Boys, set your calculator to polar mode, and draw a graph of r=4sin(3θ), then set the first interval of [0,π] and the second one: [0, 2π]
They looks exactly the same, isn’t it?
And if we take of the integral of these two to find the area, for the graph with interval [0,π], area = 4π, but for the graph with [0,2π] the area is 8π, thus, which one is the “true” one we want to find?
I. The Form of Trig Polar Equation
For a trig polar equation, the form is looks like: r=asin(bθ) or cos etc…
a decide the “size of the graph”
b decide the number of “blades”
II. Odd number b and Even number b
The thing is vital :
*The number of blades = 2b
-For even number of b, the blades created are separated(here b=4, so there are 8 “blades”):
-For odd number b, the “blades” are covered with each other, take example of r=4sin(3θ) with [0, 2π], it actually graphed 6 blades, but each two are in the same position covered the another one, so we could only see three. If we do the integral to calculate its area, we get the area of 6 blades.
Thus, if we want to find the area of 3 blades, you can set the interval of [0, π], or just simply divide the area with interval [0, 2π] by two
It’s a very interesting trick dealing with polar equation.
Landing has been a big problem to Falcon. To achieve the goal of recycling space rocket, SpaceX engineers calculate orbit by many factors, many times the problem is at the moment of landing, people could see the rocket is already on the landing pad on the ocean, but the rocket just not stable and fall down then blow up.
So, if that is possible to set a pad with electromagnet to stick the rocket while landing?
Deemo Chen, grade 11 student. Barstow School – Ningbo Campus.
(‘Oumuamua), a recently discovered asteroid in a hyperbolic orbit, is likely
the first macroscopic object of extrasolar origin identified in the solar
system. This paper mainly gives reasons why ‘Oumuamua should be a comet, by
explaining three unusual behaviors of ‘Oumuamua which are unnormal ratio,
strange track eccentricity, and Peculiar Acceleration;
disproving four main hypothesizes to say ʻOumuamua is not a
comet, then provide an evidence of reflectivity of surface of ʻOumuamua to
prove ʻOumuamua in a regular interval of comet group. Thus, ʻOumuamua is most
likely to be a comet.
ʻOumuamua (1I/2017 U1) is an object from outside of the solar system by first observed byPanoramic Survey Telescope and Rapid Response System(Pan-STARRS) located at Haleakala Observatory, Hawaii, USA (“IAU MPC”), from the IAU website, we can find that ʻOumuamua is also the first object human ever observed from outside the solar system. We are trying to figure out where exactly it came from. The thing is, there are several things shows it is not an ordinary object and can possibility to be a spacecraft may be made by aliens. However, I do not agree with this statement; I tend to describe it is an unusual comet by against three main theories exist which say ʻOumuamua is not a comet.
The ratio and shape of ʻOumuamua
Since we cannot use an optical telescope to truly see how ʻOumuamua looks like, the only way we can use it make the scientific hypothesis of the shape of it. For a rigid body, it will rotate around the maximum inertia axis(MIA), which is the axis with the most considerable moment of inertia. Dynamically, it can be proved that the rotation of the rigid body around other inertia spindles is unstable if we consider the slight deformation damping of the object.
Figure 1Explorer_1 From NASA Accessed 2019/1/13 https://heasarc.gsfc.nasa.gov/Images/explorer/explorer1.gif
Just like the graph of Explorer_1, which is the
first artificial satellite launched by America, it has a long and thin shape.
The blue line points the long spindle of Explorer_1, and that is the minimum
inertia spindle of this structure, the red line shows the maximum inertia
spindle. People at that time hope the satellite could rotate around the long axis, but the result of that be the Explorer_1
cannot stabilize its posture at all, even
they did successfully launch the satellite, but Explorer_1 cannot do many planned tasks.
In this way, we can assume that ʻOumuamua is rotating around its MIS. And the diagram below helps us to figure what could be the possible shape of ʻOumuamua.
Light curve of ʻOumuamua.All of the magnitudes have been scaled using the measured colors; the error bars indicate 1σ photometric errors. (Meet, et al. 2)
According to the Light curve of ʻOumuamua, the red dotted line corresponds 10:1:1 triaxial ellipsoid with a 20% variation in albedo across the surface (Meech et al. 2), and the dotted curve made by observation data from different telescopes of ʻOumuamua shows a nearly perfect fitness to the light curve model of a 10:1:1 triaxial ellipsoid. By analyzing the interval between two relative curve minima points, we could approximate the rotating period as 7.34 h. Also, by the hydrostatic balance theory, the size of ʻOumuamua cannot construct itself to be a sphere.
