Download My capstone project

Project Title: Discovery of Neptune: activities and simulations.
Participant Name: Hezi Yizhaq
Home Country: Israel
Host Country: U.S
Fulbright Program Year: 08/14/12-12/14/12
Mentor: Dr. Matthew Bobrowsky, Physics Department, University of Maryland.
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1. Project Description
The project aimed to develop new activities on the discovery of Neptune for high
school physics students. These activities can be used as an introduction for teaching
about dark matter in the universe which is one of the great mysterious in
contemporary astrophysics.
1.1 Introduction
Unlike the planet Uranus, Neptune was not discovered accidentally. It was proposed
that the planet beyond Uranus could account for the irregularities in Uranus' orbit.
Independently, two astronomers, John Couch Adams in England and Urbain Jean
Josef Le Verrier in France, calculated the position of this yet unknown planet.
Adams was a young Cambridge undergraduate, 26 years old, who seems to have
taken on a own personal quest to search for an explanation for the apparent
misbehavior of Uranus. Adams was an unknown, a youngster working on his own
initiative, with unproven credentials and almost without any mentor. In October
1845, Adams wrote to George Airy, the Astronomer Royal of Greenwich
Observatory, claiming that he had solved the problem of Uranus' orbit, and stating
the position where the unknown planet could be found. Now, if Airy had pointed a
telescope at that spot, he might have found Neptune (however, not at the exact spot
that Adams had pinpointed). Although, he tried to conceal it, Airy had a strong
negative reaction to Adams paper. His altitude later turned out to be of critical
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importance to the fact that Neptune was not discovered in England. The problem
was that Airy was strongly opposed to theoretical investigations and skeptical of the
abilities of younger scientists. He was not the sort of man to take a leap into the
scientific unknown. So it is not surprising that Airy was unreceptive to the
provocative notion of a new planet orbiting beyond Uranus. What Airy did was ask
Adams a technical question about his calculations, which was designed to try to
distinguish between the two conflicting theories. Airy believed that the errors in
Uranus orbit could be explained by more detailed recalculations of the perturbations
due to Saturn’s gravity, whereas Adams predicted an existence of a new planet
beyond Uranus. Adams could have given a simple reply, but for whatever reason, he
didn’t. So Airy, not surprisingly, ignored Adams’ claim. Neptune was ultimately
discovered by the German astronomer Johann Galle, on September 29, 1846, using
Le Verrier’s predictions. The most important implication of this story for science
teachers is that teachers have to help develop their students' imaginations and
encourage them to propose new ideas that even might contradict their existing
beliefs. Developing proactive new ideas by students is as important as rigorous
studying of the conventional curriculum, because that is where real creativity occurs.
The discovery of Neptune is only briefly mentioned in the physics text books and
there are almost no suitable activities for high school students. There are many
resources devoted to the history of the discovery (Grosser, 1962; Baum and
Sheehan, 1997; Standage, 2000) which can be regarded as popular texts. In the
other extreme there are much technical papers which mathematically discuss the
inverse problem of finding Neptune (Lai et al., 1997). There are almost no references
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in the intermediate level that would enable high school physics to get a feeling about
the complexity of the problem. In my capstone project I developed three such
activities. The activities were developed with my mentor help and using the
references that I found, but there are all original.
1.2 Activities
Activity 1: Finding the year of conjunction
In the first activity the students will use the data of the discrepancies between the
observed longitude of Uranus and calculated longitude (see Fig.1) to find the year of
conjunction, i.e. that the year where Uranus and Neptune were line up.
Fig. 1 Discrepancies between the observed and the calculated longitudes of Uranus
after known causes have subtracted (adapted from Lyttelton, 1960).
The student can calculate where the first derivative of the data in Fig.1 has a
maximum which has to be near the year of conjunction. This information alone
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makes the prediction of Neptune location within less than 15 degrees on the Basis of
Bode law (a rule which apply to the distances of planets from the sun) and assuming
that the orbit is circular.
Fig.2 First derivative of the data in Fig.1. The maximum occurs near the 1822 which
is the year where Uranus and Neptune were line up.
Activity 2: Numerical simulation of the orbits of Uranus and Neptune.
This activity is based on a method suggested by the famous physicist Richard
Feynman (Feynman, 1963). Feynman showed how to solve iteratively the coupled
equations of motions of planets:
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m1v1 x  
Gm1ms ( x1  xs ) Gm1m2 ( x1  x2 )

