Department of Aerospace EngineeringIndian Institute of technology
Kanpur
Project report
Submitted for
AE673A-Rocket and Missile structuresguided by Dr.S.Kamle
by
VIGNESHWARAN R14101060M.Tech
Department of Aerospace Engineering
April 3, 2015
Contents
1 Introduction 11.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Brief Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Horizon coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Absolute coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Part A(Stellarium) 22.1 How to find . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 RA/dec,Az/Alt data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Part B(Kepler’s Law) 33.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2.1 Perihelion passage Time: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.2 Orbital Period: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.3 Mean Motion: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.4 Mean Anamoly: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.5 Eccentric Anamoly: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.6 True Anamoly and Radius vector: . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.7 Heliocentric ecliptic coordinates of Sun: . . . . . . . . . . . . . . . . . . . . . . . 63.2.8 Heliocentric ecliptic coordinates of Earth: . . . . . . . . . . . . . . . . . . . . . . 63.2.9 Coordinate transformation to Earth center: . . . . . . . . . . . . . . . . . . . . . 63.2.10 Geocentric Equatorial coordinates: . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.11 RA/dec Calculation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Comparision of values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Part C(Rosetta Space Mission details) 74.1 Mission Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Historic mission: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 The cosmic billiard ball: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Specifications: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4.1 The Rosetta Orbiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4.2 The Rosetta Lander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.5 Future Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Conclusion 10
6 References and Sources 10
Appendices 11
A MATLAB program 11
List of Figures
1 Horizon coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 RA/dec coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Snapshot of Stellarium data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Rosetta mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Rosetta orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1 Introduction
1.1 Objective
The Aim is to find position of Sun, Moon and all planets of our solar system with Earth as reference ata time using any software(Eg:Stellarium). And comparing those values for Sun with the values foundusing Kepler’s law.And to collect information of space vehicle system and trajectory for any one spacemission(Here Rosetta space mission is taken).
1.2 Brief Explanation
There are three parts in this project. In part a, Stellarium softwarte is used to find Right Ascension,Declination, Azimuth, Altitude of Sun, Moon and all other planets of our solar system at the date andtime of my birth(September 9,1993). In part b, Right Ascension, Declination of Sun is calculated usingkepler’s laws and compared it with the values found using Stellarium software. In part c, Space vehiclesystem and trajectory of Rosetta space mission is briefly presented. The parameters which are used forfinding position of astral objects is explained below.
1.2.1 Horizon coordinate system
Figure 1: Horizon coordinate system
Coordinates in the horizon system are known asazimuth and altitude.Azimuth is the angular distance measured alongthe horizon from the north point. The secondaryreference circles in this system are known as verti-cal circles. Thus azimuth is measured to the footof the vertical circle passing through the object ofinterest.Altitude is the angular distance of the objectabove or below the horizon measured along a ver-tical circle.The altitude of the zenith is +90◦, and the nadiris -90◦.The azimuth of the north point is 0◦; the east point, 90◦; the south point, 180◦; and the westpoint, 270◦. The main disadvantage of the alt-az system is that it is a local coordinate system - i.e.two observers at different points on the Earth’s surface will measure different altitudes and azimuthsfor the same star at the same time. In addition, an observer will find that the star’s alt-az coordinateschanges with time as the celestial sphere appears to rotate.
1.2.2 Absolute coordinate system
Right Ascension and Declination serve as an absolute coordinate system fixed on the sky, rather thana relative system like the zenith/horizon system.
Figure 2: RA/dec coordinate system
Right ascension (symbol α, abbreviated RA)measures the angular distance of an object east-ward along the celestial equator from the vernalequinox to the hour circle passing through the ob-ject. The vernal equinox point is one of the twowhere the ecliptic intersects the celestial equator.Analogous to terrestrial longitude, right ascensionis usually measured in sidereal hours, minutes andseconds instead of degrees, a result of the methodof measuring right ascensions by timing the pas-sage of objects across the meridian as the Earth
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rotates. There are (360◦ / 24h) = 15◦ in one hourof right ascension, 24h of right ascension aroundthe entire celestial equator.Declination (symbol δ, abbreviated dec) measures the angular distance of an object perpendicular tothe celestial equator, positive to the north, negative to the south. For example, the north celestial polehas a declination of +90◦. The origin for declination is the celestial equator, which is the projection ofthe Earth’s equator onto the celestial sphere. Declination is analogous to terrestrial latitude.
