N = R *! f p! n e! f l! f i! f c! L

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Terence Davidson
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1 Astonomy 330 Pesentations Raeed Chowdhuy: The Science of AVATAR Music: What s the Fequency Kenneth? REM We have a numbe! Outline What does it mean? Okay, now how to communicate with them? we can contact in ou Galaxy today Dake Equation That s 190,540 civs!!! N = R *! f p! n e! f l! f i! f c! L Sta fomation ate 10 stas/ y of stas with 0.75 systems/ sta Eathlike pe system 1.5 x 0.11 = / system on which life aises 0.47 life/ planet that evolve intelligence 0.42 intel./ life Fank Dake that communicate Lifetime of.78 1x10 6 comm./ ys/ intel. comm.

2 = 2.5 x Communicating Civilizations Dake Equation Fo Optimist 62.5% of all stas in ou Galaxy. = 7.5 x 10-6 Communicating Civilizations Dake Equation Fo Pessimist Must wait 10 7 yeas fo one! N = R *! f p! n e! f l! f i! f c! L we can contact Rate of fomation of Sunlike stas of stas with Eathlike pe system on which life aises that evolve intelligence that communicate Lifetime of x 10 9 N = R *! f p! n e! f l! f i! f c! L we can contact Rate of fomation of Sunlike stas of stas with Eathlike pe system on which life aises that evolve intelligence that communicate Lifetime of Bithate of 50/yea! Astonomy 330 Sping 2008 Bithate of 7.5 x 10-8 / yea! Astonomy 330 Sping 2008 = 930,000 Communicating Civilizations we can contact Dake Equation Fo Aveage N = R *! f p! n e! f l! f i! f c! L Rate of fomation of Sunlike stas of stas with Eathlike pe system on which life aises that evolve intelligence that communicate Lifetime of x10 6 Hmm.. None of these esults ae wong. The aveage bithate esult of aound 1/yea would suggest that any life that is contacted is pesumable olde and theefoe moe. It is inteesting to note that fo ou values, lifetimes geate than aound 100 yeas give about 2 with which to talk. Ou total numbe was 190,540 yeas. Bithate of 0.93 /yea! Astonomy 330 Sping 2008

3 Distance to Neaest Distance to Neaest Assume that the alien ae unifomly scatteed in ou galaxy and N > We can then assume spheical volume to find ET, i.e. flatness of Galaxy not an issue. Assume N > 8000 Aveage Galactic Volume Numbe of Civilizations = " 2 galaxyh galaxy N = alien volume (ly 3 /civ) Neaest Neaest lys lys 100,000 light yeas 100,000 light yeas Distance to Neaest Assume N > 8000 Distance to Neaest Assume N > 8000 alien volume = " 2 galaxyh galaxy N = 7.85 #1012 lys 3 N alien volume = " 2 galaxyh galaxy N = 7.85 #1012 lys 3 N Neaest Neaest lys lys 100,000 light yeas 100,000 light yeas

4 Distance to Neaest Assume N > 8000 Distance to Neaest Assume N > 8000 Then Neaest Neaest lys lys 100,000 light yeas alien volume = " 2 galaxyh galaxy N 100,000 light yeas Distance to Neaest Distance to Neaest Assume that the alien ae unifomly scatteed in ou galaxy and N < Then, the flatness of Galaxy is an issue. Assume N < 8000 Aveage Galactic Volume Numbe of Civilizations = " 2 galaxyh galaxy N = alien volume Neaest Neaest lys lys 100,000 light yeas 100,000 light yeas

5 Distance to Neaest Assume N < 8000 Distance to Neaest Assume N < 8000 Then Neaest Neaest lys lys 100,000 light yeas 100,000 light yeas The s Inteesting Points We need to look at evey sta within ~200 lys fo one detection! ing N=190,540 1.! We assumed unifom density of. Undeweights the galactic cente, but maybe that s okay supenovae. 2.! Distance away is the aveage. Could be close, but unlikely to be much close. 3.! Note that is bette defined than N. R depends on N 1/2 o N 1/3. If we ae wong in N by a facto of 100, then only off in by factos of 10 o 4, espectively. 4.! Fo communication, it may be that the distance thee and back is longe than L.

