Foces Lectue 3 Basic Physics of Astophysics - Foce and Enegy http://apod.nasa.gov/apod/ Momentum is the poduct of mass and velocity - a vecto p = mv (geneally m is taken to be constant) An unbalanced foce is capable of poducing a change in momentum F = d p dt (n.b., a mass times an acceleation) dv F = m = ma dt Intuitively, foce is the push o pull on an object. A foce doesnt always poduce motion. It may be balanced by an equal opposite foce. Units: gm cm dyne = s 1 dyne =.48 x 10-6 pounds foce Consevation of momentum in the absence of foces Note that a foce may also be equied to balance anothe foce even when nothing is moving. Fo example a block setting on a table is pulled downwads by gavity but suppoted by the table. Definition of foce F= d p dt Action-eaction
Figue fom Nick Stobels site Examples A foce is equied to change eithe the speed of an object o the diection in which it is going. Two othe foces have only been ecognized duing the last centuy. They only affect phenomena on the scale of nuclei and individual paticles, i.e., they ae shot anged. All of chemisty is due to the electic foce. So ae the stength of solids and all pocesses elated to the emission and absoption of light. q 1 = Z 1 ; q = Z The stong foce is esponsible fo binding togethe the neutons and potons in the atomic nucleus. On an even smalle scale, the stong foce binds togethe quaks to make the neuton and poton and othe paticles on the sub-atomic scale. The typical ange of the stong foce is 10-13 cm. The stong foce is stong enough at shot ange to ovecome the epulsion of electically chaged potons in the nucleus (as well as the degeneacy enegy of the nucleus itself). But outside the nucleus it falls apidly to zeo. At vey shot ange the nuclea foce is actually epulsive.
The weak inteaction (actually much stonge than gavity but weake than the stong o electic inteaction) is in some sense analogous with the electic foce, but is a shot ange inteaction that acts on a quantum mechanical popety called isospin. It allows one kind of quak to tun into anothe. * Its chief effects ae that it allows neutinos to be poduced and to inteact with matte and it allows neutons to change into potons and vice vesa if enegy consevation allows it. A fee neuton outside the nucleus will decay into a poton in 10.3 minutes by the weak inteaction (udd) (uud) n p + e + e Note conseved quantities chage, bayon numbe, lepton numbe, enegy, momentum http://hypephysics.phy-ast.gsu.edu/hbase/paticles/expa.html SUMMARY FORCE STRENGTH RANGE EXAMPLE Figue fom Nick Stobels electonic text. See his website. Gavity Stong 1 10-13 cm nucleus Electic 10-1/ chemisty Weak 10-6 < 10-13 cm n p+ e + e Gavity 10-38 1/ binds eath to sun Fo spheically symmetic objects, gavity acts as if all the mass wee concentated at the cente of the sphee (moe geneally at the cente of mass). [often we Use instead of d fo distance]
The calculations on the pevious page would suggest that the sun is actually moe influential on the Eath (tides, etc.) than the moon, but the Eath is in fee fall aound the sun. = 154 pound foce 1 dyne =.48 x 10-6 pound fpce almost 00 times that of the moon. d What mattes is the diffeential foce because the Eath has finite size. df d = GMm i.e., d 3 d 1 = 3 Fo an object of diamete d, the diffeence in foce fom one side to the othe is F = df d (d) = F The distance ( in this equation) to the sun is 390 times that to the moon so the tidal foces fom each ae compaable d F tide (moon) = GM moon m F tide (sun) = F tide (moon) F tide (sun) moon GM sunm sun d eath moon d eath sun = F gav (moon) F gav (sun) = 390 176 =. sun moon distance to sun = 33 1.5 10 13 4.110 4 3.84 10 10 distance to moon Tides subtact the ed vecto fom the yellow ones The moon pulls on all pats of the Eath. It pulls stongest on the pat that is closest, less on the cente, and least of all on the fa side. Subtacting the foce at the cente of mass fom all components leads to a bulge in the oceans both on the nea and fa side of the Eath.
Q: Ae thee tides at the eaths poles? CENTRIFUGAL FORCE A: Yes. But the biggest tides ae in the topics and nea the equato. Remembe that it is the sun and moon causing the tides. These ae not geneally in the celestial equato. The path of the sun is the ecliptic, which we discussed. The moons obit is inclined by about 5 o to the ecliptic. The biggest tides ae when the moon and sun ae in the same diection o in opposite diections, i.e. at new moon and full moon. Just enough speed that centifugal foce balances the foce on the sting. Centifugal foce is sometimes called a fictitious foce o an inetialfoce. Because two bodies ae esponsible and thei position with espect to the equato changes duing the yea thee is no place on eath with no tides. Thee ae also small foces due to the otation of the eath. y x v y Centifugal Foce = x+ y x= cos xˆ y= sin yˆ x v = d dt Assume: d = (sin) dt ˆx + (cos) d dt ŷ d dt =constant, = constant * half dp F = = dt When deived coectly the / is not thee. v = d d dt dt = v F = m d dt = m d dt (diected along.) = m v = mv a = d v dt = d dt = (cos) d dt = ( x + y) d dt d + (sin) dt = d dt ˆx ŷ
