Murcia - Astronomy Lecture Notes

7/17/2019 Murcia - Astronomy Lecture Notes

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2. SPHERICAL TRIGONOMETRY

Spherical trigonometry is a mathematical branch that threads numerical relations between sides and angles fo

spherical triangles.

We define the angle between 2 planes that cut themselves as “died!al angle", and the straight line where the

intersect is called “died!al edge"

Is called “#a$i#%# &i!&%#'e!en&e" the circumference that yields from te inte!se&ti(n between a spe!e andplane tat pass t!(%g te spe!e &ente!

Is defined as p(les )P and P*+ of an specific (maximum or minimum circumference) the p(ints in te spe

s%!'a&e that yields from the inte!se&ti(n of it and a st!aigt line pe!pendi&%la! t( te plane that create succircumference.

“Spe!i&al angle" is called te angle &!eated b, tw( #a$i#%# &i!&%#'e!en&es

-The dihedral angle created by !"# is e$ual to the plane angle "# and also corresponds to the arc #.

“Spe!i&al t!iangle" is called that region (e! a spe!e s%!'a&e that is b(%nded b, t!ee #a$i#%

&i!&%#'e!en&es, the arcs correspond to the triangle %sides&, and the three spherical angle vertices are calle

spherical triangle %vertices&.

a/ b/ and & sides of an spherical triangle are measured by the angle faces #"', '", "#, respectively from th

trihedral angle.

or spherical triangles with spherical angles or sides between 0 and 103

-The summation of whatever * sides are greater than the third side.

-The summation of the + sides is lower than +-

-If * sides are e$ual, the opposite angles are e$ual too.

-The + angles summation is greater than /0- but lower than +-



Sine law for spherical triangles

Cosine law for spherical triangles

Sine by cosine theorem for spherical triangles



4. THE EARTH PLANET

It is a planet that is distant 150 #illi(ns of 1ilometers to one medium star that we called “S%n".

It has a uni$ue satellite called “M((n" located at 46 600 7il(#ete!s of distance.

In order of distance this is the ti!d planet from inside out around the sun.

2ecent techni$ues show that it is a e!, an&ient planet that e$ual to the solar system has an age of 6800 #illi(n

,ea!s.

The planet rotation is responsible to the light mass accumulation over the e$uator, in the earth the differenc between e$uator and pole radio is 9%st 21 45 #ete!s.

Is called “ge(id" the figure which tries to represent the real earth shape, doing coincide it with the #ean sea le

)MSL+ and continuing over the continental region li1e an imaginary surface, in practice 3eoid is almost impossibto identify with an easy geometric figure, due to it turns out in a completely irregular shape, therefore a g((

app!($i#ati(n is a revolution ellipsoid that is called also a “spe!(id": 3eoid can be above or below of th

revolution ellipsoid as 100 #, that difference is called “ge(id %nd%lati(n&

2otational axis coincides approximately with the earth main inertia axis

The plane tat is pe!pendi&%la! t( te !(tati(n a$is and inte!se&ts te spe!(id s%!'a&e and passes through th

center of mass is called “te!!est!ial e;%at(!"

Geocentric coordinates

This coordinate system has as (!igin the center the earth &ente! (' #ass



<* 4 geocentric latitude (angle between the line that passes through the point and the earth center and th

terrestrial e$uator)

=* 4 geocentric longitude (angle between #eas%!ed (e! te e;%at(! from the reference meridian and th

corresponding point meridian)

>* 4 radial distance from the earth center of mass.

Geodesic coordinates

This coordinate system is referred by !e(l%ti(n ellips(id

In this system are made the ma5ority of maps, atlas and geographic dictionaries.

< 4 geodesic latitude (angle between the normal of the spheroid and the terrestrial e$uator) it can defer to thgeocentric latitude until //.6 arc minutes at 76 of latitude.

= 4 geodesic longitude (angle between measured over the e$uator from the reference meridian and thcorresponding point meridian) same as in geocentric =? =*

4 observer height ( is the distance above the ellipsoid measured along the normal of that spheroid)

Latitudes transformation

Ge(&ent!i& altit%de t( ge(desi& latit%de

"r



Where ' ? (a-b) /a = 1/298.257.

@ and , &(#p(nents '!(# ge(desi& latit%de.