How to Communicate? Okay, ou estimate is optimistic. So, how do we go about detecting ou neighbos?! About 200 lys away (maybe pehaps ) Ae we seiously sending out messages now? No. How to Communicate?

6 How to Communicate? Okay, ou estimate is optimistic. So, how do we go about detecting ou neighbos?! About 200 lys away (maybe pehaps ) Ae we seiously sending out messages now? No. How to Communicate? We ae elatively a young civilization, with adio technology fo only a hunded yeas. Right now, we ae mostly a passive luke civilization. Okay, so what will an civilization use? Had to figue out.. They ae aliens! Light me up Visible light is only a tiny potion of the full electomagnetic spectum Red light has longe wavelength and lowe fequency than blue light. Divisions between egions ae fom biology o technologies. Question I want to communicate with aliens at a distance of 100 light yeas. What is the fastest way to do that? a)! X-ays b)! Radio c)! Visible light d)! Gamma-ays e)! All of the above ae light, so tavel the same speed.

Fequency The fequency of light depends on its colo.

Fo adio waves, we nomally use lage units! 1 khz = Hz! 1 MHz = 10 6 Hz!

fequency? a)! 100 Hz b)! 100 khz. c)! 100 MHz d)!

the following has the longest wavelength? a)! 100 Hz b)! 100 khz. c)!

We can t boadcast ove the whole ange too much powe = expensive.

7 Fequency The fequency of light depends on its colo. The unit is Hetz, equivalent to 1 cycle a second. Fo adio waves, we nomally use lage units! 1 khz = Hz! 1 MHz = 10 6 Hz! 1 GHz = 10 9 Hz Fequency Question Which of the following is the highest fequency? a)! 100 Hz b)! 100 khz. c)! 100 MHz d)! 100 GHz Wavelength Ap 10, 2008 Astonomy 330 Sping 2008 Question Which of the following has the longest wavelength? a)! 100 Hz b)! 100 khz. c)! 100 MHz d)! 100 GHz What s the Fequency Kenneth? We can t boadcast ove the whole ange too much powe = expensive. So what kind of easoning can we use to limit ou seach o any boadcasts? Keep in mind that ET must make the same decisions. May be vey alien decisions.

What s the Fequency Kenneth? Want biggest bang fo the buck. Intestella dust is in the Galactic plane Attenuates light that is shote than infaed wavelengths a few micons. O need vey high fequency.

In fact, light deceases in amplitude as the squae of the distance taveled. And like you adio, thee can be noise fom competing stations o noise fom the adio eceives.

! The galaxy emits lots of emission at low fequencies. 2.! The Big Bang backgound noise CMB. 3.! Noise of eceives. The pefect eceive has a quantum limit of one photon noise. 4.

8 What s the Fequency Kenneth? Want biggest bang fo the buck. Intestella dust is in the Galactic plane Attenuates light that is shote than infaed wavelengths a few micons. O need vey high fequency. Enegy equied fo the photon inceases with fequency. Agues fo low fequency o long wavelength opeation adio. Feq Show Keep in mind that adio stations fade as you get futhe away. In fact, light deceases in amplitude as the squae of the distance taveled. And like you adio, thee can be noise fom competing stations o noise fom the adio eceives. emits lots of emission at low fequencies. The best place to listen in the quiet pat of the spectum 1.! The galaxy emits lots of emission at low fequencies. 2.! The Big Bang backgound noise CMB. 3.! Noise of eceives. The pefect eceive has a quantum limit of one photon noise. 4.! The Eath s atmosphee blocks many fequencies. Feq-ing Out. Wavelengths of 3 to 30 cm! Fequencies of 1 to 10 GHz! O Lases? Chalie Townes has pointed out that sending pulses of lase light could be competitive. A numbe of seaches ae now undeway using visible light optical SETI. The light must be distinguishable fom the sta. It is easy fo to ovewhelm thei suns in adio waves, but not visible. But, poweful lases have a cetain defined wavelength. Lase fo adaptive optics, not optical SETI.

ETs with Lases? How to Communicate? Reines and Macy in 2002 seached 577 neaby stas with sensitivity to detect >60 kw lases focused fom a 10m telescope. Nothing was detected.

