1) The document describes a mathematical model for predicting the reentry and impact of satellites into Earth's atmosphere. It uses Newton's second law of motion and models the forces of gravity and drag.
2) By solving the resulting differential equation numerically, the model can determine the impact time and velocity of a satellite. This allows estimates of the damage caused and time for evacuation if the satellite were to impact a populated area.
3) The document provides an example application of the model to Sputnik 1 and graphs the results, showing a impact time of about 5.5 minutes and velocity of 47 m/s.
This unit carry information of Acceleration Due to the Gravity (g), Satellite and Planetary Motion and Gravitational Field, Potential Energy, Kinetic Energy and Total energy of the satellite. in each section, there is an example so as you could be able to manipulate those equations that are associated with this unit. Also, there is problem practice so as to straighten the understanding of this module.
Optimal trajectory to Saturn in ion-thruster powered spacecraftKristopherKerames
In this document, I derive the equations of motion for an ion-thruster powered spacecraft and use numerical methods to calculate its optimal trajectory to Saturn. I did this work within 48 hours for the University Physics Competition in 2020.
Digital Library of GLT Saraswati Bal Mandir. Gravitation is a natural phenomenon by which all physical bodies attract each other. It is most commonly experienced as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped.
This unit carry information of Acceleration Due to the Gravity (g), Satellite and Planetary Motion and Gravitational Field, Potential Energy, Kinetic Energy and Total energy of the satellite. in each section, there is an example so as you could be able to manipulate those equations that are associated with this unit. Also, there is problem practice so as to straighten the understanding of this module.
Optimal trajectory to Saturn in ion-thruster powered spacecraftKristopherKerames
In this document, I derive the equations of motion for an ion-thruster powered spacecraft and use numerical methods to calculate its optimal trajectory to Saturn. I did this work within 48 hours for the University Physics Competition in 2020.
Digital Library of GLT Saraswati Bal Mandir. Gravitation is a natural phenomenon by which all physical bodies attract each other. It is most commonly experienced as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped.
Discuss the law of universal gravitation and satellite motion.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
The pearled solar eclipse of 1912.04.17 occurred 60 hours after the TITANIC disaster had cast its shadow upon this exciting event. The data collected during this most elusive eclipse are compared to those generated by Xavier JUBIER's 5MCSE, the most up-to date ergonomical solar eclipse simulation freeware, which allows the choice of the DeltaT parameter, as well as the exact GPS Coordinates of the observation site such as the balloon Globule at 900 meter over Rethondes.
Gravitation has been the most common phenomenon in our lives but somewhere down the line we don't know musch about it. So here is a presentation whic will help you out to know what it is !! I'll be makin it available for download once i submit it in school :P :P ! Coz last one of the brats showed the same presentation that i uploade and unfortunatele his roll number fell before mine ! I was damned..:D :D :P
Gravity Gravitation English Presentation
Tugas Fisika
Tugas Bahasa Inggris
oleh :
Kelas 12 IPA 6 SMA Negeri 1 Yogyakarta tahun 2014
Semangat!!!!!!! SUKSES
This is the NCERT CBSE syllabus ppt on the topic Gravitation. It will be helpful for students studying in that class and will enable them to understand better.
Discuss the law of universal gravitation and satellite motion.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
The pearled solar eclipse of 1912.04.17 occurred 60 hours after the TITANIC disaster had cast its shadow upon this exciting event. The data collected during this most elusive eclipse are compared to those generated by Xavier JUBIER's 5MCSE, the most up-to date ergonomical solar eclipse simulation freeware, which allows the choice of the DeltaT parameter, as well as the exact GPS Coordinates of the observation site such as the balloon Globule at 900 meter over Rethondes.