Where e ? )'-)2'++ B )12+

7. '898STI9 :;9TFundamental concepts



'ontemplating celestial bodies the dept pe!&epti(n and te distan&e esti#ati(n disappea! and every celesti bodies give the sensation to be attached to a surface, which extend in all directions and create the illusion

conform a pe!'e&t spe!e where all bodies loo1 are attached at the same distance from the center where th

observer is located, this sphere is called “&elestial a%lt".

To be located in a planet s%!'a&e yields that the observer 5ust can see the al' (' te &elestial a%lt due to th

same planet prevent to see the other half.

n observer located at %ge distan&e '!(# a planet surface, doesn<t have not any issue to see the &(#ple

&elestial a%lt.

The earth has a !(tati(nal #(e#ent in westeast di!e&ti(n that can be seen by an observer located over th

planet surface as a rotational movement eastwest (' te &elestial a%lt.

The following concepts are independent of the (bse!e! l(&ati(n

Celestial N(!t P(le )PNC+ and &elestial S(%t P(le )PSC+ re the points that yield from the intersection of th

earth rotational axis and the celestial vault surface.

Celestial e;%at(!D )EC+D Is the maximum circumference that yields from the intersection of the Terrestrial e$uat

and the celestial vault dividing the complete sphere in two celestial =orth !ole and celestial South !ole.

Celestial #e!idiansD They are maximum semi>circumferences that come from the celestial =orth !ole to th

celestial South !ole? also they can be explained as the pro5ection of terrestrial meridians in the celestial vault.

The following concepts are dependent of the (bse!e! l(&ati(n

Cenit (! enit )C+D Is a celestial vault point located above the observer head.

Nadi! )C*+D Is the diametrical opposed point of 'enit over the celestial vault, in other words it is a celestial vau

point located directly below the observer feet.

Obse!e! (!iF(n Is the plane perpendicular to the line that exists from the observer and his cenit.

Obse!e! #e!idianD Is a celestial meridian that passes through the observer cenit.



Ca!dinal p(intsD re those points (=orth) (South) (8ast) (West) located in the hori@on that has the followin

features

N(!tD Is the pro5ection of celestial =orth !ole to the hori@on through the observer meridian

S(%tD 8$ual than =orth but using the celestial South !ole.

EastD When the observer is loo1ing to the north, at his right hand the east is located

estD 9ocated at /0- from east

Ast!( e!ti&alD Is the semi>circumference that comes from the observer 'enit (') to the observer nadir ('

through the specific astro.

Ast!( de&linati(n &i!&le (! Ast!( #e!idianD Is the semi>circumference that comes from 'elestial =orth !ole

the 'elestial South !ole through the astro.

Sky observation from the earth



Is not the same to observe the s1y in the Terrestrial =orth !ole (!=T) than in the Terrestrial South pole (!ST), ithe =orth !ole the observer<s hori@on coincide with the 'elestial e$uator an observer can see every stars in th

celestial north hemisphere but never the stars in the celestial south hemisphere.

PNC eigt (! Celestial N(!t P(le eigtD Is the angle between the observer hori@on and the 'elestial =or

!ole !='.

t !=T !=' height is A-

t !ST !=' height is >A-

t e$uator !=' height is - and &an see te enti!e &elestial a%lt b%t n(t si#%ltane(%sl,.

The N(!t P(le eigt !=' height (angle between !=' and observer hori@ont) is e;%al t( te latit%de where th

observer is located.

There is a star called “P(la!is" that is located almost at the same point over the celestial vault (near 76<) to th

'elestial =orth !ole !=', and #eas%!e te angle between te (bse!e! and “P(la!is" #aes te 'i!st atte#p

t( esti#ate te (bse!e! latit%de.

The ecliptic and Seasons

We as universe observers see the translation movement of the earth reflected in the Sun movement respect to th

“'i$ed sta!s"

The earth moves in counter cloc1wise direction seeing from the 'elestial =orth !ole !='

The maximum circumference produced by the inte!se&ti(n (' te ea!t a!(%nd te s%n #(e#ent plane an

te &elestial a%lt is &alled “e&lipti&"

The angle between the normal to the ecliptic plane and the earth rotational axis is called “e&lipti& (bli;%it," )

and is approximately 24.53 but due to Boon, Sun, and planetary gravitation interaction this angle slightly changes

The :ernal e$uinox is one of * points where the e&lipti& plane and &elestial e;%at(! plane inte!se&t te &elesti

a%lt specifically where the sun crosses the e$uator from the celestial South !ole to the celestial =orth !ole, th

other point is called the anti>vernal point and is located /0- to the vernal e$uinox.