Ae aliens tying to contact us with LASERs? http://www.insomniacmania.com/news/news_771_1.jpg Radio is pobably best. 1.! Dust extinction is educed. 2.! Lowe fequency means less enegy/photon, so cheape.

html Big Band The Magical 1420 Still, 1-100 GHz o even 1-10 GHz is a lot of fequency to seach. Remembe, we have to tune to the pope adio station. What s the ight channel size?

9 ETs with Lases? How to Communicate? Reines and Macy in 2002 seached 577 neaby stas with sensitivity to detect >60 kw lases focused fom a 10m telescope. Nothing was detected. Lase is a vey small beam of light, only a few stas in tansmission beam. But stength of lase does not decease as quickly as adio. Lase seems an unlikely communication tool though. Ae aliens tying to contact us with LASERs? Radio is pobably best. 1.! Dust extinction is educed. 2.! Lowe fequency means less enegy/photon, so cheape. 3.! Thee is a natual dip fom about 1 to 10 GHz in the adio whee the atmosphee and the galaxy ae the quietest. Big Band The Magical 1420 Still, GHz o even 1-10 GHz is a lot of fequency to seach. Remembe, we have to tune to the pope adio station. What s the ight channel size? Many ague that we should use 1 Hz channels, then in the 1-10 GHz band thee ae 9 x 10 9 channels! Is thee a magic fequency that would choose? Moison and Cocconi (1959) suggested the fist magical fequency of 1420 MHz o GHz. It s the fequency at which H atoms in space emit and absob adiation (21 cm line). Not a bad choice as H is the most abundant atom in the Univese. But, now we have detected ove 100 molecula tansitions, some cucial to life, so maybe not as an impotant agument as it once was.

Cal Sagan and Fank Dake suggested that species on Eath always gatheed aound the wate hole. Thee is a molecula fagment of OH that absobs at 4 fequencies between 1.612 and 1.720 GHz.

10 Cal Sagan and Fank Dake suggested that species on Eath always gatheed aound the wate hole. Thee is a molecula fagment of OH that absobs at 4 fequencies between and GHz. These molecules wee well studied at the time, so it was biased. And, now we know about moe exciting tansitions at highe fequencies. The Wate Hole? Wavelengths of 3 to 30 cm! Fequencies of 1 to 10 GHz! Fundamental Feqs What ae constants that evey civilization would be awae of? Speed of light Fine stuctue constant (1/137) Divide the speed of light as many times as necessay to get a fequency in the adio ange. In that case you get GHz. Fist suggested by Kuipe and Mois. No. Magical Fequency? Nothing is eally obvious. So, we e scewed. We have to look though a lot of adio fequencies. So, we bette undestand adio techniques a little. Radio The basic concept of adio astonomy, adio communications, television, mobile phones, etc. is the same. Infomation is tansmitted by low enegy light.

11 Radio telescopes Radio Pioneeing wok by Gote Rebe in back yad, Wheaton, Illinois. (He died in 2002). How does the antenna on you ca wok? The electo-magnetic wave cause electons to move up and down in you antenna. That signal is amplified and decoded. Fo fequencies in the band of inteest, paabolic antennas ae common used. Aecibo Obsevatoy, Pueto Rico Lagest adio telescope 300 metes. The Geen Bank Telescope W.V. The lagest fully steeable dish in the wold 100 metes

Geenbank WV Vey Lage Aay, nea Magdalena, NM Decode

time) ually analog, but digital is moe obust Can

Afte eceiving and amplifying the signals, one has

Natually ceated signals do not usually vay with

12 Geenbank WV Vey Lage Aay, nea Magdalena, NM Decode Ring Decode Ring Nomally, atificial signals encode data:! FM : fequency modulation (fequency vaies with time)! AM : amplitude modulation (bightness vaies with time) ually analog, but digital is moe obust Can tun on/off to signify 1 o 0 (most likely fo ET) Afte eceiving and amplifying the signals, one has to decode the signals. Natually ceated signals do not usually vay with time and ae unpolaized. Note, most astonomes do not look fo fast vaying signals, but weak nonvaying signals.

LGM But, astonomes studying the shot vaiations in the intestella medium did look at fast vaying signals. Jocelyn Bell noticed a egulaly epeating signal. Pefect timing, but no eal encoding.

html Jocelyn Bell Bunell Anthony Hewish Built to be Decoded If a signal is found, how do we decode it?