2 Part A(Stellarium)
Using Stellarium software Right ascension, Declination, Azimuth and Altitude at September 9,19938.45am is found with the observer at Kanpur,UP.
2.1 How to find
Step 1:Open stellarium software and set location to Kanpur,UP.Step 2:On moving mouse to the lower left corner, a menu will open. Click date & time option or pressF5.Step 3:Then enter your Data and Time of birth(September 9,1993 8.45am).Step 4:Again using same menu click search or press F3 to search for any astral object to get its location.
Figure 3: Snapshot of Stellarium data
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2.2 RA/dec,Az/Alt data
Object Right Ascension Declination Altitude Azimuth
Sun 11h10m30s 5°18’26” 104°51’54” 37°58’46”
moon 4h52m45s 21°42’23” 273°03’26” 49°27’25”
Mercury 11h47m11s 2°29’59” 102°17’48” 28°25’21”
Venus 9h10m17s 16°40’25” 113°37’16” 68°58’48”
Mars 13h07m19s -6°52’32” 100°54’35” 6°28’32”
Jupiter 13h03m44s -5°35’26” 100°12’38” 7°53’51”
Saturn 21h53m27s -14°26’54” 286°20’45” -58°42’19”
Uranus 19h20m12s -22°38’29” 62°18’33” -82°12’04”
Neptune 19h20m4s -21°29’39” 55°39’47” -81°32’26”
Pluto 15h37m15s -4°57’20” 82°03’47” -26°13’04”
3 Part B(Kepler’s Law)
3.1 Algorithm
The Algorithm to find Right Ascension, Declination using Kepler’s law is presented in the form of flowchart below:
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Start calcula-tion for astral
object(Eg:Sun)
Start calculationfor Reference
object(Eg:Earth)
Find Time of Per-ihelion passage ofobject with center
of solar system
Calculate theOrbital Period
Calculatemean motion
CalculateMean anomaly
Calculate Eccentricanomaly usingBessel function
Calculate Trueanomaly andradius vector
Stop
Calculate Heliocentriccoordinates of
astral object(Sun)
Calculate Heliocentriccoordinates of refer-ence object(Earth)
Calculate Ra/dec
Transfer the originHeliocentric coordi-nate of astral object
to Earth center
Calculate Geo-centric equatorial
coordinates of planet
3.2 Procedure
The procedure for calculating RA/dec will be explained with Sun as astral object and reference planetas Earth. For calculating true anamoly and radius vector the following data for Earth is taken fromEarth fact sheet of NASA website.It is based on J2000 i.e Jan 1,2000 12:00PM.
Orbital inclination(i)=0.00005°;
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Longitude of perihelion(p)=102.94719°;
Longitude of ascending node(o)=-11.26064°;
Mean longitude(ml)=100.46435°;
Max. orbital Velocity(ps)=30.29°;
Perihelion(pr)=147.09*106 km;
Semimajor axis(a)=1.00000011 AU;
Orbital eccentricity(e)=0.01671022;
3.2.1 Perihelion passage Time:
The time of perihelion passage of Sun is calculated by the following formula:
T0 = datenum(′January1, 2000; 12 : 00 : 00PM ′)
T = T0 +
p−mlps/p
24 ∗ 60 ∗ 60
where datanum-MATLAB function converts the date to a numerical value;
The explanation for using above formula is explained in http://bado-shanai.net/Astrogation/
Calcperihelion.htm;
3.2.2 Orbital Period:
Orbital period is calculated using following formula:
P = a3/2 ∗ 365.25
3.2.3 Mean Motion:
Mean motion is calculated using
n =360
P
3.2.4 Mean Anamoly:
Mean anamoly is calculated using
M = (t− T ) ∗ n
where t = datenum(′September9, 1993; 8 : 45 : 00AM ′)
3.2.5 Eccentric Anamoly:
The following kepler’s equation has to be solved to get Eccentric anamoly.Bessel’s function is used tosolve the kepler’s equation:
E − esin(E) = (t− T )n
E = M +∞∑j=1
2
k∗ Jk(ke) ∗ sin(kM)
where Jk(ke) is the bessel function function of ke of kth order.
