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Title: Elementi di Astronomia e Astrofisica per il Corso di Ingegneria Aerospaziale VI settimana


1
Elementi di Astronomia e Astrofisica per il Corso
di Ingegneria AerospazialeVI settimana
L'Atmosfera terrestre Un esercizio di meccanica
celeste (in English)
2
The terrestrial atmosphere - 1
This chapter is devoted to the examination of the
influence of the Earths atmosphere on the
apparent coordinates of the stars and on the
shape of their images the discussion will be
limited essentially to the visual band. The
discussion of the effects of the atmosphere on
photometry and spectrophotometry are deferred to
a later chapter.
The figure gives a schematic representation of
the vertical structure of the atmosphere the
visual band is mostly affected by what happens in
the troposphere, namely in the first 15 km or so
of height, where some 90 of the total mass of
the atmosphere is contained.
3
The terrestrial atmosphere -2
Na Layer
4
The terrestrial atmosphere - 3
The temperature profile in the troposphere is
actually more complicated than shown in the
Figure. The height of the tropopause (a layer of
almost constant temperature) from the ground
ranges from 8 km at high latitudes to 18 km above
the equator it is also highest in summer and
lowest in winter. The average temperature
gradient is approximately 6 C/km, but often,
above a critical layer situated in the first few
km, the temperature gradient is inverted, with
beneficial effects on astronomical observations,
thanks to the intrinsic stability of all layers
with temperature inversion (such as the
stratosphere and the thermosphere), essentially
because convection cannot develop. This is the
case for instance of the Observatory of the Roque
de los Muchachos (Canary Islands, height 2400 m
a.s.l.), where the inversion layer is usually few
hundred meters below the telescopes at the top of
the mountain.
5
Chemical composition and structure
The chemical composition of the troposphere is
mostly molecular Nitrogen N2 and molecular Oxygen
O2 (approximately 34 and 14 respectively), with
traces of the noble gas Argon and of water vapor
(the water vapor concentration may be as high as
3 at the equator, and decreases toward the
poles). Above the tropopause, at higher heights
in the stratosphere, the temperature raises
considerably thanks to the solar UV absorption by
the Ozone (O3) molecule with the process UV
photon O3 O2Oheat. The mesosphere ranges
from 50 to 80 km in this region, concentrations
of O3 and H2O vapor are negligible, hence the
temperature is lower than in the stratosphere.
The chemical composition of the air becomes
strongly height-dependent, with heavier gases
stratified in the lower layers. In this region,
meteors and spacecraft entering the atmosphere
start to warm up.
6
The ozone O3
O3 is a molecule containing 3 O atoms. It is blue
in color and has a strong odor. Normal molecular
O2, has 2 oxygen atoms and is colorless and
odorless. Ozone is much less common than normal
oxygen. Out of each 10 million air molecules,
about 2 million are normal oxygen, but only 3 are
ozone.
Most atmospheric ozone is concentrated in a layer
in the stratosphere, about 15-30 kilometers above
the Earth's surface. Even this small amount of
ozone plays a key role in the atmosphere,
absorbing the UVB portion of the radiation from
the sun, preventing it from reaching the planet's
surface.
7
Water vapor nomenclature - 1
Water vapor is water in the gaseous phase. The
actual amount is the concentration of water vapor
in the air, the relative concentration is the
ratio between the actual amount to the amount
that would saturate the air. Air is said to be
saturated when it contains the maximum possible
amount of water vapor without bringing on
condensation. At that point, the rate at which
water molecules enter the air by evaporation
exactly balances the rate at which they leave by
condensation. The partial pressure of a given
sample of moist air that is attributable to the
water vapor is called the vapor pressure. The
vapor pressure necessary to saturate the air is
the saturation vapor pressure. Its value depends
only on the temperature of the air. (The
Clausius-Clapeyron equation gives the saturation
vapor pressure over a flat surface of pure water
as a function of temperature.) Saturation vapor
pressure increases rapidly with temperature the
value at 32C is about double the value at 21C.
The saturation vapor pressure over a curved
surface, such as a cloud droplet, is greater than
that over a flat surface, and the saturation
vapor pressure over pure water is greater than
that over water with a dissolved solute.