The true reason because there are seas(ns is the existence of the e&lipti& (bli;%it,

round the year te s%n &!(sses 2 ti#es te e;%at(!/ t(se da,s a!e &alled “e;%in($es" and occ

approximately */ or ** of Barch and */ or ** of September? once the sun crosses the e$uator moves away unt

reach a #a$i#%# sepa!ati(n with it that is e$ual to )24.53+, those * point are called “s(lsti&es".

S%##e! is expressed has the ti#e we!e te!e a!e #(!e s%n !adiati(n in te!#s (' da, d%!ati(n it means th

observers in this earth hemisphere see the sun for more than /* hours



or us is called “&(nstellati(n" (ne (' te pa!ts in wi& te &elestial a%lt is diided and usually represenmythological figures of different cultures through the history.



6. '898STI9 '""2CI=T8S

To avoid confusion and express the astro location over the celestial vault astronomers commonly use

Horizontal coordinates

s reference plane it has the (bse!e! (!iF(n, those coordinates allow locating the apparent location of an astrfor an observer in a specific latitude and longitude.

AFi#%t )A+ ? this is the angle measured over the hori@on from the &a!dinal N(!t p(int t( te ast!( e!ti&al the east direction, it has values between - to +-

Heigt )+ ? Is the measured angle t!(%g te ast!( e!ti&al between te (!iF(n and te ast!(, it has valu between >A- to A-.

“Cenital distan&e )F+" is the complement of the height angle namely @ 4 A->h

Cue to diurnal movement the coordinates change, te!e'(!e is needed t( spe&i', te ti#e wit a %g

e$a&tit%de.

Hourly euatorial coordinates

They have the &elestial e;%at(! as !e'e!en&e plane

H(%!l, angle )H+ ? is the angle measured over te e;%at(! from the (bse!e! #e!idian in west di!e&ti(n to th

ast!( de&linati(n &i!&le (or astro meridian)? this angle is expressed in terms of time units (/6 4 /hour)

Je&linati(n angle )K+ 4 is the angle measured al(ng te de&linati(n &i!&le (astro meridian) from the e;%at(!

te ast!(, if the astro is located in the =orth hemisphere D is positive contrarily negative.

lthough declination angle is e$ual for an observer located in whatever geographical location and time, te (%!

angle n(t



!uatorial coordinates or "bsolute coordinates

Eis reference plane is the celestial e$uator.

St!aigt as&ensi(n )+ ? is the angle measured al(ng te e;%at(! '!(# te e!nal p(int )e!nal e;%in($+ t( t

ast!( de&linati(n &i!&le (astro meridian), in &(%nte!&l(&wise direction loo1ed from the celestial =orth !ole !='

Je&linati(n angle )K+ ? e$ual to the hourly e$uatorial coordinates, is the angle measured along the declinatio

circle from the e$uator to the astro.

8$ual than the declination circle straight ascension is commonly measured in time units (/64/hr)

Those coordinates a!e abs(l%tes and independent t( te ge(g!api&al l(&ati(n (! ti#e, and therefoastronomical almanac express the location of planets, star, etc.. in such coordinates



!cliptic coordinates

They have as reference plane the e&lipti& plane.

E&lipti& l(ngit%de )=+ 4 ngle measured al(ng te e&lipti& plane '!(# e!nal P(int (:ernal e$uinox) to thsemi>circumference that passes through the e&lipti& N(!t P(le )+/ te ast!( and 'inises in te e&lipti& S(%t

P(le )*+/ measure in counter>cloc1wise direction seen from the 'elestial =orth !ole (!='), it can ta1es valu

from - to +-.

E&lipti& latit%de )+ ? is the angle measured from the semi>circumference that passes through the ecliptic pol

and the astro (ecliptic astro meridian) to the astro.

Coordinates transformation #horizontal to hourly euatorial or vice versa#



Coordinates transformation #hourly euatorial to absolute euatorial or vice versa#

Cue to declination (D) is e$ual for both only must to be considered is the relation between straight ascension an

hourly angle

9ocal sidereal time (TS9) is defined as the hourly angle to the vernal point for an observer in specific instant o

time.

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Murcia - Astronomy Lecture Notes