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3.2.6 True Anamoly and Radius vector:
The true anamoly and radius vector is calculated is found using following equation:
tan(v
2) =
√1 + e
1− e∗ tan(
E
2;
r = a ∗ (1− e ∗ cos(E));
v = v+ 180 is the final should be taken for Sun. Because the true anamoly calculated now is the angleat which earth is viewed from from Sun but we need true anamoly when Sun is viewed from earth.Hence 180deg is added to calculated value. But radius vector will remain the same.
3.2.7 Heliocentric ecliptic coordinates of Sun:
The Heliocentric coordinates of Sun is calculated using following formulae:
X = r ∗ [cos(o) ∗ cos(v + p− o)− sin(o) ∗ sin(v + p− o) ∗ cos(i)];
Y = r ∗ [sin(o) ∗ cos(v + p− o) + cos(o) ∗ sin(v + p− o) ∗ cos(i)];
Z = r ∗ [sin(v + p− o− 2 ∗ pi) ∗ sin(i)];
3.2.8 Heliocentric ecliptic coordinates of Earth:
The entire procedure should be done again for calculating true anamoly and radius vector of Earth.And 180deg should not be added this time as we need only data of Earth from our center of solarsystem.
Xe = r ∗ [cos(o) ∗ cos(v + p− o)− sin(o) ∗ sin(v + p− o) ∗ cos(i)];
Y e = r ∗ [sin(o) ∗ cos(v + p− o) + cos(o) ∗ sin(v + p− o) ∗ cos(i)];
Ze = r ∗ [sin(v + p− o− 2 ∗ pi) ∗ sin(i)];
3.2.9 Coordinate transformation to Earth center:
Coordinate transformation to earth center is relatively simple as it is just subtraction of two values:
Xprime = X −Xe;
Y prime = Y − Y e;
Zprime = Z − Ze;
3.2.10 Geocentric Equatorial coordinates:
After transforming to Earth center, the values are corrected for Obliquity of Earth(23.439deg) to get
Xq = Xprime;
Y q = Y prime ∗ cos(23.439 ∗ pi/180)− Zprime ∗ sin(23.439 ∗ pi/180);
Zq = Y prime ∗ sin(23.439 ∗ pi/180) + Zprime ∗ cos(23.439 ∗ pi/180);
3.2.11 RA/dec Calculation:
This is the last step in which Right ascension and declionation is calculated from Geocentric equatorialcoordinates:
RA = atan(Y q/Xq);
dec = atan(Zq/sqrt(Xq2 + Y q2));
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3.3 Comparision of values
MethodRight
AscensionDeclination
Stellarium 11h10m30s 5°18’26”
Kepler’s Method 11h11m29s 5°12’18”
From the above table it can be noticed that the values are almost same from both methods.
4 Part C(Rosetta Space Mission details)
Rosetta is a robotic space probe built and launched by the European Space Agency. Along with Philae,its lander module, Rosetta is performing a detailed study of comet 67P/Churyumov–Gerasimenko(67P). It also performed a flyby of the planet Mars and asteroids 21 Lutetia and 2867 Steins. On 12November 2014 the mission performed the first soft landing on a comet and returned data from thesurface.
4.1 Mission Overview
Rosetta was launched on 2 March 2004 on an Ariane 5 rocket and reached the comet on 6 August2014, becoming the first spacecraft to orbit a comet.The spacecraft consists of the Rosetta orbiter,which features 12 instruments, and the Philae lander, with nine additional instruments. The Rosettamission will orbit 67P for 17 months and is designed to complete the most detailed study of a cometever attempted. The spacecraft is controlled from the European Space Operations Centre (ESOC), inDarmstadt, Germany.The planning for the operation of the scientific payload, together with the dataretrieval, calibration, archiving and distribution, is performed from the European Space AstronomyCentre (ESAC), in Villanueva de la Canada, near Madrid, Spain. It has been estimated that in thedecade preceding 2014, some 2,000 people assisted in the mission in some capacity.The spacecraft performed two asteroid flyby missions on its way to the comet. In 2007, Rosetta alsoperformed a Mars swing-by (flyby).The craft completed its flyby of asteroid 2867 Steins in September2008 and of 21 Lutetia in July 2010.On 20 January 2014, Rosetta was taken out of a 31-month hiber-nation mode as it approached the comet.Rosetta-Philae lander successfully made the first soft landing on a comet nucleus when it touched downon 67P on 12 November 2014.
