Mathematical Modeling Lecture

Some Applications I– Various mathematical models and its applications

2009. 10. 27

Sang-Gu Lee, Duk-Sun KimSungkyunkwan University

sglee@skku.edu

In physics, an orbit is the gravitationally curved path of one object around apoint or another body, for example the gravitational orbit of a planet around astar.Historically, the apparent motion of the planets were first understood in termsof epicycles, which are the sums of numerous circular motions. This predictedthe path of the planets quite well, until Johannes Kepler was able to show thatthe motion of the planets were in fact elliptical motions. Isaac Newton wasable to prove that this was equivalent to an inverse square, instantaneouslypropagating force he called gravitation. Albert Einstein later was able to showpropagating force he called gravitation. Albert Einstein later was able to showthat gravity is due to curvature of space-time, and that orbits lie upongeodesics.

Two bodies with a slight difference in mass orbiting around a common barycenter. The sizes, and this particular type of orbit are similar to the Pluto–Charon system.

t : timer : distance (between earth and sun)

지구지구지구지구

달달달달

moonearth xxx +=

태양태양태양태양

( ) ( )( )ttearth ππ 2sin,2cos=x

( ) ( )( )mtrmtrmoon ππ 2sin,2cos=x

t : time / (m=12month)r : distance (between earth and sun) / (moon-earth : 0.3)

moonearth xxx +=

=B5+D5, =C5+E5

( ) ( )( )ttearth ππ 2sin,2cos=x

( ) ( )( )mtrmtrmoon ππ 2sin,2cos=x

=cos(2*pi()*A5), =sin(2*pi()*A5)

=$B$1*cos(2*pi()*$B$2*A5), =$B$1*sin(2*pi()*$B$2*A5)

=B5+D5, =C5+E5

t : time / (12month)r : distance (between earth and sun) / (moon-earth : 0.3)

moonearth xxx +=

=B5+D5, =C5+E5

m=12, r=0.3

( ) ( )( )ttearth ππ 2sin,2cos=x

( ) ( )( )mtrmtrmoon ππ 2sin,2cos=x

=cos(2*pi()*A5), =sin(2*pi()*A5)

=$B$1*cos(2*pi()*$B$2*A5), =$B$1*sin(2*pi()*$B$2*A5)

=B5+D5, =C5+E5

m=9, r=0.01

M=4, r=0.2

In astrodynamics or celestial dynamics orbital state vectors (sometimes state vectors) are vectors of position and velocitythat together with their time (epoch) uniquely determine the state of an orbiting body.State vectors are excellent for pre-launch orbital predictions when combined with time (epoch) expressed as an offset tothe launch time. This makes the state vectors time-independent and good general prediction for orbit.

Orbital position vector and orbital velocity vector and other orbit's elements

Please note that the following is a classical (Newtonian) analysis of orbital mechanics, which assumes the more subtleeffects of general relativity (like frame dragging and gravitational time dilation) are negligible. General relativity does,however, need to be considered for some applications such as analysis of extremely massive heavenly bodies, preciseprediction of a system's state after a long period of time, and in the case of interplanetary travel, where fuel economy, andthus precision, is paramount.

22

2

h

GMu

d

ud=+

θm

hGr

uH=±== , /kgNm 10 × 0.001)(6.6742 ,

1 2211-

M is the mass of the central body (the Sun),m is the mass of the orbiting body (planet),G : constant of universal gravitation

http://orinetz.com/planet/animatesystem.php?sysid=QUQTS2CSDQ44FDURR3XD6NUD6&orinetz_lang=1#

http://www.ioncmaste.ca/homepage/resources/web_resources/CSA_Astro9/files/multimedia/unit4/planetary_orbits/planetary_obits.html

In probability theory, the birthday problem, or birthday paradox pertains to the probabilitythat in a set of randomly chosen people some pair of them will have the same birthday. Ina group of at least 23 randomly chosen people, there is more than 50% probability thatsome pair of them will have the same birthday. Such a result (for just 23 people,considering that there are 365 possible birthdays) is counter-intuitive to many.For 57 or more people, the probability is more than 99%, and it reaches 100% when thenumber of people reaches 366 (by the pigeonhole principle, ignoring leap years). Themathematics behind this problem lead to a well-known cryptographic attack called thebirthday attack.

A graph showing the approximate probability of at least two people sharing a birthdayamongst a certain number of people.

