Volume and listening times
A) Speed of sound propagation in the gas.
The speed of sound propagation in any gas can be obtained from the following formula: v=RTM (where it is the ratio between the specific heat cp of the gas in isobaric transformations, and the specific heat cv of the gas in isochoric transformations. The velocity is directly proportional to the temperature of the gas and inversely proportional to its molar mass.Taking into consideration as temperature the one used in the calculation of the gas density (87.15 K, average surface temperature of Saturn) we have respectively:
Helium case:
vprop. sound, He=20,812,58,31487,154,003=17355ms
Hydrogen case :
vprop. sound, H2=28,720,48,31487,152,016=24455 ms
Consequently, the time taken by the sound waves (i.e. the time that elapses between the emission of sound waves generated by the solicitation of the musical instruments by the orchestra at the center of the Solar System and their reception at the maximum possible distance inside the ellipsoid, which corresponds to the semimajor axis) is:
tHe=ssemimajor axisvHe=8.228101217355=4.74108 s=15 years
tH2=xsemimajor axis vH2=8.228101224455=3.3108 s=10.7 years
B) Listening volume.
To calculate how loud our musicians should play, it is first necessary to understand what the required energy would be, to do this we use the formula for sound intensity:
E=I4r2 t
Where "r" is set as the distance between the planet and the orchestra, consequently the Sun, and I corresponds to 80 dB, the average volume held during a music concert
classical. However, it is not possible to use the magnitude in dB directly, but the transformation into W/m 2 is required :
W/m2=10-12(108010)=10-4
The origin of this calculation derives from the dB scale, a logarithmic scale based on 10 which starts from the audibility threshold, the 10 -12 W/m 2 which appears in the formula.
To find the necessary time, for example up to Pluto (which being the most distant planet would require the maximum energy) just divide the space, aphelion of Pluto 7.37593 x 10 12 m, by the speed, let's assume 17355 ms in helium :
E=(7.375931012)/17355=425,003,169.115s13.477 years
It is therefore now possible to calculate the result of the energy needed, for example, to play Holst's good music up to Pluto's friends:
10-44(7.375931012)24.25003169115108=2.905561031Ws
The volume, in dB, that a spectator sitting in the front row of this crazy concert would hear when the orchestra finally had to sing the notes of "Pluto the Renewer", would therefore be:
10log10(2.90556103110-12)= 434.6323dB
The loudest sound ever produced was that of the eruption of Krakatoa on August 27, 1883, an eruption that destroyed ⅔ of the island and created tsunamis 46m high, with a volume of about 310 dB (small curiosity, if it had occurred at inside our pressurized ellipsoid would have been heard all the way to Pluto with a volume of about 88 dB; next time I hear a truck go by I'll wonder if it isn't actually a destructive volcanic eruption millions of km away), our good orchestra it should play more than 83 times louder, definitely a good effort.
Also the value of energy required is absolutely remarkable, on the Kardashev scale, already mentioned above, here we would need a 2.54 type civilization, i.e. capable of exploiting the energy of several star systems.
For the other evaluations of required energy, volume in dB and level on the Kardashev scale to "play" up to the planets, we attach the link to the spreadsheet with the missing data:
Let's reflect for a moment on the meaning of the value of energy required? But request for what?
It is the quantity necessary not for a loudspeaker or a strange machine to produce such a sound, but rather that required for musicians to play.
As we all know, to perform a physical effort we need energy and we take our energy in the form of calories from food. So let's calculate how many and how much food our formidable musicians should eat so that no good grandmother sees them "wasted":
2.905561031Ws=6.939671027Kcal
Let us now consider the average caloric requirement of a person in the space of 57 minutes (duration of the concert). For this hypothesis we take into consideration the data for a person weighing 62 kg (world average weight) aged between 30-59 years;
FabgD=2247.2342857142858 cal - FabgU=2603.4242857142854 cal
FabgM=2425.32857142 cal
Considering that this is the daily requirement and that a day is made up of 86400 s, in proportion to the duration of the concert, the amount of calories burned by each musician is about 96 cal, which is more or less as many as a good mandarin. In order for each player to consume only this small amount of energy, we would need this number of players:
# musicians=(6.939671027)/0.0967.228821028
This means that at least a couple of the mothers of the 3.8 x 10 35 workers involved in building the megastructure will have to be good musicians capable of teaching their 1.04 x 10 26 children how to play Holst perfectly.
And it also means that mandarin growers will have to do a lot: considering that a mandarin plant can produce up to 600 mandarins in a year, we would need approximately 1.2 x 10 23 mandarin plants , which , being able to reach a height of even 4.5m while maintaining a parallebiped shape and covering an area of approximately 9m 2 , if planted they would require land equal to 7,248,320,000 times the globe, taking into account the cultivation of the entire emerged land ( 149 million km2 ) . That's a lot of tangerines…
Avoiding exploiting this huge orchestra, and filling the bellies of a few good musicians a little more, let's evaluate how much the orchestra should eat if we want to exploit only 55 people, and therefore the exact organic for which the Planets were composed. which comprises:
woodwinds: 4 flutes (3rd also 1st piccolo , 4th also 2nd piccolo and bass flute in C ), 3 oboes (3rd also bass ), English horn , 3 clarinets in B flat and A, bass clarinet in B flat , 3 bassoons , contrabassoon ;
brass: 6 horns in F, 4 trumpets in C, 3 trombones (two tenors and one bass), 2 tubas ( tenor in B flat and one bass)
percussion: 6 timpani (two players); triangle , drum , tambourine , cymbals , bass drum , tam-tam , bells , glockenspiel (three players); celesta and xylophone (two players)
2 harps
bows .
plus a small six-part women's choir which only fits into the seventh concert (Uranus, the Mystic).
Let's first evaluate the value of calories that everyone should consume:
Kcal=(6.939671027)/55 1.261026
This value corresponds approximately to the calories that the entire world population would consume in 17.906 billion years.
A margherita pizza gives our body an average of 700 Kcal, therefore every musician should devour, within 57 minutes, in order to sustain physical effort, 1.80 x 10 14 billion margherita pizzas ( or 1.3 x 10 18 billion tangerines…) which by far beats the record of Tom Waes, an eater able to eat a daisy in 11.6 s, but which in the time of the concert would have stopped at only 52 portions, being crushed by our more than appetizing musicians for whom it is imperative to swallow a daisy every 1.9 x 10 -11 nanoseconds (1 nanosecond = 0.000000001 s).