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Figure 4: Rosetta mission
4.2 Historic mission:
The Rosetta mission will achieve many historic firsts.
� Rosetta will be the first spacecraft to orbit a comet’s nucleus.
� It will be the first spacecraft to fly alongside a comet as it heads towards the inner Solar System.
� Rosetta will be the first spacecraft to examine from close proximity how a frozen comet is trans-formed by the warmth of the Sun.
� Shortly after its arrival at Comet 67P/Churyumov-Gerasimenko, the Rosetta orbiter will despatcha robotic lander for the first controlled touchdown on a comet nucleus.
� The Rosetta lander’s instruments will obtain the first images from a comet’s surface and makethe first in situ analysis to find out what it is made of.
� On its way to Comet 67P/Churyumov-Gerasimenko, Rosetta will pass through the main asteroidbelt, with the option to be the first European close encounter with one or more of these primitiveobjects.
� Rosetta will be the first spacecraft ever to fly close to Jupiter’s orbit using solar cells as its mainpower source.
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4.3 The cosmic billiard ball:
Rosetta will bounce around the inner Solar System like a ‘cosmic billiard ball’, circling the Sun almostfour times during its ten-year trek to Comet 67P/Churyumov-Gerasimenko. Along this roundaboutroute, Rosetta will enter the asteroid belt twice and gain velocity from gravitational ‘kicks’ providedby close fly-bys of Mars (2007) and Earth (2005, 2007 and 2009).On 25 February 2007, the craft was scheduled for a low-altitude bypass of Mars, to correct the trajectory.This was not without risk, as the estimated altitude of the flyover manoeuvre was a mere 250 kilometres(160 mi). During that encounter, the solar panels could not be used since the craft was in the planet’sshadow, where it would not receive any solar light for 15 minutes, causing a dangerous shortage ofpower. The craft was therefore put into standby mode, with no possibility to communicate, flying onbatteries that were originally not designed for this task.This Mars manoeuvre was therefore nicknamed”The Billion Euro Gamble”. The flyby was successful, with Rosetta even returning detailed imagesof the surface and atmosphere of the planet, and the mission continued as planned.
Figure 5: Rosetta orbit
4.4 Specifications:
4.4.1 The Rosetta Orbiter
Rosetta is a large aluminium box with dimensions 2.8 x 2.1 x 2.0 metres. The scientific instrumentsare mounted on the ’top’ of the box (Payload Support Module) while the subsystems are on the ’base’(Bus Support Module). On one side of the orbiter is a 2.2-metre diameter communications dish – thesteerable high-gain antenna. The lander is attached to the opposite face. Two enormous solar panel’wings’ extend from the other sides. These wings, each 32 square metres in area, have a total spanof about 32 metres tip to tip. Each of them comprises five panels, and both may be rotated through
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+/-180 degrees to catch the maximum amount of sunlight.Rosetta’s instruments In the vicinity of Comet 67P/Churyumov-Gerasimenko, the scientific instru-ments almost always point towards the comet, while the antennae and solar arrays point towards theSun and Earth (at large distances, they are more or less in the same direction). In contrast, the orbiter’sside and back panels are in shade for most of the mission. Since these panels receive little sunlight,they are an ideal location for the spacecraft’s radiators and louvres. They will also face away from thecomet, so damage from comet dust will also be minimised.Propulsion At the heart of the orbiter is the main propulsion system. Mounted around a verticalthrust tube are two large propellant tanks, the upper one containing fuel, and the lower one containingthe oxidiser. The orbiter also carries 24 thrusters for trajectory and attitude control. Each of thesethrusters pushes the spacecraft with a force of 10 Newtons, about the same as experienced by someoneholding a large bag of apples. Over half the launch weight of the entire spacecraft is taken up bypropellant.