r13s
r123
m1v1 y  
Gm1ms ( y1  ys ) Gm1m2 ( y1  y2 )

r13s
r123
Gm2ms ( x2  xs ) Gm1m2 ( x2  x1 )
m2v2 x  

r13s
r123
m2v2 y  
(1)
Gm2ms ( y2  ys ) Gm1m2 ( y2  y1 )

r13s
r123
where subscript 1 denotes Uranus, subscript 2 denotes Neptune and subscript s
denotes t the Sun and G is the universal constant of gravitation. I developed a
MATLAB code (see Appendix 1) that solve the equations and calculate few physical
quantities like the total energy of the system (Fig. 3). This code can be used by
teachers and student in class to get insight to the calculations of perturbations. They
can further the developed the code and other planets (like Jupiter) or they can
change the time step and study the accuracy of the code. They can compare the
orbit
of
Uranus
with
and
without
the
perturbation
of
Neptune.
Fig.3 The calculated orbits of Uranus (black) and Neptune (blue). Note that while
Uranus finished one revolution around the Sun, Neptune still didn't finish its
revolution.
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Activity 3: A simplifying calculation of the distance of Uranus from the Sun when
they were line up.
As the full analysis is very complex and far beyond the level of high school physics,
using some simple assumption can make the problem more tractable for high school
students. Our assumptions are:
1. The orbits of Uranus and Neptune are circular (before and after the perturbation)
and the two planets are near the conjunction and the Sun is fixed.
2. Conservation of the total energy and total angular momentum before and after
the perturbation.
Using the above assumption we can write the following equations:

Gm1ms Gm2ms
Gm1ms 1
Gm2ms 1
Gm1m2


 m1v32 
 m2v42 
2 R1
2 R2
R3
2
R4
2
R4  R3
m1v1R1  m2v2 R2  m3v3 R3  m4v4 R4
Gm1ms
Gm1m2
m1v32


R32
( R4  R3 ) 2
R3
(2)
Gm2ms
Gm1m2
m2v42


R42
( R4  R3 )2
R4
where R1 and R2 are the unperturbed radius of the orbits of Uranus and Neptune
respectively and R3 and R4 are the perturbed ones. Doing the algebra we can get
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the following cubic equation for R3 :
AR33  BR32  CR  D  0
A  m1 ' R2 R4  m2 ' R1R4  m2 ' R1R2
B  2m1 ' R2 R42  2m2 ' R1R42  m1 ' R1R2 R4  2m2 ' R1R2 R4  m1 ' m2 ' R1R2 R4
C  m1 ' R2 R43  m2 ' R1R43  2m1 ' R1R2 R42  m2 ' R1R2 R42  m1 ' m2 ' R1R2 R42
(3)
D  m1 ' R1R2 R43
and
m '
R4   1
 m2 '