8
Water vapor nomenclature - 2
Relative humidity is the ratio of the actual
vapor pressure to the saturation vapor pressure
at the air temperature, expressed as a
percentage. Because of the temperature dependence
of the saturation vapor pressure, for a given
value of relative humidity, warm air has more
water vapor than cooler air. The dew point
temperature is the temperature the air would have
if it were cooled, at constant pressure and water
vapor content, until saturation (or condensation)
occurred. The difference between the actual
temperature and the dew point is called the dew
point depression. The wet-bulb temperature is
the temperature an air parcel would have if it
were cooled to saturation at constant pressure by
evaporating water into the parcel. (The term
comes from the operation of a psychrometer, a
widely used instrument for measuring humidity, in
which a pair of thermometers, one of which has a
wetted piece of cotton on the bulb, is
ventilated. The difference between the
temperatures of the two thermometers is a measure
of the humidity.) The wet-bulb temperature is the
lowest air temperature that can be achieved by
evaporation. At saturation, the wet-bulb, dew
point, and air temperatures are all equal
otherwise the dew point temperature is less than
the wet-bulb temperature, which is less than the
air temperature.
9
Water Vapor Mixing ratio
Specific humidity is the ratio of the mass of
water vapor in a sample to the total mass,
including both the dry air and the water vapor.
The mixing ratio is the ratio of the mass of
water vapor to the mass of only the dry air in
the sample. As ratios of masses, both specific
humidity and mixing ratio are dimensionless
numbers. However, because atmospheric
concentrations of water vapor tend to be at most
only a few percent of the amount of air (and
usually much lower), they are both often
expressed in units of grams of water vapor per
kilogram of (moist or dry) air. Absolute humidity
is the same as the water vapor density, defined
as the mass of water vapor divided by the volume
of associated moist air and generally expressed
in grams per cubic meter. The term is not much in
use now.
10
Water reservoir
Water vapor is constantly cycling through the
atmosphere, evaporating from the surface,
condensing to form clouds blown by the winds, and
subsequently returning to the Earth as
precipitation. Heat from the Sun is used to
evaporate water, and this heat is put into the
air when the water condenses into clouds and
precipitates. This evaporation - condensation
cycle is an important mechanism for transferring
heat energy from the Earth's surface to its
atmosphere and in moving heat around the Earth.
Water vapor is the most abundant of the
greenhouse gases in the atmosphere and the most
important in establishing the Earth's climate.
Greenhouse gases allow much of the Sun's
shortwave radiation to pass through them but
absorb the infrared radiation emitted by the
Earth's surface. Without water vapor and other
greenhouse gases in the air, surface air
temperatures would be well below freezing.
11
Aerospace devices
A multitude of systems exist for observing water
vapor on a global scale and at high altitudes,
supplementing the instruments on the ground, that
measure in special sites and at ground level.
Each has different characteristics and
advantages. To date, most large-scale water vapor
climatological studies have relied on analysis
of radiosonde data, which have good resolution in
the lower troposphere in populated regions but
are of limited value at high altitude and are
lacking over remote oceanic regions.
12
The Water Vapor content in 1992
NASA Water Vapor Project (NVAP) Total Column
Water Vapor 1992 The mean distribution of
precipitable water, or total atmospheric water
vapor above the Earth's surface, for 1992. This
depiction includes data from both satellite and
radiosonde observations.
13
Cloud effects on Earth Radiation
14
The outer layers
Following the smooth decrease in the mesosphere,
the temperature raises again in the thermosphere,
because the solar UV and X-rays, and the
energetic electrons from the magnetosphere can
partly ionize the very thin gases of the
thermosphere. The weakly ionized region which
conducts electricity, and reflects radio
frequencies below about 30 MHz is called
ionosphere it is divided into the regions D
(60-90 km), E (90-140 km), and F (140-1000 km),
based on features in the electron density
profile. Finally, above 1000 km, the gas
composition is dominated by atomic Hydrogen
escaping the Earths gravity, which is seen by
satellites as a bright geocorona in the resonance
line Ly-? at ? 1216 Å.