For a rigid body, it is tough to rotate in this way, it will quickly be rapture, and it from outer space and we even cannot figure out it from which galaxy (Ye, et al. 3), so it must have traveled in space for a long time, and that means ʻOumuamua should keep rotating like this for a long time, the possibility of that should be very small, because if P(x)= . P(d) should be minimal (d refers to the distance ʻOumuamua travel, x= ʻOumuamua keep this long and thin shape), and because this shape is too unusual for human, so many people believe ʻOumuamua is an artificial object or spacecraft.
However, the evidence of this hypothesis has some vulnerabilities.
First, the model of the shape of ʻOumuamua have some problem, since we do not have the graphic evidence of the shape, if only according to the light curve, there are much more shapes could fit this model, or there is a light absorbing material covered on, and the following statement about “rigid body” cannot works because other shapes could prevent them from rapture.
Second, we do need to assume that ʻOumuamua is a rigid body, because for a rigid body, they will rupture because of their inside force, so if ʻOumuamua is not a rigid body but made by relatively elasticity materials, in this way, ʻOumuamua could keep a long and thin shape but not broke.
Third, I think even ʻOumuamua
truly looks long and thin, that is not a strange thing to human, because if one
thing could exist in this world by way we know about physics, then it is
is the sole criterion for testing knowledge”, we should believe in data,
whatever how impossible it looks like.
In this way, I think the shape of ʻOumuamua has nothing to doubt about.
The Orbit of ʻOumuamua
Figure 2 Meech, et al. 3
The path of ‘Oumuamua (1I/2017 U1)
through the Solar System.
“The orbit of a
typical Halley-type comet is shown for comparisonas the solid grey
line. The inset shows the inner Solar System; the solid linesegment
along the trajectory taken by ‘Oumuamua
(dashed)indicates the short window of two weeks during which it was
bright” (Meech, et al. 1), As we can see that its track
eccentricity is only 1.1956 ± 0.0006, it looks got into the solar system
vertically to the zodiac and soon go back the same way, it just like aim to the sun. So, it could easily be seems as a
spacecraft want to use the gravity of the sun to redirect its orbit because it
is the best way we can think to save the energy for a shuttle.
Marco Micheli’s team analyzed the full
observational dataset, which includes 177 ground-based and 30 HST based
astrometric positions (for a total of 414 scalar measurements), applying the
procedures and assumptions discussed in Methods. Their analysis shows that the
observed orbital arc cannot be fitted in its entirety by a trajectory governed
solely by gravitational forces due to the Sun, the eight planets, the Moon,
Pluto, the 16 most massive bodies in the asteroid main belt and relativistic
effects14. As shown in diagram D1-a, the residuals in right ascension and
declination of the best-fitting gravity-only trajectory are incompatible with
the formal uncertainties: ten data points deviate by more than 5σ in at least one coordinate,
and 25 are discrepant by more than 3σ. Furthermore, the offsets (as large as 22″ for the 2017 October 14
Catalina observation) are not distributed randomly but show clear trends along
the trajectory. (Micheli 1)
The most reasonable reason for the offset of the
orbit is the force of comet dust which will give a power to the object when it
spread out of the object, however, by the data from Micheli’s
team (Micheli 3), they built a model of comet could have this kind of behavior,
they found even 2kg of the dust from the object is enough for telescope to observe, the thing is, from the actual
diagram, we cannot see any dust from ʻOumuamua.
Is that means ʻOumuamua is not a comet but could change its orbit
by itself and become a spacecraft? It could be a reasonable guess, but I think there
is a better explanation that the object has released only a small quantity of
dust, probably in the form of larger grains which are hard to see.