R1  R3  R2 


2
where m1 '  m1 / ms and m2 '  m2 / ms . Equation 3 can be solved numerically (using
MATLAB or Excel) but we have t assume values for R2 and m2 which are the
unperturbed radius of Neptune and its mass. For the orbit we can use Bode law as
have done by Adams and Le Verrier and take R2  38.8AU where AU is
astronomical unit (the average distance from Earth to the Sun).
Fig. 4 shows the perturbed distance of Uranus from the Sun for different masses
(expressed in Uranus masses). The larger the Neptune mass the larger the
perturbation. The idea is that despite the simplified assumption we made the
calculation which is based on general physical laws allow the student to get insight
on the order of the perturbation and that it linearly depends on the Neptune mass as
shown in Fig. 5
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Fig. 4 Numerical solutions to Eqs.3. The perturbed distances of Uranus from the Sun,
can be found from the intersections of the lines with the zero line ( f ( R3 )  0 .
Fig.5 The corrected radius of the orbit of Uranus due to the perturbation applied by
Neptune (located at 38.8 AU from the Sun) for different masses (expressed in Uranus
masses. The unperturbed orbit is 19.22 AU.
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3. The discovery of Neptune and dark matter in the universe
A galaxy is a collection of billions of stars and clouds of gas. Spiral galaxies are flat,
like a an egg fried sunny (no pun intended) side up. The central part of the galaxy,
round like the yolk of an egg, is called the bulge. The outer parts of the galaxy, where
the spiral arms are seen, is like the white of the egg, and is called the disc. The egg
analogy is useful in another way. Much like most of the mass of an egg is in the yolk,
most of the mass of the stars in a galaxy is concentrated in the bulge. Unlike egg
whites, stars in the disc move around the central bulge of a spiral galaxy in orbits
shaped like the orbits of planets around our sun. Stars farther from the central bulge
should move slower than stars closer to the center if the only matter in the galaxy is
the stuff we can see. Planets in our solar system behave that way. Mercury, closest
to the Sun, moves at a brisk 48 km/sec (107,000 mi/hr), while Neptune, the planet
most distant from the Sun, moves at a relatively leisurely 5.4 km/sec (12,150 mi/hr).
This is because almost all the mass in the solar system is located in sun.
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Fig.6. The rotation curve of planets in the solar system (left panel) shows that the
further the planet the lower its orbital velocity. The situation is completely different
in the Milky Way galaxy where the velocity of the stars around the center of the
galaxy is almost constant.
Now we get to the mystery. The actual motion of stars in a galaxy is very different
than this prediction based on the Newtonian dynamics. Stars farther from the center
move faster than stars closer to the center! What could the solution to this mystery
be? As with the mysterious motion of Uranus, there are two possibilities. Either
there is matter we have not seen or Newton’s law of gravitation is incorrect. The
matter we haven’t seen is called dark matter. The theory of the Modified Newtonian
Dynamics is called MOND and was suggested by the Israeli physicist Mordehai
Milgrom (Milgrom, 2002).
Thus, the activities on the discovery of Neptune can be connected to the more
advanced subject of dark matter. The codes I developed will be available for