15
Refraction Index
As is well known, the light propagates in a
straight line in any medium of constant
refraction index n, with a phase velocity v given
by
where ? is the dielectric constant and ? the
magnetic permeability of the medium. All these
quantities are wavelength dependent. The group
velocity u is instead
At the separation surface between two media of
different refraction index (say vacuum/air), the
ray changes direction, so that the observer
immersed in the second medium sees the light
coming from an apparent direction different from
the true one (see Figure)
16
The atmospheric refraction - 1
Suppose that the atmosphere can be treated as a
succession of parallel planes (hypothesis of
plane-parallel stratification), by virtue of its
small vertical extension with respect to the
Earths radius. According to Snells laws, when
the ray coming from the region of index of
refraction n0 encounters the separation surface
with a medium of refraction index n1gt n0, part of
the energy will be reflected to the left, on the
same hemi-space with the same angle r0 with
respect to the normal. This part will not be
considered here, it only implies a dimming of the
source. The remaining fraction will be refracted,
in the same plane as the incident ray, to an
angle r1 lt r0. Indeed, in a clear atmosphere
without clouds, no sharp air-vacuum separation
surface exists, the refraction index gradually
increases from 1 to a final value nf near the
ground, with typical scale lengths much greater
than the wavelength of light (as already said, we
limit our considerations to the visual band), so
that the continuously varying direction can be
considered as a series of finite steps in the
plane passing through the vertical and the
direction to the star.
17
The atmospheric refraction - 2
By following each refraction in cascade we have
where ni1gt ni, and ri1lt ri. By equating each
term
Therefore in a plane-parallel atmosphere the
total angular deviation only depends on the
refraction index close to the ground, independent
of the exact law with which it varies along the
path.
18
The atmospheric refraction - 3
The net effect is as shown in the figure the
star is seen in direction z smaller than the
true direction z, namely closer to the local
Zenith, by an amount R which is the atmospheric
refraction z z R
By virtue of
and for small Rs (in practice, if z lt 45)
19
The atmospheric refraction - 4
and finally
In the visual band, for average values of
temperature and pressure (T 273 K, P 760 mm
Hg), nf ? 1.00029, so that in round numbers
R(15) ? 16, R(45) ? 60 Already for Zenith
distances as small as 20, the refraction is
larger than the annual aberration, and of any of
the effects discussed in previous chapters that
alter the apparent direction of a star.
20
The atmospheric refraction - 5
For zenith distances larger than 45, the path of
the ray inside the atmosphere is so long that the
curvature of the Earth cannot be ignored, and the
mathematical treatment becomes more intricate,
even restricting it to successive refraction in
the same plane with n decreasing outwards with
continuity.
After several mathematical steps
21
Effect of the refraction on the coordinates
The main effect of refraction is to move the
star closer to the Zenith in the vertical plane,
thus raising its elevation h but leaving
essentially unchanged its azimuth A. XX R
?h PXX PXZ q ZX z, ZX z PX
90-? XU ??
For an object in meridian, the refraction is all
in declination, and in particular this is true
for the Sun at true noon.
22
Approximate formulae for refraction
For Zenith distance not greater than
approximately 45, after several passages we
finally get
by means of which formulae we can derive the true
(or the apparent, according to the sign)
topocentric positions. Obviously no such
correction is necessary for a telescope in outer
Space.
23
The chromatism of the refraction
The refraction index n depends from the
wavelength, diminishing from the blue to the red,
and the same will be true for the refraction
angle R the image on the ground of the star is
therefore a succession of monochromatic points
aligned along the vertical circle the blue ray
will be below the red one, and thus the blue star
will appear to the eye above the red one
The atmosphere behaves therefore like a prism
producing a short spectrum in the vertical plane,
whose length increases with the zenith distance,
reaching several arc seconds at low elevations.
The relationships n(?) can be expressed by the
so-called Cauchys formula
(? in micrometers), corresponding to a variation
of about 2 over the visible range, namely to
about 1.2 at 45.
24
Density - temperature relationship
Once we have fixed ?, the refraction index n
depends from the density ? according to
Gladstone-Dales law
and with the hypothesis of a perfect gas of
pressure P, temperature T and molecular weight ?

(where R is now the gas universal constant)
(P in mm Hg, T in K)
25
Vertical gradients of temperature
Calling H the height over the ground, we have
The variation of pressure with the height is
equal to the weight of the air in the elementary
volume having unitary base and height dH,
so that
where the constant g/R equals approximately 3.4
K/km, and is called adiabatic lapse. Hence the
conclusion that the variations of the refraction
index depend from the vertical gradients of the
temperature. A practical consequence is that all
effort must be made to control and minimize those
gradients over the accessible volume of the
telescope enclosure.