Combining the definition of centifugal foce and Newtons equation fo gavitational attaction we get http://en.wikipedia.og/wiki/keple's_laws_of_planetay_motion geneal equation not cicula obits v = P v as we deived GM = P Keples Fist Law Keples Second Law The obit of a planet aound the sun is an ellipse with the sun at one focus of the ellipse. a is called the semi-majo axis of the ellipse. e is the eccenticity Fo eath e = 0.0167. Fo Mecuy e = 0.06. A line connecting the obiting both with one focus of the ellipse (e.g., the sun) sweeps out equal aeas in equal times. x a + y b = constant The equation fo an ellipse The equation fo a cicle is x + y =
Keples Thid Law The squaes of the peiods of the planets ae popotional to the cubes of thei semi-majo axes Planet P(y) a(au) P a 3 Mecuy 0.4 0.39 0.058 0.058 P 4 GM = 4 GM (1 y) 1 AU 3 Venus 0.6 0.7 0.38 0.38 Eath 1.0 1.0 1.0 1.0 Mas 1.88 1.5 3.5 3.5 This only woks when compaing small masses, each of which obits the same big mass, M (e.g., planets aound the sun). THE ASTRONOMICAL UNIT (AU) The astonomical unit (AU) is the distance fom the eath to the sun. It is measued fom the cente of the eath to the cente of the sun. Actually the distance fom the eath to the sun vaies about 1.7% duing the yea since its obit is elliptical. Pio to 1976 the AU was defined as the semi-majo axis of the eaths obit but in 1976 the IAU edefined it to be the aveage distance to the sun ove a yea. Its standad (1996) value is 1.495978707 x 10 13 cm http://en.wikipedia.og/wiki/astonomical_unit
WEIGHING THE UNIVERSE P = 4 GM 3 M = 4 GP 3 and the length of the yea and AU ae known assumes Mas obit is in the same plane as the Eaths. fom cente of mass of the eath to cente of mass of the sun M = = 4 G(1 y) (AU) 3 (4)(3.14) (gm) 1.50 (1 (6.67 10 8 )(dyne cm AU)3 )(1 y) (4)(3.14) (1.50 10 13 ) 3 = (6.6710 8 )(3.16 10 7 ) gm =.0010 33 gm 10 13 cm 1 AU 3 dyne s 1 y gm cm 3.16 10 7 s Late we shall see how to get the masses of othe stas (and planets) if those stas ae in binay systems 4 M = GP 3 (5.977 x 10 7 )
How massive is the Milky Way Galaxy? (and how many stas ae in it?) THE MASS OF THE MILKY WAY GALAXY Measuements of the paallax of numeous sta foming egions using adio (Vey Long Baseline Aay) have given (009) accuate measuements of the distance fom the galactic cente to the eath (8,000 ly) and the speed of the sola system in its obit aound the cente (54 km/s; used to be 0 km/s pio to Januay, 009) COBE (1990) - the galaxy as seen in fa infaed 1.5,., and 3.5 micons (optical = 0.4-0.7 micons) stas ae white; dust is eddish http://www.nao.edu/p/009/mwotate/ THE MASS OF THE MILKY WAY GALAXY Teat Milky Way as a disk. Gavitationally, the sun sees only the mass inteio to its obit and that mass acts as if it wee all located at the cente of mass, i.e., the cente of the Galaxy. Milky Way inteio to suns obit * Sun The matte outside the suns obit exets zeo net foce on the sun. http://www.melyn.demon.co.uk/gavity1.htm - FiSSh 4 M MW = GP 3 P = v ( )( v ) ( )( ) 4 M MW = G 4 3 = v G (.56 10 ) 7.65 10 = 6.67 10 8 =.51 10 44 gm = 1.9 10 11 M ( ) Peiod not known but do know the speed and the distance (peiod is actually about 00 My) 8,000 ly is.65 x 10 cm (1 ly = 9.46 x 10 17 cm) inteio to suns obit
Whee is the edge of the Milky Way, i.e. whee does the amount of mass enclosed each a constant? Once M = constant then as inceases GMm = mv v 1/ > R M * R m Suns position GMm mv = v = const. M. If density wee independent of, M 3 (sphee), M (disk) so the density is indeed declining
http://www.sdss.og/news/eleases/008057.mwmass.html May 7, 008 - Sloan sky suvey using a sample of ove 400 Population II (blue hoizonal banch) stas infes a mass of slightly less than 10 1 sola masses
What is Dak Matte? * Anything with a lage mass to light atio Bayonic dak matte: (made of neutons, potons and electons) White dwafs Black holes (lage and small) Neuton stas Bown dwafs and planets Gas eithe in small cold clouds o a hot inte-galactic cluste gas But Big Bang nucleosynthesis limits the amount of bayonic mass. The value infeed is less than that equied to bind galaxies and especially clustes of galaxies togethe. * Big Bang nucleosynthesis implies seveal times moe bayonic matte than we see in galaxies and stas. Whee is the est? Pobably in an ionized hot integalactic medium. But obsevations of the dynamics of galactic clustes suggest even moe matte than that. About 5 times moe mass than BB nucleosynthesis suggests
Best indications ae that dak matte is composed of two pats bayonic dak matte which is things made out of neutons, potons and electons, and non-bayonic dak matte, which is something else. Of the bayonic matte, stas that we can see ae at most about 50% and the est (the dak stuff) is in some othe fom. Pat of the bayonic matte may be in the hot integalactic medium and in ionized halos aound galaxies (50%?). The non-bayonic matte has, in total, 5 times moe mass than the total bayonic matte. I.e. bayonic matte is 1/6 of matte and non-bayonic matte is 5/6. Moe Exotic Anything with a lage mass to light atio Non-bayonic dak matte: Massive neutinos Axions (paticle that might be needed to undestand absence of CP violation in the stong inteaction) Photinos, gavitinos,... WIMPs unknown... In geneal, paticles that have mass but little else * The univese was not bon ecently ( i.e., thee is nothing special about the pesent We ae not at the cente of the sola system Ou sola system is not at the cente of the Milky Way Galaxy Ou galaxy is not at the cente of the univese (and thee ae many othe galaxies) We ae not made of the matte which compises most of the univese (as thought <30 yeas ago) [Still to come matte is not the dominant constituent of the univese]