teachers in my website:
http://www.boker.org.il/meida/negev/desert_biking/school/dark_matter/darkmatter-003.htm
5. Conclusion
We developed three activities on the discovery of Neptune for high school physics
that can help students to understand the nature of the problem and to understand
the way of thinking of scientists and how they approach a difficult problem. The
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numerical simulation can be further developed and improved by teachers and
students. I hope that these activities will be useful in the physics class.
5. References used in the capstone project:
Baum, R. and Sheehan, W. (1997). In the search of the planet Vulcan. Plenum Trade,
New York.
Bollobás, B. Ed. 1986. Littlewood's Miscellany. Cambridge University Press,
Cambridge.
Craig, M. and Schultz, S. 2007. Invisible Galaxies: The Story of Dark Matter. The
universe in the classroom, no. 72. The Astronomical society of the Pacific.
390 Ashton Avenue, San Francisco, CA 94112.
Grosser, M. 1962. The Discovery of Neptune. Harvard University Press.
Feynman, R. P., Leighton, R. B. and Sands, M. 1963. The Feynman Lectures on
Physics, Chapter 9. Addison Wesley Publishing Company, Reading, Massachusetts.
Lai, H. M., Lam. C. C. and Young, 1990. Perturbation of Uranus by Neptune: A
modern perspective. Am. J. Phys. 58, 946 (1990); doi: 10.1119/1.16307.
Lyttleton, R. A., 1960. The rediscovery of Neptune. Vistas in Astronomy 3.
Milgrom, M. 2002. Does dark matter really exist? Scientific America, August , 43-52.
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Panek, R. 2011. The 4 Percent Universe. Houghton Mifflin Harcourt, Boston, New
York.
Price, F. W. 2001. The Planet Observer's Handbook. Cambridge University Press.
Roy, A. E. and Clarke D. 2003. Astronomy: Principles and Practice. Institute of Physics
Publishing, Bristol and Philadelphia.
Rubin. V. 1997. Bright Galaxies Dark Matters American Institute of Physics,
Woodbury, New York.
Sanders, R. H. 2010. The dark matter problem- A historical perspective. Cambridge
University Press.
Standage, T. 2000. The Neptune File. Walker & Company, New York.
Turner, H. T. 1963. Astronomical Discovery. University of California Press.
Appendix 1:
The MATLAB code which I developed for the numerical simulations of the coupled
orbits of Uranus and Neptune. The code should be run by MATLAB software and I
sent it separately as M script.
clear all
%The code is part of my capstone project in the 2012 Distinguished
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%Fulbright Awards in Teaching hosted by the University of Maryland.
%This code simulate the orbit of a planet around the sun according to
%Feynman (Chapter 9: Newton's laws of Dynamics, in Feynman Lectures on
%Physics (Feynman, Leighton and Sands) ,1963. Addison-Wesley Publishing
%Company, Reading Massachusetts.
%algorithm. with taking into account the mutual perturbations between Uranus and
Neptune.
%The initial velocity is very important to give the right ellipse and that
%should be found by try and error as to give the right number in A.U
%We assume that the sun is fixed and located in (0,0) and that GM_s =1 as
%was assumed by Feynman to make the calculations simple.
%Note that is quite straight forward to add another planet (like Jupiter)
%but remember to the right terms in the equations for Neptune and
%Uranus.
%For comments and questions please contact:
%Hezi Yizhaq [email protected]
%____________________________________________________________________