26
Turbulence, Scintillation, Seeing
  • The Earth's atmosphere is turbulent and
    variations in the index of refraction cause the
    plane wavefront from distant objects to be
    distorted. This distortion introduces amplitude
    variations, positional shifts and image
    degradation.
  • This causes two astronomical effects
  • scintillation, which is amplitude variations,
    which typically varies over scales of few cm
    generally very small for large aperture
    telescopes
  • seeing positional changes and image quality
    changes. The effect of seeing depends on aperture
    size for small apertures, one sees a diffraction
    pattern moving around, while for large apertures,
    one sees a set of diffraction patterns (speckles)
    moving around on scale of 1 arcsec.
  • These observations imply
  • wavefronts are flat on scales of small apertures
  • instantaneous slopes vary by 1 arcsec.
  • The typical time scales are few milliseconds and
    up.
  • The effect of seeing can be derived from theories
    of atmospheric turbulence, worked out originally
    by Kolmogorov, Tatarski, Fried.

27
Structure function
The structure of the refraction index n in a
turbulent field can be described statistically by
a structure function
where x is separation of points, r is position.
Kolmogorov turbulence gives
where Cn is the refractive index structure
constant. From this, one can derive the phase
structure function at the telescope aperture
where the coherence length r0 (also known as the
Fried parameter) is
where z is zenith angle, ? is wavelength. Using
optics theory, one can convert D? into an image
shape.
28
The Fried parameter
Notice that r0 increases with ??6/5 ??1. 2.
Physically, the image size d from seeing is
(roughly) inversely proportional to r0
as compared with the image size from a
diffraction-limited telescope of aperture D
Seeing dominates when r0 lt D a larger r0 means
better seeing. Seeing is more important than
diffraction at shorter wavelengths, diffraction
more important at longer wavelengths effect of
diffraction and seeing cross over in the IR (at ?
5 microns for 4m) the crossover falls at a
shorter wavelength for smaller telescope or
better seeing. Frieds parameter r0 varies from
site to site and also in time. At most sites,
there seems to be three regimes called surface
layer (wind-surface interactions and manmade
seeing), planetary boundary layer ( influenced
by diurnal heating), free atmosphere (10 km is
tropopause high wind shears)
29
An example of Cn2
A typical site has r0 ? 10 cm at 5000Å ,
namely a seeing of 1". On rare occasions, in the
best sites, the seeing can be as low as 0".3.
30
The isoplanatic angle
We also have to consider the coherence of the
same turbulence pattern over the sky coherence
angle call the isoplanatic angle
where H is the average distance of the seeing
layer
For r0 ? 10 cm, H 5000 m , ? ? 1.3 arcsec.
In the infrared r0 ? 70 cm, H 5000 m , ? ? 9
arcsec. Note however, that the isoplanatic
patch for image motion" (not wavefront) is ?
0.3D/H. For D 4m, H 5000 m, ?kin ? 50 arcsec.
-             Another useful parameter is the
correlation time ?0, which is approximately the
dimension of the typical air bubble divided by
the velocity of the wind. As r0, also ?0
increases with ?6/5.
31
The seeing
Bubbles of air having slightly different
temperatures, and therefore slightly different
refractive indexes, are carried by the wind
across the aperture of the telescope. The Fried
parameter r0 can be used to simplify the
description of a very complex rapidly varying
medium, namely the typical size of the bubble.
Values vary from few centimeters (a poor site) to
some 30 cm (a very good site). r0 can be
understood also as the effective diameter of the
diffraction limited telescope in that site (with
respect to the angular resolution).
32
Representation of the seeing
  • There are two main components of the seeing
  • one coming from high altitudes (choice of site)
  • one due to ground layers (it can be actively
    controlled by shape of dome and proper
    thermalisation of structure)
  • The spectral power of the air turbulence is
    appreciable over a large interval of frequencies
    , say 1 to 1000 Hz, with a 1/f distribution.

The angles are exaggerated, actually AdOpt
correction can be made over small fields of view.
Another useful parameter is the maximum angle
over which fluctuations are coherent (isoplanatic
angle). Both Frieds parameter and isoplanatic
angle improve with increasing wavelength, the
correction is better in the IR than in the
Visible.
33
A first remedy Speckle Interferometry
  • a very large number of short duration exposures
    are taken with very long focal length (say 100m)
    and narrow bandwidth (say 1 nm) in each exposure
    the seeing is frozen, each speckle represents the
    diffraction figure of the aperture
  • Fourier Transforms allow the reconstruction of
    the true image
  • The technique works well for simple structures
    (e.g. double or multiple stars, disks).