Gravitation has been the most common phenomenon in our lives but somewhere down the line we don't know musch about it. So here is a presentation whic will help you out to know what it is !! I'll be makin it available for download once i submit it in school :P :P ! Coz last one of the brats showed the same presentation that i uploade and unfortunatele his roll number fell before mine ! I was damned..:D :D :P
Gravity Gravitation English Presentation
Tugas Fisika
Tugas Bahasa Inggris
oleh :
Kelas 12 IPA 6 SMA Negeri 1 Yogyakarta tahun 2014
Semangat!!!!!!! SUKSES
This is the NCERT CBSE syllabus ppt on the topic Gravitation. It will be helpful for students studying in that class and will enable them to understand better.
In this presentation the following topics are covered:
- Active debris removal techniques
- Tethered space tug
- Mathematical model
- Numerical simulation and analysis
- Results and conclusion
Presentation for the 5th Eucass - European Conference for Aerospace Sciences - Munich, Germany, 1-4 July 2013.
Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite...Sérgio Sacani
With interstellar mission concepts now being under study by various space agencies and institutions,
a feasible and worthy interstellar precursor mission concept will be key to the success of the long
shot. Here we investigate interstellar-bound trajectories of solar sails made of the ultra lightweight
material aerographite. Due to its extremely low density (0.18 kgm−3) and high absorptivity (∼1), a
thin shell can pick up an enormous acceleration from the solar irradiation. Payloads of up to 1 kg can
be transported rapidly throughout the solar system, e.g. to Mars and beyond. Our simulations consider
various launch scenarios from a polar orbit around Earth including directly outbound launches as well
as Sun diver launches towards the Sun with subsequent outward acceleration. We use the poliastro
Python library for astrodynamic calculations. For a spacecraft with a total mass of 1 kg (including
720 g aerographite) and a cross-sectional area of 104 m2, corresponding to a shell with a radius of 56m,
we calculate the positions, velocities, and accelerations based on the combination of gravitational and
radiation forces on the sail. We find that the direct outward transfer to Mars near opposition to Earth
results in a relative velocity of 65 kms−1 with a minimum required transfer time of 26 d. Using an
inward transfer with solar sail deployment at 0.6AU from the Sun, the sail’s velocity relative to Mars
is 118 kms−1 with a transfer time of 126 d, whereMars is required to be in one of the nodes of the two
orbital planes upon sail arrival. Transfer times and relative velocities can vary substantially depending
on the constellation between Earth andMars and the requirements on the injection trajectory to the Sun
diving orbit. The direct interstellar trajectory has a final velocity of 109 kms−1. Assuming a distance
to the heliopause of 120AU, the spacecraft reaches interstellar space after 5.3 yr. When using an
initial Sun dive to 0.6AU instead, the solar sail obtains an escape velocity of 148 kms−1 from the
solar system with a transfer time of 4.2 yr to the heliopause. Values may differ depending on the
rapidity of the Sun dive and the minimum distance to the Sun. The mission concepts presented in this
paper are extensions of the 0.5 kg tip mass and 196m2 design of the successful IKAROS mission to
Venus towards an interstellar solar sail mission. They allow fast flybys atMars and into the deep solar
system. For delivery (rather than fly-by) missions of a sub-kg payload the biggest obstacle remains in
the deceleration upon arrival.
Why Does the Atmosphere Rotate? Trajectory of a desorbed moleculeJames Smith
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Kármán Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
3.1.3 Relate gravitational acceleration, g on the surface of the Earth with the universal gravitational constant, G
3.1.4 Justify the importance of knowing the values of gravitational acceleration of the planets in the Solar System.
3.1.5 Describe the centripetal force in the motion of satellites and planets system.
Centripetal Force, F = 푚푣2푟
3.1.6 Determine the mass of the Earth and the Sun using Newton’s universal law of gravitation and centripetal force.
Solar radiation management with a tethered sun shieldSérgio Sacani
This paper presents an approach to Solar Radiation Management (SRM) using atethered solar shield at the modified gravitational L1 Lagrange point. Unlike previousproposals, which were constrained by the McInnes bound on shield surface density,our proposed configuration with a counterweight toward the Sun circumvents thislimitation and potentially reduces the total mass by orders of magnitude. Furthermore,only 1% of the total weight must come from Earth, with ballast from lunar dustor asteroids serving as the remainder. This approach could lead to a significant costreduction and potentially be more effective than previous space-based SRM strategies.