______
%Initial conditions and parameter definitions
delta=6.2648;%delta is a parameter for adjusting the initial velocity.
delta1=6.5;
eps=0.04; %eps is the time step
eps2=eps/2; %eps2 is used for computing the first time step.
rr1=19.22; % rr1 is the initial distance of Uranus from the Sun.
rr2=30.8; % rr2 is the initial distance of Neptune from the Sun.
x1(1)=rr1; %x1 is the x coordinate of Uranus.
y1(1)=0; % y1 is the y coordinate of Uranus.
x2(1)=rr2; %x2 is x coordinate of Neptune
y2(1)=0; %y2 is the y coordinate of Neptune
m1=4.373e-5; %m1 is the mass of Uranus (the ratio between Uranus mass and the
Sun mass.
m2=5.178e-5; %m2 is the mass of Neptune (the ratio between Neptune mass and
the Sun mass.
m12=m1*m2; %m12 is defined to make the calculation faster.
m22=1/m2; % m22 is needed in order to compute the ratio of the gravity forces.
tu=84.01;% Uranus revolution time (in years).
tn=164.8;% Neptune revolution time (in years).
%_______________________________________________________
% Calculating the first time step for Uranus
vx1(1)=0; %Initial velocity in the x direction.
vy1(1)=2*pi*rr1/(tu*delta); %Initial velocity in the y direction.
r1(1)=(x1(1)^2+y1(1)^2)^0.5;% calculating the initial distance from Neptune to the
Sun.
r13(1)=1/(r1(1)^3);
run(1)=((x2(1)-x1(1))^2+(y2(1)-y1(1))^2)^0.5;%calculating the initial distance
between Uranus and Neptune.
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run3(1)=1/(run(1)^3);
vsqr1(1)=vx1(1)^2+vy1(1)^2; %calculation of the squared velocity of Uranus.
v1(1)=vsqr1(1)^0.5;%calculating the initial velocity of Uranus
r1(1)=(x1(1)^2+y1(1)^2)^0.5;
ax1(1)=-x1(1)*r13(1)-m2*(-x2(1)+x1(1))*run3(1);% the acceleration of Uranus in the
x direction.
ay1(1)=-y1(1)*r13(1)-m2*(-y2(1)+y1(1))*run3(1);% the acceleration of Uranus in the
y direction.
vvx1(1)=vx1(1)+ax1(1)*eps2;%these values are used for calculation of the next time
step.
vvy1(1)=vy1(1)+ay1(1)*eps2;%these values are used for calculation of the next time
step.
energy1(1)=-m1/r1(1)+0.5*m1*vsqr1(1);%the total initial energy of Uranus
j1(1)=m1*v1(1)*r1(1);%the total angular momentum of Uranus
%_______________________________________________________
%Calculating the first time step for Neptune
%the same explanations written above are applied for Neptune.
vx2(1)=0;
vy2(1)=2*pi*rr2/(tn*delta1);
r2(1)=(x2(1)^2+y2(1)^2)^0.5;
r23(1)=1/(r2(1)^3);
vsqr2(1)=vx2(1)^2+vy2(1)^2;
v2(1)=vsqr2(1)^0.5;
r2(1)=(x2(1)^2+y2(1)^2)^0.5;
ax2(1)=-x2(1)*r23(1)-m1*(x2(1)-x1(1))*run3(1);;
ay2(1)=-y2(1)*r23(1)-m1*(y2(1)-y1(1))*run3(1);;
vvx2(1)=vx2(1)+ax2(1)*eps2;
vvy2(1)=vy2(1)+ay2(1)*eps2;
energy2(1)=-m2/r2(1)+0.5*m2*vsqr2(1);
%energy12 is the potential energy between Neptune and Uranus
energy12(1)=-m12/run(1);
energy_total(1)=energy1(1)+energy2(1)+energy12(1);% the total energy of the
system: Uranus and Neptune.
j2(1)=m2*v2(1)*r2(1);%the angular momentum of Neptune.
f(1)=m22*(run(1)/r1(1))^2;% the ratio of the forces applied by the Sun to Uranus to
that of Neptune on Uranus.
x1(2)=x1(1)+eps*vvx1(1);
y1(2)=y1(1)+eps*vvy1(1);
x2(2)=x2(1)+eps*vvx2(1);
y2(2)=y2(1)+eps*vvy2(1);
%end of calculating the first step
%Starting the main loop of the program
%n is the number of time steps
%____________________________________________________________________
______
%n=53860;
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n=40000;
for i=2:n
%calculating the distances necessary for the calculation.
r1(i)=(x1(i)^2+y1(i)^2)^0.5;
r13(i)=1/(r1(i)^3);
r2(i)=(x2(i)^2+y2(i)^2)^0.5;
