L’atmosfera terrestre

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The terrestrial

atmosphere - 1

This  chapter  is  devoted  to  the  examination  of  the  influence  of
the  Earth’s  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

the

atmosphere

photometry

and

spectrophotometry are deferred to a later chapter.

The figure gives a
schematic
representation

the

vertical

structure

the

atmosphere;

the

visual band is mostly
affected by what
happens

the

troposphere

namely in

the first

15  km  or  so  of
height,  where  some
90%  of  the  total
mass

the

atmosphere

contained.

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Chemical composition and

structure

The chemical composition of the troposphere is mostly
molecular

Nitrogen

and

molecular

Oxygen

(approximately 3:4 and 1:4 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 (

) molecule with the process:

UV photon

+O+

heat

The

mesosphere

ranges from 50 to 80 km; in this region,

concentrations of

and

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.

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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
32°C is about double the value at 21°C. 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.

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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 d

ew 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.

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Water reservoir

Water

vapor

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.

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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

(60-90 km),

(90-140 km), and

(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 Earth’s gravity, which is seen
by satellites as a bright

geocorona

in the resonance line

Ly-





= 1216 Å.

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Refraction Index

As is well known, the light propagates in a straight line in any
medium of constant refraction index

, with a phase velocity

given by

1/2

1/( )

c n

where



is the dielectric constant and



the magnetic

permeability of the medium. All these quantities are
wavelength dependent. The group velocity

is instead:

dv/d

= -

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):

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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
Earth’s radius. According to

Snell’s laws

, when the ray

coming from the region of index of refraction

encounters the

separation surface with a medium of refraction index

part of the energy will be reflected to the left, on the same
hemi-space with the same angle

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

. 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

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.

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The atmospheric

refraction - 2

sin

1 1

sin

r n

sin

where

i+1

, and

i+1

. By equating each term:

sin

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.

By following each refraction in cascade we have:

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The atmospheric

refraction - 3

By virtue
of

The  net  effect  is  as
shown  in  the  figure:
the  star  is  seen  in
direction

z’

smaller

than the true direction

, namely closer to the

local Zenith, by an
amount

which is the

atmospheric
refraction:

z’

–

sin

and for small

’s (in practice, if

< 45°):

sin ' sin

sin( '

) sin 'cos

cos 'sin

sin '

cos '

z R

+ =

�

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The atmospheric

refraction - 4

(

1)tan '

In the visual band, for average values of temperature and
pressure (

= 273 K,

= 760 mm Hg),

 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.

and finally:

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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

decreasing outwards

with continuity.

sin '

R a n

n d n

�

= �

� -

�

tan '

tan ' (

1) (1

)tan '

tan '

R A

z B

�

After

several

mathematical

steps:

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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

but

leaving essentially unchanged
its azimuth

XX’ = R = h

PXX’ = PXZ = q

ZX = z, ZX’ = z’

PX = 90-



XU = 



cos

( '

)cos

sin

cos

a a

d d d

- D

�

D = -

�

cos cos

sin cos

cos sin cos

sin sin

cos

sin

cos sin

For an object in
meridian,

the

refraction  is  all  in
declination,  and  in
particular  this  is
true  for  the  Sun  at
true noon.

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Approximate formulae for

refraction

For Zenith distance not greater than approximately 45°, after
several passages we finally get:

sec sin

(

cos

tan tan

tan

tan cos

(

cos

tan tan

�

D =

�

�D = -

�

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.

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The chromatism of the

refraction

The refraction index

depends from the wavelength,

diminishing from the blue to the red, and the same will be true
for the refraction angle

: 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



)

can

expressed

the

so-called

Cauchy’s formula:

0.00566 0.000047

( ) 0.00028 1

n l

�

(



in micrometers), corresponding to a

variation of about 2% over the visible
range, namely to about

1”.2 at 45°.

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Density - temperature

relationship

Once we have fixed



, the refraction index

depends from the

density



according to Gladstone-Dale’s law:

- =

and with the hypothesis of a perfect gas of pressure

temperature

and molecular weight



r =

(where

is now the gas universal

constant)

- =

P T

1 78.7 10

- �

�

( /760)

60".4

tan

( / 273)

�

(

in mm Hg,

in K)

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Vertical gradients of

temperature

Calling

the height over the ground,

we have:

n k

�

1 d

P T

P T P

T H T

P H

�

The variation of pressure with the height is equal to the weight
of the air in the elementary volume having unitary base and
height

g H

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.

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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.

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Structure function

The structure of the refraction index

in a turbulent field can

be described statistically by a

structure function

( )

(

)

( )

D x

n r x n r

=� + -

�

where

is separation of points,

is position. Kolmogorov

turbulence gives:

2 2

( )

D x

C x

where

is the refractive index structure constan

t. From this,

one can derive the

phase structure function

at the

telescope aperture:

5/2

6.88

where the coherence length

(also known as the Fried

parameter) is:

3/5

6/5

3/5

0.185

cos

z C h

�

where

is zenith angle,



wavelength. Using optics theory,
one can convert D



into an

image shape.

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The Fried parameter

Notice that

increases with 



6/5





1. 2

Physically, the image size

from seeing is (roughly) inversely

proportional to

�

as compared with the image size from a diffraction-limited
telescope of aperture

�

Seeing dominates when

; a larger

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. Fried’s parameter

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)

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The isoplanatic angle

We also have to consider the coherence of the same
turbulence pattern over the sky: coherence angle call the

isoplanatic angle

0.314 /

r H

q �

where

is the average distance of the seeing layer:

For

 10 cm,

= 5000 m ,



 1.3 arcsec.

In the infrared

 70 cm,

= 5000 m ,



 9 arcsec.