34
A better remedy Adaptive Optics
  • The fairly complex techniques that are nowadays
    implemented on the largest telescopes to contrast
    the seeing are known collectively as Adaptive
    Optics devices.
  • A suitable reference wavefront is also
    necessary. Suitably bright stars are rare.
  • An artificial laser star is a possible solution.

35
The artificial laser star
36
Before and after AdOpt
If one freezes the image with short exposure
times (say less than 0.01 sec) and a narrow
filter, the seeing image breaks up in large
number of speckles, each having dimension of
the order of the diffraction figure of the
telescope. The number of speckles is of the order
of (seeing diameter/diffraction figure)2
37
The Galactic Center with the Keck AdOpt
Without AdOpt
With AdOpt
38
Quality of the image -1
The quality of an image can be described in many
different ways. The overall shape of the
distribution of light from a point source is
specified by the point spread function (PSF).
Diffraction gives a basic limit to the quality of
the PSF, but any aberrations or image motion add
to structure/broadening of the PSF. Another way
of describing the quality of an image is to
specify it's modulation transfer function (MTF).
The MTF and PSF are a Fourier transform pair.
Turbulence theory gives
where ? is the spatial frequency. Note that a
Gaussian goes as ? 2, so this MTF is close to a
Gaussian. The shape of seeing-limited images is
roughly Gaussian in core but has more extended
wings. This is relevant because the seeing is
often described by fitting a Gaussian to a
stellar profile.
A potentially better empirical fitting function
is a Moffat function
39
Quality of the image -2
Probably the most common way of describing the
seeing is by specifying the full-width-half-maximu
m (FWHM) of the image, which may be estimated
either by direct inspection or by fitting a
function (usually a Gaussian) note the
correspondence of FWHM to ? of a Gaussian FWHM
2.355? . The FWHM doesn't fully specify a
PSF, and one should always consider how
applicable the quantity is. Another way of
characterizing the PSF is by giving the encircled
energy as a function of radius, or at some
specified radius. A final way of characterizing
the image quality, more commonly used in adaptive
optics applications, is the Strehl ratio SR. The
Strehl ratio is the ratio between the peak
amplitude of the PSF and the peak amplitude
expected in the presence of diffraction only. In
practice, in the visible it is already very good
reaching SR 0.1 .
40
The EE of the Rosetta WAC
The WAC is in space, so there is no seeing to
worry about, only the vibrations of the
spacecraft or thermal distortions of the jitter
of the attitude.
41
Effects of the atmosphere at radiofrequencies - 1
The ionosphere will introduce a delay on the
arrival time of the wave, given by
seconds, being I the path along the line of sight
and Ne the electron density (cm-3). This density
will vary with the night and day cycle, with the
season and also with the solar cycle.
42
Effects of the atmosphere at radiofrequencies -2
The tropospheric delay can be resolved in two
components, a dry one and a wet one. The dry
component amounts to about 7 ns at the Zenith,
and varies with the modified cosec z we have
discussed for the optical observations
The wet component depends on the amount of water
vapour, and amounts to about 10 of the dry one,
but it varies rapidly and in unpredictable
way.   Finally, two other mediums affect the
propagation of the radio waves, namely the solar
corona and the ionized interstellar medium.
43
Extinction and spontaneous emission by the
atmosphere
In addition to chaotic refraction effects, the
atmosphere absorbs a fraction of the incident
light, both in the continuum and inside atomic
and molecular lines and bands. Furthermore, the
atmosphere spontaneously emits in particular
atomic and molecular bands (this is in addition
to scattering of artificial lights, see later).
The molecular oxygen O2 in particular is so
effective at blocking radiation around 6800A and
7600A that Fraunhofer could detect by eye two
dark absorption bands in the far red of the solar
spectrum, bands he called respectively B and A
(he examined the spectra from red to blue, the
current astronomical practice is from blue to
red).