In physics, gravity (from Latin gravitas 'weight'[1]) is a fundamental interaction which causes mutual attraction between all things that have mass. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong interaction, 1036 times weaker than the electromagnetic force and 1029 times weaker than the weak interaction. As a result, it has no significant influence at the level of subatomic particles.[2] However, gravity is the most significant interaction between objects at the macroscopic scale, and it determines the motion of planets, stars, galaxies, and even light.
On Earth, gravity gives weight to physical objects, and the Moon's gravity is responsible for sublunar tides in the oceans (the corresponding antipodal tide is caused by the inertia of the Earth and Moon orbiting one another). Gravity also has many important biological functions, helping to guide the growth of plants through the process of gravitropism and influencing the circulation of fluids in multicellular organisms.
The gravitational attraction between the original gaseous matter in the universe caused it to coalesce and form stars which eventually condensed into galaxies, so gravity is responsible for many of the large-scale structures in the universe. Gravity has an infinite range, although its effects become weaker as objects get farther away.
Gravity is most accurately described by the general theory of relativity (proposed by Albert Einstein in 1915), which describes gravity not as a force, but as the curvature of spacetime, caused by the uneven distribution of mass, and causing masses to move along geodesic lines. The most extreme example of this curvature of spacetime is a black hole, from which nothing—not even light—can escape once past the black hole's event horizon.[3] However, for most applications, gravity is well approximated by Newton's law of universal gravitation, which describes gravity as a force causing any two bodies to be attracted toward each other, with magnitude proportional to the product of their masses and inversely proportional to the square of the distance between them.
Current models of particle physics imply that the earliest instance of gravity in the universe, possibly in the form of quantum gravity, supergravity or a gravitational singularity, along with ordinary space and time, developed during the Planck epoch (up to 10−43 seconds after the birth of the universe), possibly from a primeval state, such as a false vacuum, quantum vacuum or virtual particle, in a currently unknown manner.[4] Scientists are currently working to develop a theory of gravity consistent with quantum mechanics, a quantum gravity theory,[5] which would allow gravity to be united in a common mathematical framework (a theory of everything) with the other three fundamental interactions of physics.
It is always amazing to see the interaction of planets, Sun, Stars, and other celestial objects in space which leads to astronomical events. In this chapter we will learn certain laws of physics which explains gravitation between celestial objects, free fall of body, mass and weight of the objects.
SUMMARY OF CHAPTER:-
Definition of Gravitation
Acceleration Due to Gravity
Variation Of “G” With Respect to Height And Depth
Escape Velocity
Orbital Velocity
Gravitational Potential
Time period of a Satellite
Height of Satellite
Binding Energy
Various Types of Satellite
Kepler’s Law of Planetary motion
The mass of_the_mars_sized_exoplanet_kepler_138_b_from_transit_timingSérgio Sacani
Artigo da revista Nature, descreve o trabalho de astrônomos para medir o tamanho e a massa de um exoplaneta parecido com Marte, além de caracterizar por completo o sistema planetário da estrela Kepler-138.
The mass of_the_mars_sized_exoplanet_kepler_138_b_from_transit_timing
Write up Final
1. 1
Reentry of Satellites into the Atmosphere
Math 238 – Spring 2015
Rick Snell
Edgar Mendoza-Seclen
Hannah Carson
2. 2
Abstract
The reentry of a satellite can be modeled using Newton’s Second Law of motion. The
equation used in our model was based on the famous Russian satellite, Sputnik 1. The model
assumes the mass is constant, the Earth is flat and non-rotating, and neglects solar and weather
activity.
With this model, we are able to find the impact velocities and determine the time at
which the space debris, or in our case a satellite, will hit the ground. Assuming the satellite does
not burn up in the atmosphere, we can predict how long a populated area has in order to
evacuate and predict how much damage the space debris or satellite will cause, based on the
speed.