To date thee has only been one seious message sent fom Eath.

100 kw beamed o 20 tillion watts of powe if omni-diectional. Stongest man-made signal eve sent.!!!!!!!!!!"!"!"!"!!!!!!#!!"!!!!!!!"!"!!!!!"!"!!#!"!!""!"!!"!!!"!!!"!!!"#!!"!!"!!"!"!"!"!"!"!"!"#!!!!!!!!!""!!!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!"!"!!!!!!!!!#!!!!!!!!!"""""!

13 LGM But, astonomes studying the shot vaiations in the intestella medium did look at fast vaying signals. Jocelyn Bell noticed a egulaly epeating signal. Pefect timing, but no eal encoding. Jokingly called LGMs, then Pulsas. Eventually ealized to be fom neuton stas. The lighthouse beam fom the apid otato sometimes intesecting the Eath. Jocelyn Bell Bunell Anthony Hewish Built to be Decoded If a signal is found, how do we decode it? Most coding is meant to hide the signal, but in this case we want it to be decoded by any intelligence. Obviously this is not tivial. Many suggestions that evolve aound mathematics have been made. To date thee has only been one seious message sent fom Eath. The Aecibo Message On Novembe 16, 1974 Cal Sagan and Fank Dake sent a message fo 3 minutes Fequency used was 2380 MHz, with fequency modulation (10 Hz) ed the Aecibo telescope with a lage tansmitte 100 kw beamed o 20 tillion watts of powe if omni-diectional. Stongest man-made signal eve sent.!!!!!!!!!!"!"!"!"!!!!!!#!!"!!!!!!!"!"!!!!!"!"!!#!"!!""!"!!"!!!"!!!"!!!"#!!"!!"!!"!"!"!"!"!"!"!"#!!!!!!!!!""!!!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!"!"!!!!!!!!!#!!!!!!!!!"""""!!!!!!!!!#!!!""!!!!""!!!"""!!!!""#!!!!"!!""!!!!!!!!!!!!!"#!"!""!!!!""!!!""!!!"!""# """""!"""""!"""""!"""""#!"!!!!!!!!!!!!!!!!!"!!!# "!!!!!!!!!!!!!!!!!"!!!!# """""!!!!!!!!!!!!!"""""#!!!""!!"""!!!!!""!!!!""#!!!!"!!!!!!!!!"!!!!!!!"#!"!""!!"""!!!""!!!!"!""# """""!"""""!"""""!"""""#!"!!!!!!!!!""!!!!!!"!!!#!!!!!!!!!!!""!!!!!!!!!!# "!!!!!!!!!!""!!!!!"!!!!# """""!!!!!!""!!!!!"""""#!!!!!!!!!!!""!!!!!!!!!!#!!"!!!!!!!!"!!!!!!!!"!!#!!!"!!!!!!!""!!!!!!"!!!#!!!!"!!!!!!""!!!!""!!!!#!!!!!""!!!!"!!!""!!!!!!#!!!!!!!""!!""!!!!!!!!!!#!!!!!""!!!!"!!!""!!!!!!#!!!"!!!!!!!""!!!!""!!!!#!!!"!!!!!!!!"!!!!!!"!!!#!!"!!!!!!!!""!!!!!!!"!!#!!"!!!!!!!!""!!!!!!!!"!#!!!"!!!!!!!"!!!!!!!!!"!#!!!!"!!!!!!!"!!!!!!!"!!#!!!!!""!!!!!!!!!!!!"!!!#!!!!!!!""!!!!!!!!""!!!!#!!!!!!!!!""!"!"""!!!"!!#!!!!!!!!!!!!"!!!!!!!"!!#!!!!!!!!!!"""""!!!!!"!!# ""!""!"!!"!"""!"!!!!"!!# """"""!!"!!"""!!"!!!!!!# """!""!!!!!"""!!!!"""!"# ""!"""!!!!!"!"!!!!!!!!!# """"""!!!!!"!"!!!!!!"!!#!!!!""!!!!!"!"!!!!!!"!!#!!!!!!!!!!""!""!!!!!"!!#!!!!!!!!!!!!"!!!!!"""!!# "!"!"!"!"!"!!!"!"!"""!!#!!"!"!"!"!!!!!!!!!"""!!#!!!!!!"!"!!!!!!!!!!!!!!#!!!!!!!!!!"""""!!!!!!!!#!!!!!!!!"""""""""!!!!!!#!!!!!!"""!!!!!!!"""!!!!#!!!!!""!!!!!!!!!!!""!!!#!!!!""!"!!!!!!!!!"!""!!#!!!""!!""!!!!!!!""!!""!#!!!"!!!"!"!!!!!"!"!!!"!#!!!"!!!"!!"!!!"!!"!!!"!#!!!!!!!"!!!"!"!!!"!!!!!#!!!!!!!"!!!!"!!!!"!!!!!#!!!!!!!"!!!!!!!!!"!!!!!#!!!!!!!!!!"!"!!"!!!!!!!#!!!""""!!"!"""""!!""""!# Ap 15, 2008 Can you Figue it out? See if you can decode anything. By fequency modulation, they sent 1679 bits of 1/on and 0/off is the poduct of 2 pime numbes 23 and 73. ET should be able to ty aanging them into a pictue.