4.4.2 The Rosetta Lander
The 100-kilogram Rosetta lander is provided by a European consortium under the leadership of theGerman Aerospace Research Institute (DLR). The box-shaped lander is carried on the side of theorbiter until it arrives at Comet 67P/Churyumov-Gerasimenko. Once the orbiter is aligned correctly,the lander is commanded to self-eject from the main spacecraft and unfold its three legs, ready for agentle touchdown at the end of the ballistic descent. On landing, the legs damp out most of the kineticenergy to reduce the chance of bouncing, and they can rotate, lift or tilt to return the lander to anupright position. Immediately after touchdown, a harpoon is fired to anchor the lander to the groundand prevent it escaping from the comet’s extremely weak gravity. The minimum mission target is oneweek, but surface operations may continue for many months.Lander design The lander structure consists of a baseplate, an instrument platform, and a polygonalsandwich construction, all made of carbon fibre. Some of the instruments and subsystems are beneatha hood that is covered with solar cells. An antenna transmits data from the surface to Earth via theorbiter. The lander carries nine experiments, with a total mass of about 21 kilograms. It also carries adrilling system to take samples of subsurface material.
4.5 Future Mission
The orbiter continues to orbit Comet 67P/Churyumov-Gerasimenko, observing what happens as theicy nucleus approaches the Sun and then travels away from it. The mission ends in December 2015.Rosetta will once again pass close to Earth’s orbit, more than 4000 days after its adventure began.
5 Conclusion
Thus from Part a and part b, Right Ascension , declination are calculated and compared. In part c,Space mission details about rosetta and Philae is presented which can be useful to get Astro Knowledge.
6 References and Sources
[1] S.Kamle:Lecture notes of Rocket and missile structures(AE673A).
[2] http://bado-shanai.net/Astrogation/Calcperihelion.htm-for perihelion passage calculation
[3] http://www.stargazing.net/kepler/ellipse.html#twig04-for calculating Right Ascension anddeclination from true anamoly and radius vector.
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Appendices
A MATLAB program
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4/3/2015 code
file:///D:/PG2ndsem/Rocket%20and%20missile%20structures/Report/code.htm 2/2
%Eccentric anomalyE=M*pi/180;%bessel function solverfor k=1:100E=E+(2/k)*besselj(k,e*k)*sin(k*M*pi/180);end%true anomalyv=2*atan(sqrt((1+e)/(1‐e))*tan(E/(2)));v=v*180/(pi);%radius vectorr=a*(1‐e*cos(E));v=v*pi/180;Xe=r*[cos(o)*cos(v+p‐o)‐sin(o)*sin(v+p‐o)*cos(i)];Ye=r*[sin(o)*cos(v+p‐o)+cos(o)*sin(v+p‐o)*cos(i)];Ze=r*[sin(v+p‐o‐2*pi)*sin(i)];
%‐‐‐‐Transfer to earth center‐‐‐%Xprime=X‐Xe;Yprime=Y‐Ye;Zprime=Z‐Ze;%‐‐Geocentric equatorial coordinates of planet‐‐‐%Xq=Xprime;Yq=Yprime*cos(23.439*pi/180)‐Zprime*sin(23.439*pi/180);Zq=Yprime*sin(23.439*pi/180)+Zprime*cos(23.439*pi/180);RA=atan(Yq/Xq);dec=atan(Zq/sqrt(Xq^2+Yq^2));%‐‐Right Ascension‐‐‐‐%if (Xq>=0&&Yq<0) RA=(RA+2*pi)*180/(pi*15)%‐‐‐‐‐changeelseif(Xq<=0) RA=(RA+pi)*180/(pi*15)%‐‐‐‐‐changeelse RA=(RA)*180/(pi*15)%‐‐‐‐‐changeend%Declinationdec=dec*180/pi
RA =
11.1914
dec =
5.2051
Published with MATLAB® 7.10
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