Introduction
There is an old saying, “what goes up, must also come down.” Today, scientists are
facing the issue of how man-made objects, such as satellites, will make it safely back to the
ground. Over the past several decades, satellites and other spacecraft have scattered debris
totaling to about 6,700 tons into space around the Earth. All of this space debris is merely
orbiting our planet, waiting to come back down.
Reentry into the atmosphere can cause an object to travel hundreds of thousands of
kilometers in a matter of minutes. Because of the vast uninhabited regions of the planet, space
debris tends to land in either the ocean or a desert area, causing no damage to human life or
property. Also, some objects are small enough, or structurally weak enough to simply burn up
upon reentry. But if a satellite or other piece of space debris lands in a populated area, it could
3. 3
potentially cause immense damage to buildings or people. This debris is neither controlled nor
stable, so it is difficult for scientists to predict the exact location of impact. But, with both lives
and structures at risk, it is imperative to try to model their trajectories.
The main concern this project will focus on is finding the impact velocities and
determining the time at which the space debris or in our case, a satellite, will hit the ground.
With these two results, we will be able to predict how much time a populated area may have to
evacuate and how much damage the space debris or satellite will cause.
Explanation of Model
A mathematical model of a satellite reentering Earth’s atmosphere and falling toward
the ground, same as any physical body in motion, is best described using Newton’s 2nd law of
motion:
∑ 𝐹 =
𝑑
𝑑𝑡
(𝑚𝑣)
For our purposes, we have chosen to focus on two forces acting on the satellite i.e. the forces
due to drag and gravity. For gravity, we once again turn to Newton for a description. The law
of universal gravitation states:
𝐹𝐺 = 𝐺
𝑚1 𝑚2
𝑟2
where G is the universal gravitational constant, m1 and m2 are the masses of the respective
bodies acting on each other, and r is the distance between the centers of mass of the objects.
For drag, we rely on information gathered from the University of Texas website. This resource
describes the force of drag on artificial satellites as follows1:
1. Equation fromsource is givenperunit mass.
4. 4
𝐹𝐷 = −
1
2
𝜌0 𝑒
−
𝑟−𝑅
𝐻 𝐶 𝐷 𝐴𝑣| 𝑣|
where ρ0 is the atmospheric density at ground level (approx. 1.3 kg/m3), r is the distance of the
satellite from the Earth’s center, R is the radius of the Earth (6.4x106 m), H is the atmospheric
scale height (approx. 8.5x103 m), CD is coefficient of drag, A is the cross-sectional area of the
satellite perpendicular to the direction of motion, and v is the velocity of the object. The
exponential term in this equation allows us to model the air density based on altitude, as
density decreases roughly exponentially with height. Inserting these equations into Newton’s
2nd Law of Motion and assigning up as the positive direction, we have:
−𝐺
𝑚1 𝑚2
𝑟2
−
1
2
𝜌0 𝑒
−
𝑟−𝑅
𝐻 𝐶 𝐷 𝐴𝑣| 𝑣| =
𝑑
𝑑𝑡
(𝑚𝑣)
Our next step is to substitute the appropriate symbols and dependent variables, setting the
ground as the origin and remembering that we have oriented up as positive:
−𝐺
𝑚𝑀
(𝑅 + 𝑦)2
−
1
2
𝜌0 𝐶 𝐷 𝐴𝑒
−
𝑦
𝐻 𝑦̇| 𝑦̇| = 𝑚𝑦̈
having m as the mass of the satellite, M as the mass of the Earth, y as the height above the
Earth’s surface, 𝑦̇ as velocity, and 𝑦̈ as acceleration.
5. 5
We selected a convenient historical example, Sputnik 1, as the satellite in question. The
convenience lies in the fact that Sputnik was spherical. This means that the value A in our
equation will be the same for all orientations of the satellite, and according to the same
University of Texas source, CD is approximately 2.2 for spherical satellites. Using the
specifications of Sputnik we can determine:
𝑚 = 83.6 kg , 𝐴 = 0.0841𝜋 m2
This leaves initial height and velocity as the only remaining values to be determined arbitrarily.