!!!!!!!!!!"!"!"!"!!!!!!#!!"!!!!!!!"!"!!!!!"!"!!#!"!!""!"!!"!!!"!!!"!!!"#!!"!!"!!"!"!"!"!"!"!"!"#!!!!!!!!!""!!!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!"!"!!!!!!!!!#!!!!!!!!!"""""!

!"!!"""!!"!!!!!!# """!""!!!!!"""!!!!"""!"# ""!"""!!!!!"!"!!!!!!!!!# """"""!!!!!"!"!!!!!!"!!#!!!!""!!!!!"!"!!!!!!"!!#!!!!!!!!!!""!""!!!!!"!!#!!!!!!!!!!!!"!!!!!"""!!# "!"!"!"!"!"!!!"!"!"""!!#!!"!"!"!"!!!!!!!!!"""!!#!!!!!!"!"!!!!!!!!!!!!!!#!!!!!!!!!!"""""!!!!!!!!#!!!!!!!!"""""""""!

!"!"""""!!""""!# Ap 15, 2008 We attempted Contact a zeo a one Now, we wait. Sent towad the globula cluste M13 which is 21,000 lys away. If they e looking, any SETI expeiment will detect this.

14 !!!!!!!!!!"!"!"!"!!!!!!#!!"!!!!!!!"!"!!!!!"!"!!#!"!!""!"!!"!!!"!!!"!!!"#!!"!!"!!"!"!"!"!"!"!"!"#!!!!!!!!!""!!!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!""!!!!!!!!!!#!!!!!!!!!"!"!"!!!!!!!!!#!!!!!!!!!"""""!!!!!!!!!#!!!""!!!!""!!!"""!!!!""#!!!!"!!""!!!!!!!!!!!!!"#!"!""!!!!""!!!""!!!"!""# """""!"""""!"""""!"""""#!"!!!!!!!!!!!!!!!!!"!!!# "!!!!!!!!!!!!!!!!!"!!!!# """""!!!!!!!!!!!!!"""""#!!!""!!"""!!!!!""!!!!""#!!!!"!!!!!!!!!"!!!!!!!"#!"!""!!"""!!!""!!!!"!""# """""!"""""!"""""!"""""#!"!!!!!!!!!""!!!!!!"!!!#!!!!!!!!!!!""!!!!!!!!!!# "!!!!!!!!!!""!!!!!"!!!!# """""!!!!!!""!!!!!"""""#!!!!!!!!!!!""!!!!!!!!!!#!!"!!!!!!!!"!!!!!!!!"!!#!!!"!!!!!!!""!!!!!!"!!!#!!!!"!!!!!!""!!!!""!!!!#!!!!!""!!!!"!!!""!!!!!!#!!!!!!!""!!""!!!!!!!!!!#!!!!!""!!!!"!!!""!!!!!!#!!!"!!!!!!!""!!!!""!!!!#!!!"!!!!!!!!"!!!!!!"!!!#!!"!!!!!!!!""!!!!!!!"!!#!!"!!!!!!!!""!!!!!!!!"!#!!!"!!!!!!!"!!!!!!!!!"!#!!!!"!!!!!!!"!!!!!!!"!!#!!!!!""!!!!!!!!!!!!"!!!#!!!!!!!""!!!!!!!!""!!!!#!!!!!!!!!""!"!"""!!!"!!#!!!!!!!!!!!!"!!!!!!!"!!#!!!!!!!!!!"""""!!!!!"!!# ""!""!"!!"!"""!"!!!!"!!# """"""!!"!!"""!!"!!!!!!# """!""!!!!!"""!!!!"""!"# ""!"""!!!!!"!"!!!!!!!!!# """"""!!!!!"!"!!!!!!"!!#!!!!""!!!!!"!"!!!!!!"!!#!!!!!!!!!!""!""!!!!!"!!#!!!!!!!!!!!!"!!!!!"""!!# "!"!"!"!"!"!!!"!"!"""!!#!!"!"!"!"!!!!!!!!!"""!!#!!!!!!"!"!!!!!!!!!!!!!!#!!!!!!!!!!"""""!!!!!!!!#!!!!!!!!"""""""""!!!!!!#!!!!!!"""!!!!!!!"""!!!!#!!!!!""!!!!!!!!!!!""!!!#!!!!""!"!!!!!!!!!"!""!!#!!!""!!""!!!!!!!""!!""!#!!!"!!!"!"!!!!!"!"!!!"!#!!!"!!!"!!"!!!"!!"!!!"!#!!!!!!!"!!!"!"!!!"!!!!!#!!!!!!!"!!!!"!!!!"!!!!!#!!!!!!!"!!!!!!!!!"!!!!!#!!!!!!!!!!"!"!!"!!!!!!!#!!!""""!!"!"""""!!""""!