44
Extinction
Let us consider the absorption due to a thin
layer of atmosphere at height between h and hdh
in the usual simple model of a plane-parallel
atmosphere. The light beam from the star makes an
angle z with the Zenith, so that the traversed
path is dh/cosz secz?dh. If I?(h) is the
intensity at the top of the layer, at the exit it
will be reduced by the quantity
In total, if I?(?) is the intensity outside the
atmosphere, at the elevation h0 of the
Observatory the intensity will be reduced to
45
Optical Depth
where we have introduced the a-dimensional
quantity ?? called optical depth
The variable k? (dimensionally, cm-1) represents
the absorption per unit length of the atmosphere
at that wavelength. Astronomers use a particular
measure of the apparent intensity, namely the
magnitude, defined by m m0 -2.5logI (see in a
later lecture), so that
D? is called the optical density of the
atmosphere, while the variable X(z) secz is
called air-mass. The minimum value of the airmass
is 1 at the Zenith, and 2 at z 60 (the limit
of validity of the present approximate
discussion).
46
The Bouguer line
Suppose we start observing the star at its upper
transit, and then keep observing it while its
Hour Angle (and therefore also its Zenith
distance) increases we would notice a linear
increase of its magnitude in agreement with the
previous equation, namely a straight line with
slope 2.5D? in a graph (m, secz). It is common
practice to plot the m-axis pointing down. This
straight line is known as Bouguer line, from the
name of the XVIII century French astronomer who
introduced it. The extrapolation of this line to
X 0 (a mathematical absurdity) gives the
so-called loss of magnitude at the Zenith, or
else the magnitude outside atmosphere. According
to the formulae of the first lectures we have
where ? is the latitude of the site, ? and HA the
coordinates of the star.
47
The least continuous extinction
The Table shows the continuous extinction of the
atmosphere above Mauna Kea, whose elevation above
sea level (4300 m) is higher than that of most
observatories so that the transparency of the sky
is at its best, in the extended visible region.
48
Figures of the extinction from the visible to the
near IR
The figure on the left gives the optical depth,
the one on the right the transmission (one is the
reverse of the other). In the violet region, the
transparency quickly goes to zero, essentially
because of the ozone O3 molecular absorption at
the other end of the spectrum the transparency is
reasonably good until about 2.4 micrometers, when
the H2O and CO2 molecules heavily absorb the
light. The astronomical photometric wide bands
(U,B,V, R, I, J, H, ) are indicated.
49
Spontaneous and artificial emissions
To complete these considerations about the
influence of the atmosphere on the photometry
(and also on the spectroscopy) of the celestial
bodies, we must add that the atmosphere
contributes radiation, by spontaneous emission
and by scattering of natural and artificial
lights. If the Observatory is close to populated
areas, bright emission lines of Mercury and
Sodium from street lamps are observed Hg at ??
4046.6, 4358.3, 5461.0, 5769.5, 5790.7 Na at
5683.5, 5890/96 (the yellow D-doublet), 6154.6
Ne at 6506, and so on. Natural lines come from
the atomic Oxygen in forbidden transitions
(designated with OI) at ?? 5577.4, 6300 and
6367, and especially from the molecular radical
OH who provides a wealth of spectral lines and
bands filling the near-IR region above 6800A. The
OH comes from the dissociation of the water vapor
molecule under the action of the solar UV
radiation. Therefore, the atmosphere is a
diffuse source of radiation, whose intensity
strongly depends on the Observatory site to set
an indicative value in the visual band, a
luminosity equivalent to one star of 20th mag per
square arcsec at the Zenith can be assumed.
50
The visible spectrum of the night sky
The night sky is calibrated (see ordinate) in
surface brightness, given as mag/(arcsec)2. Mt.
Boyun is in Korea.
51
The Near-IR sky emission - 2
A very detailed section of the near-IR night sky
OH-emission obtained at ESO Paranal with UVES.
http//www.eso.org/observing/dfo/quality/UVES/uves
sky/sky_8600U_1.html
52
A second limit of the terrestrial atmosphere the
artificial lights
The full Moon has difficulties in competing with
the spectrum of artificial lights.
53
The situation in Italy
If the extrapolation is correct, in 2025 no
Italian will be able to see the Milky Way
54
Planetary light pollution
From a paper by Cinzano, Falchi e Elvidge (2001)
55
A first exercise of celestial mechanics
Consider the total energy E of a particle P2 of
very small mass m2 at the surface of a
non-rotating spherical body P1 of radius R and
mass m1
The limiting velocity Ve
is said escape velocity from body P1. If by some
means we impart to P2 a velocity V greater than
Ve in any direction, P2 will reach infinity with
final velocity greater than zero. Another useful
critical velocity is that on the circular orbit
at distance rgtR from the center of P1 from the
equilibrium between centrifugal and gravitational
forces we get
56
Escape velocities from the 9 planets
The table provides escape and circular velocities
for the 9 planets, neglecting their diurnal
rotation. The 3rd column gives the surface
gravity in comparison with that at the Earths
surface (9.78 m/s2). The first two velocities
(4th and 5th column) pertain to the equator of
each body the other two velocities (6th and 7th
column) to the circular orbit at the average
distance of the body from the Sun.
