Lecture 3. Basic Physics of Astrophysics - Force and Energy. Forces

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Lectue 3 Basic Physics of Astophysics - Foce and Enegy http://apod.nasa.

poducing a change in momentum F = d p dt (n.b., a mass times an acceleation) dv F = m = m a dt Intuitively, foce is the push o pull on an object.

1 Lectue 3 Basic Physics of Astophysics - Foce and Enegy Foces 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 = m a dt Intuitively, foce is the push o pull on an object. Units: gm cm dyne = s 1 dyne =.48 x 10-6 pounds foce 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.

eithe the speed of an object o the diection in

q 1 = Z 1 ; q = Z All of chemisty is due to

2 Figue fom Nick Stobels site Examples Consevation of momentum in the absence of foces Definition of foce F= d p dt Action-eaction A foce is equied to change eithe the speed of an object o the diection in which it is going. q 1 = Z 1 ; q = Z 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.

3 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. 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 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 + ν e Note conseved quantities chage, bayon numbe, lepton numbe, enegy, momentum * SUMMARY FORCE STRENGTH RANGE EXAMPLE Stong cm nucleus Electic 10-1/ chemisty Weak 10-6 < cm n p + e +ν e Gavity / binds eath to sun

4 Figue fom Nick Stobels electonic text. See his website. Gavity = 154 pound foce 1 dyne =.48 x 10-6 pound fpce 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] almost 00 times that of the moon. d 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. What mattes is the diffeential foce because the Eath has finite size. df d = GMm 3 Fo an object of diamete d, the diffeence in foce fom one side to the othe is ΔF = df d (d) = F d i.e., d d 1 = 3 F tide (moon) = GM moonm 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) = =. sun moon distance to sun = distance to moon The distance ( in this equation) to the sun is 390 times that to the moon so the tidal foces fom each ae compaable

The moons obit is inclined by about 5 o to the ecliptic. 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.

5 Tides subtact the ed vecto fom the yellow ones Q: Ae thee tides at the eaths poles? 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 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. 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. 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. CENTRIFUGAL FORCE Just enough speed that centifugal foce balances the foce on the sting. Centifugal foce is sometimes called a fictitious foce o an inetialfoce. half dp F = = dt When deived coectly the / is not thee.

y) dθ dt = dθ dt ˆx * ŷ Combining the definition of centifugal foce and Newtons equation fo gavitational attaction we get GM = π P v = π P v Keples Fist Law http://en.wikipedia.

6 y # v = dθ dt dθ dt = v x F = m d dt = m dθ dt = m v y (diected along.) Centifugal Foce = x + y x = cosθ xˆ y = sinθ yˆ v θ x = v = d dt Assume: mv dθ = ( sinθ) dt dθ dt =constant, ˆx + (cosθ) dθ dt ŷ = constant a = d v dt = d dt = ( cosθ) dθ dt dθ + ( sinθ) dt = ( x + y) dθ dt = dθ dt ˆx * ŷ Combining the definition of centifugal foce and Newtons equation fo gavitational attaction we get GM = π P v = π P v Keples Fist Law geneal equation not cicula obits 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 = Fo Mecuy e = as we deived x a + y b = constant The equation fo an ellipse The equation fo a cicle is x + y = i.e.,a=b

Keples Second Law Using Ratios: Keple s Thid Law fo Cicula Obits A line connecting the

P 4π GM = 4π GM (1 y) 1 AU 3 This only woks when compaing small masses, each of which obits

Keples Thid Law The squaes of the peiods of the planets ae popotional to the cubes of thei

7 Keples Second Law Using Ratios: Keple s Thid Law fo Cicula Obits A line connecting the obiting both with one focus of the ellipse (e.g., the sun) sweeps out equal aeas in equal times. P 4π GM = 4π GM (1 y) 1 AU 3 This only woks when compaing small masses, each of which obits the same big mass, M (e.g., planets aound the sun). 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 Venus Eath Mas

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.

8 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 x cm 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 Reaange WEIGHING THE UNIVERSE P = 4π GM 3 M = 4π GP 3 Late we shall see how to get the masses of othe stas (and planets) if those stas ae in binay systems M = 4π G(1 y) (AU) 3 (4)(3.14) (gm) = ( )(dyne cm )(1 y) ( cm AU)3 1 AU (4)(3.14) ( ) 3 = ( )( ) gm = gm and the length of the yea and AU ae known 3 dyne s 1 y gm cm s

How massive is the Milky Way Galaxy? (and how many stas ae in it?) 4π M = GP 3 (5.977 x 10 7 ) COBE (1990) - the galaxy as seen in fa infaed 1.5,., and 3.5 micons (optical = 0.4-0.

8 km/s] 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

Gavitationally, the sun sees mostly 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.

9 How massive is the Milky Way Galaxy? (and how many stas ae in it?) 4π M = GP 3 (5.977 x 10 7 ) COBE (1990) - the galaxy as seen in fa infaed 1.5,., and 3.5 micons (optical = micons) stas ae white; dust is eddish THE MASS OF THE MILKY WAY GALAXY [obital speed of eath aound sun is 9.8 km/s] 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) Teat Milky Way as a disk. Gavitationally, the sun sees mostly 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. If the galaxy wee a sphee this appoximation would be exact. - FiSSh Milky Way inteio to suns obit * Sun The matte outside the suns obit exets little net foce on the sun. To easonable accuacy can teat the disk mass as being concentated at the cente, especially at lage distances

f 1 = edge of disk 4π M MW = GP 3 P = π v THE MASS OF THE MILKY WAY GALAXY ( )( v ) ( )( ) 4π M MW = G 4π 3 = v G (.56 10 ) 7.65 10 = 6.67 10 8 =.51 10 44 gm = 1.

10 f 1 = edge of disk 4π M MW = GP 3 P = π v THE MASS OF THE MILKY WAY GALAXY ( )( v ) ( )( ) 4π M MW = G 4π 3 = v G ( ) = = gm = 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 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

11 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.

(blue hoizonal banch) stas infes a mass of slightly less than 10 1 sola

* Anything with a lage mass to light atio Bayonic dak matte: (made of

stas Bown dwafs and planets Gas eithe in small cold clouds o a hot

12 May 7, 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.

13 * 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 *

14 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]

Lecture 3. Basic Physics of Astrophysics - Force and Energy. Forces

Lecture 3. Basic Physics of Astrophysics - Force and Energy. Forces 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