceans into a prolate
spheroid. The residual differential force causes two high tides per
day and two low tides.
-- Rocks move a few cm; ocean move more
than 1 meter [and depending on the geometry of the coastline, which focuses
the motion of the water, by up to 7-8 meters (eg. Bay of Fundy, Canada).]
-- spring tides and neap tides:
effects of tides from the Moon and Sun either add or subtract.
* highest high tides and lowest
low tides are at new and full moon (spring tides).
-- effects of tides are seen everywhere,
not just on Earth; [e.g. Moon's locked rotation, Io's volcanoes, rings
of Saturn, binary stars, etc.]
* weaker range of tides at first and third
quarter moon (neap tides).
(should be able to also figure out what times
of day each high or low tidewould happen).
Interaction of light and matter
-- all electromagnetic radiation (composed
of oscillating electric and magnetic fields) always travels a single speed
through space, called the speed of light:
c = 3 x105km/sec
-- the longer the wavelength, the lower
the energy of the associated wavepacket (photon), and the shorter the wavelength,
the higher the energy:
E = hc/l
-- gamma rays (g),
x-rays, ultraviolet, visible light, infrared, (microwave), radio waves.
[from shortest wavelength to longest wavelength (highest energy to lowest
-- visible light wavelengths ~400nm -
700nm (4000Å -7000Å)
-- infrared radiation is what we sense
-- other than wavelength (which affects
energy), all electromagnetic radiation is really pretty much 'the same
Interaction of Light and the Atmosphere:
* Opacity (and Transparency) of the Atmosphere:
-- visible light range and
radio waves (and a small amount of the infrared and ultraviolet spectra)
are the only parts of the entire EM spectra that the Earth's atmosphere
is transparent to.
-- the atmosphere blocks or
absorbs (is opaque to) rest of the EM spectrum.
-- x-rays and g-rays
are blocked by oxygen and nitrogen in upper atmosphere.
-- UV blocked by ozone in
the upper atmosphere; protects us from skin cancer, and all life from continuous
-- water vapor and carbon
dioxide in lower atmosphere absorb most IR; causes our lower atmosphere
to be warmer than normal --
-- telescopes for visible
and radio wavelengths can be on the Earth's surface, but all others must
be carried on satellites.
* Refraction and Dispersion of Light in the
-- caused by small particles
and molecules in the atmosphere. More particles = more scattering (dispersion).
-- if there were no atmosphere,
the sky in daylight would be black.
-- stronger scattering of
shorter wavelengths of light causes the sky to look blue.
-- weaker scattering of longer
wavelengths of light causes sunsets to be brilliant orange-red.
* Twinkling of Stars: (astronomers call
-- caused by turbulence and
pockets of warmer and cooler air moving across the line of sight to the
stars, as the air pockets are moved by wind currents.
-- temperature is a way to measure of
the motion and energy of atoms.
-- the greater the internal energy of
the matter, the more the atoms vibrate or bounce off each other , and the
higher the temperature.
-- the states of matter with increasing
temperature: solid (at zero temperature -- frozen -- all atomic motion
stopped), liquid, gas, molecules break apart (atoms, free radicals), ions
(electrons strip off atoms), at highest temperature many electrons strip
off of each atom (multiply ionized).
-- Kelvin Temperature scale only
scale that starts at true (absolute) zero where all motion stops. (273K
-- freezing point of water; 373K -- boiling point of water; 300K average
* Weins Law:
-- All matter gives off radiation over
a broad wavelength range associated with its temperature, called blackbody
radiation or continuum spectra.
-- the greatest amount of energy (the
maximum intensity) is given off at a particular wavelength associated with
the emitting object's temperature by:
* Stefan-Boltzmann Law:
T(K) = 3,000,000/lmax(nm)
-- The total amount of light a star (or
any other warm or hot object) gives off, its Luminosity, L,
is related its size -- specifically its surface area 2pR2
-- and also to the objects temperature.
L = 2pR2
-- where s
is called Boltzmann's constant
5.67 x 10-8 W/m2.k4
-- blue stars are hotter that
stars, which are hotter than red stars.
-- hotter stars are brighter than
cooler stars, and the brightness goes up very fast, as the fourth power
of the temperature.
-- larger stars are brighter
than small stars.
Interaction of Light and Atoms:
-- atoms have discrete energy levels
like stair steps; the electrons in each atom can only have those precise
energies, and not in between.
-- the energy levels of each atom depend
on how many electrons the atom has; and thus each different element,
(and each different ionization state for each element) has a unique
set of energy spacings -- a "fingerprint" that can be used to identify
the element and state.
-- Absorption: atoms absorb light
(and create a dark band at one wavelength or color) when some of the incident
light wavelengths match the precise energy of available transitions in
the atom -- the electrons hop from lower energy states to higher
-- Emission: atoms emit light (and
create a bright line at one wavelength or color) if they have electrons
that have previously been excited up to higher energy states -- the electrons
will drop from the higher energy state to lower energy
-- in both of these processes, the light's
energy (and thus wavelength) is defined by the energy differences between
the energy levels in that atom.
Three Types of Spectra:
-- continuum spectra: from
hot opaque sources (Can obtain the sources temperature.)