57
Escape velocities and atmospheres - 1
These considerations on escape velocities from
the planetary surfaces are useful not only for
dynamical questions, but also for the
understanding of their atmospheres. Let T (in
Kelvin) be the temperature of such an atmosphere,
supposed in thermal equilibrium the distribution
function of molecules of mass m2 among the
velocities is given by Maxwells law
so that the mean square velocity of those
molecules will be
where k 1.38?10-16 erg/K is Boltzmann constant.
For instance, the mass of the Hydrogen atom H is
m2 ? 1.6?10-24 g, so that
km/s (T in K)
At the surface of the Earth, assuming T ? 290 K
we get VH ? 2.7 km/s ltlt Ve.
58
Escape velocities and atmospheres - 2
All other molecules being heavier than the atom
of H, we conclude that the Earth is well capable
of retaining a substantial quasi-stationary
atmosphere. However, Maxwells distribution has a
very long tail at high velocity, so that a
fraction of the Earths gases, and in particular
of H, will continuously escape to the outer
space. The observational evidence of such loss is
the so-call geocorona, well visible in the Ly-?
spectral line at ? 1216A. Mercury and the Moon
do not have such capability their tenuous
atmospheres must be continuously lost by thermal
escape and replenished by phenomena such as UV
solar photons and solar particles impinging on
the soil and extracting gases, or by meteoroid
bombardment. In the case of the Sun, the surface
gravity is 28 times that at the surface of the
Earth, and the photospheric temperature is
approximately 5800 K higher up, in the
cromosphere and in the corona, the temperatures
of the solar gases rise to tens, hundreds and
even millions of degrees, so that the thermal
escape becomes conspicuous. However, observations
prove that the loss of particles from the Sun
(the so-called solar wind) is orders of magnitude
larger than that accounted for by thermal loss
other more efficient mechanisms, whether magnetic
or electric, must act to accelerate the ionized
(electrically charged) particles escaping from
the Sun.
59
A second exercise of celestial mechanics
Let us launch from the surface of a spherical
non-rotating Earth of radius a? a satellite of
mass m2 with initial velocity V gt Ve . Its energy
will be
(m2 ltlt M?)
At an altitude H, the distance from the centre
becomes r a? H, and the energy
or else, equating the two values for the
conservation of the energy
60
Delta V
At infinity
In conclusion, if we launch with ?V 1km/s,
the satellite will reach infinity with a velocity
of approximately 4.7 km/s (ignoring the very
small losses of energy due to the atmospheric
drag). There are several practical consequences
of this gain at infinity, for instance one has
to be careful not to reach the final destination
with too high a velocity. We underline the
convenience of using in space applications the
parameter ?V instead of the energy. The circular
velocity at the surface of the Earth is around 8
km/s, which will also be the velocity of low
altitude satellites (e.g. the International Space
Station at 300 km). Their period is then of
approximately 90 minutes suppose we place such
satellite in a polar orbit it will go out of
phase with the Sun by about 30 min at each orbit,
and for several orbits it will see an almost
constant illumination (day or night) of its
Nadir. The low polar orbit is therefore used for
surveillance.
61
Geostationary orbits
At H 36.000 km the orbital period becomes of
24h, so that a satellite placed on the equatorial
plane at this altitude in a circular orbit (e.g.
the Meteosat) will be practically stationary with
respect to the ground observer. Actually,
several satellites have simply a geosynchronous
orbit (that was the case of the International
Ultraviolet Explorer), slightly different from
the rigorously defined geostationary one. At
any rate, the two body condition is a
mathematical abstraction, several perturbing
forces (like the Earth-Moon and solar tides, the
non-sphericity of the Earth potential, the
radiation pressure, etc.) will act to perturb the
orbit, and appropriate corrections must be
performed to keep the wanted position of the
satellite, for instance by occasional firings of
small thrusters.
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