# Ap 15, 2008 We attempted Contact a zeo a one Now, we wait. Sent towad the globula cluste M13 which is 21,000 lys away. If they e looking, any SETI expeiment will detect this. But do we expect anyone? Decode what? An amazing amount of infomation in 1679 bits. But human expets had touble decoding it. Question In 21,000 yeas, when the Dake message gets to M13 a)! The aliens will be able to decode it, and enjoy images and sounds of the people of Eath. b)! The aliens may not be able to decode it. c)! The aliens will not be able to decode it, but with the additional hints sent aftewads, they will figue it out. d)! The aliens will be able to decode it and build thei own telescope with the knowledge. Encounte Messages Sent by commecial company based in Houston, Texas using the Evpatoiya Deep Space Cente adio telescope in Ukaine to 5 neaby stas less than 50 lys. Dake s message had 1,679 bits of infomation. This has 300,000 bits, with built-in edundancy. If some bits ae lost to noise en oute, ET might be able to decode. Astonomes deived code: Dutil & Dumas Included names and addess of 2000 donos and pesonal messages. Astonomy 330 Sping 2008

Encounte Messages Contacting What does an civilization that wants to contact us do? Could set-up adio beacons!

Would be much bette if they could use diectional messages.

small beam. The poblem is whee to look o to tansmit. http://en.wikipedia.

One solution is to look fo unintentional leakage signals. Leakage, as it leaks fom the planet s ionosphee.

15 Encounte Messages Contacting What does an civilization that wants to contact us do? Could set-up adio beacons! Boadcast in all diections.! Boadcast at seveal fequencies.! Would equie enomous enegy souces. Would be much bette if they could use diectional messages. Existing tansmittes on Aecibo ae stong enough to communicate acoss the galaxy with simila telescopes, but with a vey small beam. The poblem is whee to look o to tansmit. Does ET Love Lucy? One solution is to look fo unintentional leakage signals. Leakage, as it leaks fom the planet s ionosphee. We can not cuently detect this, but maybe othe can. This is the scenaio exploed in the novel Contact by Cal Sagan and the movie based on the novel. What leakage do we have? TV, FM Radio, ada Television tansmission exceeds 10 7 watts (10 MW). scientist/pofile/fanswoth.html

Figure Geometry for Computing the Antenna Parameters.

Figure Geometry for Computing the Antenna Parameters. Spheical Coodinate Systems Definitions Figue 1.2.1 Geomety fo Computing the Antenna Paametes. Antenna Radiation Patten: The distibution of adiated enegy fom an antenna ove a suface of constant adius centeed