r23(i)=1/(r2(i)^3);
run(i)=((x2(i)-x1(i))^2+(y2(i)-y1(i))^2)^0.5;
run3(i)=1/(run(i)^3);
%__________________________________________
%calculating acceleration components
ax1(i)=-x1(i)*r13(i)-m2*(-x2(i)+x1(i))*run3(i);
ay1(i)=-y1(i)*r13(i)-m2*(-y2(i)+y1(i))*run3(i);
ax2(i)=-x2(i)*r23(i)-m1*(x2(i)-x1(i))*run3(i);
ay2(i)=-y2(i)*r23(i)-m1*(y2(i)-y1(i))*run3(i);
%_______________________________________
%Note that the vvx1(i) are calculated in the middle of interval at times
% (0.5, 1.5. 2.5,...)*eps/2 whereas the x and y are computed at
% (1,2,3,...)*eps such as a time delay of eps/2 between velocity and x.
% This important when computing the total energy.
vvx1(i)=vvx1(i-1)+ax1(i)*eps;
vvy1(i)=vvy1(i-1)+ay1(i)*eps;
vvx2(i)=vvx2(i-1)+ax2(i)*eps;
vvy2(i)=vvy2(i-1)+ay2(i)*eps;
%_______________________________________
x1(i+1)=x1(i)+eps*vvx1(i);
y1(i+1)=y1(i)+eps*vvy1(i);
x2(i+1)=x2(i)+eps*vvx2(i);
y2(i+1)=y2(i)+eps*vvy2(i);
vsqr1(i)=vvx1(i)^2+vvy1(i)^2;
v1(i)=vsqr1(i)^0.5;
r1(i)=(x1(i)^2+y1(i)^2)^0.5;
vsqr2(i)=vvx2(i)^2+vvy2(i)^2;
v2(i)=vsqr2(i)^0.5;
r2(i)=(x2(i)^2+y2(i)^2)^0.5;
%_______________________________________
%clculating the total energy, total angular momentum and the ration between
%the gravity forces on Uranus.
energy1(i)=-m1/r1(i)+0.5*m1*vsqr1(i);
j1(i)=m1*v1(i)*r1(i);
energy2(i)=-m2/r2(i)+0.5*m2*vsqr2(i);
energy12(i)=-m12/run(i);
j2(i)=m2*v2(i)*r2(i);
energy_total(i)=energy1(i)+energy2(i)+energy12(i);
f(i)=m22*(run(i)/r1(i))^2;
% The lines below which are commented allow to plot the location of the
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% planets in intermediate times.
% if i==(n/4)*j
% j=j+1;
% figure
% t=(0:eps:(n-1)*eps);
% plot(x1,y1,'k','LineWidth',2)
% grid on
% hold on
% plot(x2,y2,'b','LineWidth',2)
% hold on
% plot(x1(i),y1(i),'o')
% hold on
% plot(x2(i),y2(i),'o')
% xlabel('x [AU]')
% ylabel('y[AU]')
% end
end
%____________________________________________________________________
%the part of the code below plots the different results of the
%calculations
% The first figure plot the orbits of Uranus and Neptune.
figure
set(gca,'FontSize',16)
t=(0:eps:(n-1)*eps);
plot(x1,y1,'k','LineWidth',2)
grid on
hold on
plot(x2,y2,'b','LineWidth',2)
hold on
plot(x1(i),y1(i),'o')
hold on
plot(x2(i),y2(i),'o')
xlabel('x [AU]')
ylabel('y[AU]')
axis square
%___________________________________________
% This figure plot the total energy of the system and the total angular
% momentum of the system.
figure
set(gca,'FontSize',16)
subplot(1,2,1)
plot(t,energy_total,'r','LineWidth',2)
grid on
% hold on
% plot(t,energy2,'b','LineWidth',2)
xlabel('Time')
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ylabel('Total Energy')
axis square
subplot(1,2,2)
set(gca,'FontSize',16)
plot(t,j1+j2,'r','LineWidth',2)
grid on
% hold on
% plot(t,j2,'b','LineWidth',2)
xlabel('Time')
ylabel('Angular Momentum')
axis square
%__________________________________________________________________
%This figure plot the orbital velocity of the planets as a function of the
%orbit radius
figure
set(gca,'FontSize',16)
plot(r1,v1,'k','LineWidth',2')
grid on
hold on
plot(r2,v2,'b','LineWidth',2')
xlabel('r')
ylabel('Orbital Velocity')
%____________________________________________________________________
__
%This figure plot the ratio between the gravity forces acting on Uranus by
%the Sun and by Neptune.
figure
set(gca,'FontSize',16)
plot(t,f,'b','LineWidth',2')
grid on
xlabel('Time')
ylabel('Ratio between the Gravity forces on Uranus')
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