To compute the approximate probability that in a room of n people, at least twohave the same birthday, we disregard variations in the distribution, such as leapyears, twins, seasonal or weekday variations, and assume that the 365 possiblebirthdays are equally likely. Real-life birthday distributions are not uniform since notall dates are equally likely.If P(A) is the probability of at least two people in the room having the same birthday,it may be simpler to calculate P(A'), the probability of there not being any twopeople having the same birthday. Then, because P(A) and P(A') are the only twopossibilities and are also mutually exclusive, P(A') = 1- P(A).

n p(n)10 11.70%20 41.10%23 50.70%30 70.60%50 97.00%57 99.00%100 100.00% When events are independent of each other, the probability of all of the events

occurring is equal to a product of the probabilities of each of the events occurring.Therefore, if P(A') can be described as 23 independent events, P(A') could becalculated as P(1) * P(2) * P(3) * ... * P(23).

This process can be generalized to a group of n people, where p(n) is the probabilityof at least two people sharing a birthday.

)(1)(

365

11

365

21

365

111)(

npnp

nnp

−=

−−××

−×

−×= L

100 100.00%200 100.00%300 (100 − (6×10−80))%350 (100 − (3×10−129))%366 100%

−−××

−×

−×=365

11

365

21

365

111)(

nnp L

=B3*(B$1-A3)/B$1

)(1)( npnp −=

=1-B4

−−××

−×

−×=365

11

365

21

365

111)(

nnp L

)(1)( npnp −=

=B3*(B$1-A3)/B$1

=1-B4

In a group of at least 23 randomly chosen people, there is more than 50% probability that some pair of them will have the same birthday.

The Taylor series expansion of the exponential function

provides a first-order approximation.

L+++=!2

12x

xex

xex +≈1

The first expression can be approximated as

( ) ( )

( )

( )3652

1

365

121

365

1

365

2

365

1

1

1

−−

−+++−

−−−−

=

×=

××××≈

nn

n

n

e

e

eeenpL

L

Therefore,

An even coarser approximation is given by

( ) ( ) ( )3652

1

11 ×

−−

−≈−=

nn

enpnp

( ) 3652

2

1 ×−

−≈

n

enpwhich, as the graph illustrates, is still fairly accurate.

A graph showing the accuracy of the approximation.http://math1.skku.ac.kr/home/pub/432/

The white squares in this table show the number of hashes needed to achieve the given probability of collision (column) given a hashspace of a certain size in bits (row). (Using the birthday analogy, the hash space would be of size 365 (row); one desired to know the number of people that will give a 50% chance (column) of a collision; the number of people is the white square where the row and column intersect.) For comparison, 10−18 to 10−15 is the uncorrectable bit error rate of a typical hard disk. In theory, MD5, 128 bits, should stay within that range until about 820 billion documents, even if its possible outputs are many more.

If 32-bit Hash Table system was tried 77000 times then this hash table can be recognized by Hacker.

• Abell, Morrison, and Wolff (1987). Exploration of the Universe (fifth ed.). Saunders College Publishing.• orbit (astronomy) - Britannica Online Encyclopedia• Encyclopaedia Britannica, 1968, vol. 2, p. 645.• Jones, Andrew. "Kepler's Laws of Planetary Motion" (in en). about.com. http://physics.about.com/od/astronomy/p/keplerlaws.htm.

Retrieved 2008-06-01.• See Eq. 8.37 in John R Taylor (2005). Classical Mechanics. University Science Books. p. 306. ISBN 189138922X.

http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA306.• See, for example, Eq. 8.20 in John R Taylor (2005). op. cit.. Sausalito, Calif.: Univ. Science Books. pp. 299 ff. ISBN 189138922X.

http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA299.• Fitzpatrick, Richard (2006-02-02). "Planetary orbits". Classical Mechanics – an introductory course. The University of Texas at

Austin. Archived from the original on 2006-05-23.http://web.archive.bibalex.org/web/20060523200517/farside.ph.utexas.edu/teaching/301/lectures/node155.html.

• F. Varadi, B. Runnegar, M. Ghil (2003). "Successive Refinements in Long-Term Integrations of Planetary Orbits". TheAstrophysical Journal 592: 620–630. doi:10.1086/375560.

• E. H. McKinney (1966) Generalized Birthday Problem, American Mathematical Monthly 73, 385–387.• M. Klamkin and D. Newman (1967) Extensions of the Birthday Surprise, Journal of Combinatorial Theory 3, 279–282.• M. Abramson and W. O. J. Moser (1970) More Birthday Surprises, American Mathematical Monthly 77, 856–858• D. Bloom (1973) A Birthday Problem, American Mathematical Monthly 80, 1141–1142.• Shirky, Clay Here Comes Everybody: The Power of Organizing Without Organizations, (2008.) New York. 25–27.