First, the boundary between Earth’s atmosphere and “outer space” is accepted by
international standard to be the Kármán line, or 100 km above the Earth’s surface and our
problem models the reentry of a satellite into Earth’s atmosphere. So we have:
𝑦0 = 100,000 m
Second, according to Tom Henderson of the Physics Classroom, a satellite’s orbital speed
is inversely proportional to the square root of its distance from the Earth’s center. Using the
values we already know, we have:
𝑣 = √
𝐺𝑀
𝑅 + 𝑦0
≈ 7835.3 m/s
From an online article posted by the Massachusetts Institute of Technology, the controlled
reentry angle of Space Shuttles is typically 40 degrees. This angle is calculated to balance the
risks of burning up in the atmosphere and bouncing back into orbit (bouncing is caused by the
force of lift due to the Space Shuttles wings). Because our satellite lacks wings, we need not be
concerned with bouncing, therefore any angle less than 40 degrees will suit our needs. Taking
6. 6
10 degrees below the horizontal as the angle of entry into Earth’s atmosphere, we can find the
initial vertical velocity by:
|𝑣 𝑦| = | 𝑦0̇ | = 7835.3sin(10) ≈ 1360.6 m/s
Our model is now complete, bearing in mind the following assumptions. (1) Solar
activity affects the density and thickness of the Earth’s atmosphere, so for our model we must
assume no solar activity. (2) Meteorological factors, such as jet streams and high/low-pressure
systems can impact the trajectory of an object falling to Earth; therefore, we must assume no
meteorological disturbances. And (3) the curvature, rotation, and topography of the Earth
must all be taken into account for a realistic model of satellite reentry; however, the
mathematics required to incorporate these factors is very advanced. Thus we must assume a
perfectly flat, non-rotating Earth in order for our one-dimensional model to make any sense.
Derivation of the Differential Equation
Our differential equation is an ordinary, second order, non-linear differential equation.
Because our equation is non-linear, we used a computer algebra system, Maple, to solve the
equation numerically. The dependent variables are y(t), y’(t), and y’’(t). These variables stand
for position, velocity, and acceleration respectively. Our independent variable is t, representing
time measured in seconds. Our equation contained many constants, such as G, m, M, R, ρ0, CD,
A, H. These stand for the gravitational constant, mass of the satellite, mass of the earth, radius
of the earth, density of the atmosphere at sea level, the coefficient of drag, cross-sectional area
of the satellite perpendicular to the direction of motion, and the atmospheric scale height,
respectively.
7. 7
The solution was produced by the Fehlberg fourth-fifth order Runge-Kutta method. With
this approximation, we were able to plot the satellite’s velocity versus time and the height
above the Earth’s surface versus time.
A few assumptions we made were a constant mass and density of the satellite, the
satellite doesn’t burn up, no solar or weather activity, and the Earth is a flat, non-rotating
object. These assumptions overall simplify the differential equation. Assuming the satellite does
not burn up or break apart allows us to hold mass as a constant, instead of another dependent
variable. Solar and weather activity could potentially change the force of air resistance changing
the density of the air and thickness of the atmosphere. Finally, assuming the Earth is flat and
not rotating, we are able to use a one dimensional model instead of a three dimensional model
with spherical coordinates.
Solution to Differential Equation
As mentioned above, this non-linear equation was solved numerically. The graph below
represents the height of the satellite versus time as it falls to the ground. As can be seen from
the graph, the satellite hits the ground at 330.27 seconds, or about 5.5 minutes. If the satellite
is uncontrolled this leaves less than adequate time for an area to evacuate, especially if it were
over a populated city.
8. 8
The second graph represents the velocity versus time. It shows that the velocity initially
increases but then decreases due to air resistance because the air density increases as height
decreases. The force of gravity is stronger than the force of air resistance for about the first 27
seconds. After that, the drag increases, causing the satellite to start slowing down. Because the
density of the air is a function of height in our model, terminal velocity is dependent upon
altitude and therefore decreases as the satellite approaches the ground.