-- absorption spectra: from
cool or warm gas cloud between observer and hot source (Can find the elements
present in the cloud -- dark bands on continuum spectrum).
-- emission spectra: from warm
or hot gas cloud observed directly (Can find the elements present -- bright
-- used to measure the relative
motion of the source and observer.
-- works for all waves: water, sound electromagnetic
-- seen in stellar spectra as a small
shift in the (wavelength) positions of absorption or emission lines.
-- a shift to shorter wavelength (blueshift)
means they are approaching, and shift to longer wavelengths (redshift)
means receding or separating.
= l - lo = lovr/c
the wavelength shift,
the original wavelength (measured from an unmoving source),
the wavelength of the moving source,
the speed of the source radially toward or way from the observer (*movement
laterally across the line-of-sight is unobservable)
is the speed of the waves (usually the speed of light)
Refraction and Reflection of Light
-- Refraction depends on the bending of
light as it passes through a transparent medium.
* the precise shape ("figure")
of both surfaces are important.
-- Reflection depends on the precise
angles between incoming and outgoing reflected light rays.
* the density, uniformity, transparency,
and composition of the lens is critical.
* refractive telescopes suffer
from chromatic aberration (because different wavelengths focus at
* glass sags under its own weight,
and also thermally deforms under temperature changes (normally this is
undetectable, but because of the unbelievably precise requirements of telescope
optics, this sagging distorts and degrades the telescope images.
(especially true of very large refracting telescopes because the glass
is very heavy and they can only be supported around the perimeter of their
* the largest refracting telescope
is 1.04-meter telescope at Yerkes Observatory, Wisconsin)
* the "figure" or shape
of only one surface, the front surface is important -- easier to make very
large telescope optics.
* the largest refracting telescopes
are the 10.0-meter diameter Keck-I and Keck-II, each
composed of 36 segmented mirrors on Mauna Kea, Hawaii. (Advances
in very large telescope design have allowed the building of several very
large optical telescopes, with the following, all over 8.0-meters being
commissioned in the last few years:
* Gemini-North (Mauna Kea)
and Gemini-South (Cerro Pachon, Chile) -- each 8.0 meters
* Subaru (Mauna Kea) --
* Very Large Telescope (VLT)
(Cerro Paronal, Chile -- European Southern Observatory) -- four x 8.0-meter,
on same mountain peak, that can be coupled together (interferometry), or
* Hobby-Eberly (Ft. Davis,
Texas) -- 9.4-meter (spherical)
Three Powers of a Telescope:
-- the ability to detect
very faint objects:
Light Gathering Power
* the amount of light that a telescope
can gather (effectively how faint the dimmest object that can be observed)
is related to the surface area of the input aperture or objective -- pR2
* the larger the objective diameter,
the more light that can be gathered.
* improved by long time-exposures
(max. time human eye can integrate light ~1/30 sec.)
* also improved by advances in
* more recent advances by using
long time-exposures with very sensitive electronic sensors: CCD's.
* biggest problem is "light
pollution" - can't see anything dimmer than the overall glow of the
sky which is caused by upward scattered light from cities.
-- the ability to separate
the images of two objects very close together:
several things limit
-- the smallest angle between two bright
point sources whose images can be visually distinguished.
-- the smaller the resolution angle,
the intrinsic limitation due to the passage of the light through
an entrance aperture of finite size.
in the telescope "figure".
this limit (called the Dawe's limit or resolution limit)
is dependent on the diameter of the telescope's input (or objective) lens
or primary mirror.
forces the image to be smeared out into rings around each bright
spot on an image.
the larger the telescope objective, the smaller the angular separation
that can be resolved.
for visible wavelengths, the minimum
resolved angle is:
for all wavelengths (resolution
is dependent both on aperture and wavelength, with long wavelengths like
radiowaves requiring huge apertures to get good resolution.) the minimum
resolved angle is:
the most common is 'spherical aberration', where the mirror is not polished
to ideal shape (a parabola), but distorted to be more spherical in shape.
due to passage of light through the atmosphere:
also roughness, poor alignment of the telescopes optical components, jitter
in the mounting of the telescope as it track stars, etc.
often, for the largest telescopes, the resolution is not limited by diffraction,
but by "seeing", the twinkling caused by heat, wind currents, turbulence,
and density variations in the atmosphere.
-- the ability to multiply or visibly
expand the size of images:
* this last limitation
is very recently being reduced using a technique called "active" or "adaptive"
optics, where, using super-computer control, the shape of one of the
mirrors in the telescope is continually adjusted by hundreds of tiny actuators
to compensate for variations in the arrival time of different parts of
the image wavefront due to "seeing".
for visible wavelengths, the minimum
resolved angle is:
= ~2.0 (observer at sea-level)
= ~0.5 (observer on highest mountain
this is, by far, the easiest of the powers of a telescope to obtain
or to change, because although the objective optics (lens or mirror) is
fixed, the very small eyepiece lenses are easily interchangeable.
is the focal length of the objective lens (mirror),
magnification is good, and sometimes it doesn't help (as in when the object
being observed covers a large angular patch of the sky.)
magnification does not depend on the diameters of the optical elements,
but rather on their
focal length (effectively how strongly the lenses
or mirrors are curved):
and fe is the focal
length of the eyepiece lens.