Note however, that the ``isoplanatic patch for image motion"
(not wavefront) is  0.3

D/H

. For

= 4m,

= 5000 m,



kin



50 arcsec.

Another useful parameter is the correlation time



, which is approximately the dimension of the typical air

bubble divided by the velocity of the wind. As

also



increases with



6/5

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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:

5/3

3.44( / )

MTF

where



is the spatial frequency. Note that a Gaussian goes as



, 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.

(

)

(

)

y p

x p

�

A potentially better
empirical fitting function is
a Moffat function:

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Quality of the image -2

Probably the most common way of describing the seeing is by
specifying the full-width-half-maximum (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

0.1

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Effects of the atmosphere at

radiofrequencies - 1

The

ionosphere

will

introduce a delay on the
arrival time of the wave,
given by:

40.3

N s

D =

�

seconds, being

the path

along the line of sight and

the electron density

(cm

-3

). This density will

vary with the night and
day cycle, with the season
and also with the solar
cycle.

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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:

0.0014

7(cos

) ns

0.0445 cot

D �

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

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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

particular atomic and molecular bands (

this is in addition

to scattering of artificial lights

, see later).

The molecular oxygen

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).

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Extinction

Let us consider the absorption due to a thin layer of
atmosphere at height between

and

h+dh

in the usual

simple model of a plane-parallel atmosphere. The light beam
from the star makes an angle

with the Zenith, so that the

traversed path is

dh/cosz = seczdh



(h)

is the intensity at the top of the layer, at the exit it

will be reduced by the quantity:

( ) ( )sec d

I h k h

z h

In total, if



()

is the intensity outside the atmosphere, at

the elevation

of the Observatory the intensity will be

reduced to:

sec

( ) d

( ) sec

( )

z k h h

I h

�

��

�

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Optical Depth

where we have introduced the a-dimensional quantity





called

optical depth

( ) d

k h

t =

�

( ) d

k h

�

The variable



(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

-2.5log

(see in a later lecture), so that:

ground

2.5 ( ) sec

outsi e

� �



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

= 60° (the limit of

validity of the present approximate discussion).

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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 (

, sec

It is common practice to plot the

-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

= 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:

sec

( )

sin sin

cos cos cos

X z

where



is the latitude of the site,



and

the coordinates

of the star.

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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.

Wavelength (nm)

Extinction

(mag / air mass)

Wavelength

(nm)

Extinction

(mag / air mass)

310

1.37

500

0.13

320

0.82

550

0.12

340

0.51

600

0.11

360

0.37

650

0.11

380

0.30

700

0.10

400

0.25

800

0.07

450

0.17

900

0.05

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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

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  20

mag per square arcsec at the

Zenith can be assumed.

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A first exercise of celestial

mechanics

Consider the total energy

of a particle

of very small mass

at the surface of a non-rotating spherical body

of radius

and mass

1 2

The limiting velocity

2Gm

is said escape velocity from body

. If by some means we

impart to

a velocity V greater than

in any direction

will reach infinity with final velocity greater than zero.
Another useful critical velocity is that on the

circular orbit

distance

r>R

from the center of

; from the equilibrium

between centrifugal and gravitational forces we get:

1 2

( )

V r

( )

V R

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Escape velocities from the 9

planets

The table provides escape and circular velocities for the 9 planets,
neglecting their diurnal rotation. The 3

column gives the surface

gravity in comparison with that at the Earth’s surface (9.78 m/s

). The

first two velocities (4

and 5

column) pertain to the equator of each

body; the other two velocities (6

and 7

column) to the circular orbit at

the average distance of the body from the Sun.

Body

Distanc

(AU)

Mass

(g)

Radius

(km)

g/g



(km/s)

(⊙)

(km/s

)

(⊙)

(km/s

)

Sun

1.9910

6.9610

27.9

618

437

Mercur
y

0.387

3.310

2439

0.3

4.3

2.5

Venus

0.723

4.910

6051

0.9

10.4

7.3

Earth

1.000

6.010

6378

1.0

11.2

7.9

Moon

1.000

7.310

1738

0.2

2.4

1.6

Mars

1.524

6.410

3393

0.4

5.0

3.6

Jupiter

5.203

1.910

71492

2.3

59.6

42.5

Saturn

9.539

5.710

60268

0.9

35.5

25.0

Uranus

19.191

8.710

25559

0.8

21.1

15.5

Neptun
e

30.061

1.010

24764

1.1

23.6

16.0

Pluto

39.529

1.310

1150

0.04

1.1

0.8

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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

(in

Kelvin) be the temperature of such an atmosphere, supposed in
thermal equilibrium; the distribution function of molecules of
mass

among the velocities is given by Maxwell’s law:

dN V

N m

V e

( )



G IKJ



3 2



so that the mean square velocity of those
molecules will be:



where

= 1.3810

-16

erg/K is Boltzmann constant. For

instance, the mass of the Hydrogen atom H is

 1.610

-24

g, so that:

V T

( )

 



16 10

km/s (

in K)

At the surface of the Earth, assuming

 290 K we get

 2.7

km/s <<

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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, Maxwell’s distribution has a

very long tail at high velocity, so that a fraction of the Earth’s
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.

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A second exercise of celestial

mechanics

Let us launch from the surface of a spherical non-rotating
Earth of radius

⊕

a satellite of mass

with initial velocity

> V

. Its energy will be:

�

(

⊕

)

At an altitude

, the distance from the centre becomes

r = a

⊕

+ H

, and the energy:

�

or else, equating the two values for the conservation of the
energy:

�

1 1

�

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Delta V

At infinity:

(

)(

) 2

V V V V

V V

�

= -

� D

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.

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Geostationary orbits

= 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.

Document Outline