9. 9
As determined from the graph, the impact velocity is around 47 m/s downward. The
diameter of a crater is proportional to the kinetic energy of the impactor. In other words, if this
were a smaller satellite, such as Sputnik, this would not cause significant damage. But as the
mass or velocity of the satellite (or piece of space debris) increases, the size of the crater and
damage will also increase.
The two graphs below show varying initial conditions, holding mass constant. The first is
with different initial velocities. Whether the satellite starts at rest or from a speed greater than
our modeled satellite, the impact velocity, when the satellite hits the ground, is consistently
around 47 m/s.
The second graph compares different initial heights, while holding mass constant. The
same is true as above that the impact velocity is about 47 m/s, no matter where the initial
height of the satellite.
10. 10
The final two graphs both hold initial velocity and initial height constant. The first
represents the height versus velocity of satellites with different masses. As can be seen, with
increasing mass, the terminal velocity is reached at a lower height and this causes the impact
velocity to increase.
11. 11
The second graph represents the height versus velocity at different cross-sectional
areas. When the surface area increases, terminal velocity is reached at a higher altitude and
thus the impact velocity is decreased.
Conclusion
If a piece of space debris or a satellite does not burn up while reentering the
atmosphere, scientists must consider the landing. The reentry can cause space debris or
satellites to travel thousands of kilometers in a matter of minutes. If the satellite is projected to
land over a body of water or a deserted area, scientist do not have to be concerned about the
impact on the Earth. If the satellite is projected to land over a populated area or structure,
scientists must take into account the landing speed and the time of fall. This will allow them to
take precautions, such as having an area evacuate.
Newton’s Second Law of Motion is a way to model the velocity and time of impact.
Using Sputnik’s specifications as our main model, we can predict that the satellite will hit the
12. 12
ground shortly after 5.5 minutes at a speed close to 47 m/s. Because Sputnik is a fairly small
satellite, the impact on the Earth is relatively insignificant. If the size of the satellite increased,
scientists may need to be more considerate of the impact. Overall, initial conditions such as
height and velocity do not alter the speed at impact, however mass and surface area have a
significant effect on impact velocity.
13. 13
References
Deziel, C. (n.d.). Facts on reentry into the earth's atmosphere. Retrieved May 8, 2015, from
Synonym website: http://classroom.synonym.com/reentry-earths-atmosphere-
6679.html
Fitzpatrick, R. (2014, August 2). Effect of atmospheric drag on artificial satellite orbits. Retrieved
May 8, 2015, from
http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node94.html
Gaposchkin, E. M., & Coster, A. J. (1988). Analysis of satellite drag. The Lincoln Laboratory
Journal, 1(2), 203-224. Retrieved from
http://www.ll.mit.edu/publications/journal/pdf/vol01_no2/1.2.6.satellitedrag.pdf
Garber, S. (2007, October 10). Sputnik and the dawn of the space age. Retrieved May 8, 2015,
from NASA website: http://history.nasa.gov/sputnik/
German Aerospace Center. (n.d.). The re-entry of the ROSAT satellite and the risks (H. Klinkrad,
Author). Retrieved from
http://www.dlr.de/dlr/en/Portaldata/1/Resources/documents/ROSAT_klinkrad_en.pdf
Grayzeck, E., Dr. (2014, August 26). Sputnik 1. Retrieved May 8, 2015, from National
Aeronautics and Space Administration website:
http://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1957-001B
Henderson, T. (2015). Mathematics of Satellite Motion. Retrieved May 18, 2015, from The
physics classroomwebsite: http://www.physicsclassroom.com/class/circles/Lesson-
4/Mathematics-of-Satellite-Motion
14. 14
Maplesoft. (2015). dsolve/numeric/rkf45. Retrieved May 8, 2015, from Maplesoft
website: http://www.maplesoft.com/support/help/Maple/
view.aspx?path=dsolve%2frkf45
Ritter, M. (2014, March 28). How many man-made satellites are currently orbiting earth.
Retrieved May 8, 2015, from Talking Points Memo website:
http://talkingpointsmemo.com/idealab/satellites-earth-orbit