Also, by the theory of Fitzsimmons’s team,
we have another way to explain that there is a cover on ʻOumuamua’s surface which prevents the
ice inside of ʻOumuamua spread out from itself, and the cover is so thick that
even the sun cannot put effects on ʻOumuamua in that short time, that they
reported the spectroscopic characterization of ‘Oumuamua, and finding it to be variable with time but
similar to organically rich surfaces found in the outer Solar System. The
observable ISO population is expected to be dominated by comet-like bodies in
agreement with our spectra, yet the reported inactivity implies a lack of
surface ice. (Fitzsimmons, Alan, et al 4). This is consistent with predictions
of an insulating mantle produced by long-term cosmic ray exposure. An internal
icy composition cannot, therefore, be ruled out by the lack of activity, even
though ‘Oumuamua passed within 0.25 AU of the Sun.
So ʻOumuamua is more likely to be a comet in this
The theory to be a LightSailBecause of the usual orbit of ‘Oumuamua, it could consider an artificial origin; one possibility is that Oumuamua is a LightSail, floating in interstellar space as debris
advanced technological equipment (Bialy 1), As a solar sail, its propulsion is
dependent on solar
radiation alone. Solar photons exert radiation pressure on the
sail, producing a small degree of acceleration.
Thus, the solar sail will be propelled by force from sunlight itself, and not
by the charged particles of the solar wind.
Even Mr. Bialy mentioned that ‘Oumuamua could be alien craft since its
orbit is so unusual: “Alternatively, a more exotic scenario is that ‘Oumuamua may
be a fully operational probe sent intentionally to Earth vicinity by an alien
civilization.” (Bialy 4)
It is just an educated guess of the orbit
of ‘Oumuamua, but I think this hypothesis did not consider more, because we
already prove that ‘Oumuamua is rotating, but if LightSail works, it must let
one side of itself toward the sun, so the theory of a LightSail could not work.
of the surface
Since all hypothesis of saying is not a comet be proved as wrong, here
I can give relevant evidence that ‘Oumuamua is a comet.
The reflectivity of the surface of ‘Oumuamua.The
surface reflectivity of ‘Oumuamua is consistent with D-type asteroids and comets. Data are
normalized to 1 at 0.65 μ m, and the error bars reflect the 1σ standard
deviation. (Meech 3)
From the data given by Meech’s team,‘Oumuamua’s reflectivity
is pretty close to D-type asteroids and comets, compare to Comet 67P, Comet 9P,
and Comet 103P, it does not have much differentness.
One possible way to check what‘Oumuamua truly is
The challenge of
reaching the object within a reasonable timeframe is formidable due to its high
heliocentric hyperbolic excess velocity of about 26 km/s (Hein
1); Much faster than any vehicle yet launched,
Launching a spacecraft to ’Oumuamua in a reasonable timeframe of 5-10 years
requires a hyperbolic solar system excess velocity between 33 to 76 km/s for
mission durations between 30 to 5 years.
(Hein 4) It is demonstrated that based on existing technologies now
human have such as from the Parker Solar Probe, launchers such as the Falcon
Heavy and Space Launch System could send spacecraft with masses ranging from
dozens to hundreds of kilograms to 1I/’Oumuamua, if launched in 2021. A further
increase in spacecraft mass can be achieved with an additional Saturn flyby
post solar Oberth maneuver. The potential of more advanced technologies such as
laser electric propulsion, solar and laser sails would also allow for chasing
’Oumuamua, although their development will likely push launch dates farther
into the future and might be more attractive for reaching future ‘Oumuamua-like
objects. And we can chase it in 2029.
Even the shape of ’Oumuamua is strange. Its orbit changes but we cannot observe any dust, and the model is suitable to be a LightSail, but all we can find a reasonable explanation. So `Oumuamua is a comet –a tiny one, with perhaps different chemistry and an unusual shape, but colored like Solar System comets, and outgassing like our comets or covered with an organic layer. So, we were indeed lucky to find it, but we did not need to be too fortunate to find an interstellar asteroid. And we still have the chance to take a clear picture of this comet to see how exactly it looks like if we want to.
Bialy, Shmuel, and
Abraham Loeb. “Could Solar Radiation Pressure Explain ‘ʻOumuamua’s
Peculiar Acceleration?” The Astrophysical Journal Letters 868.1
Accessed on 19 December 2018.
et al. “Spectroscopy and thermal modelling of the first interstellar
object 1I/2017 U1 ‘ʻOumuamua.” Nature Astronomy 2.2
Accessed on 19 December 2018.
Hein, Andreas M.,
et al. “Project Lyra: Sending a Spacecraft to 1I/’ʻOumuamua (former A/2017
U1), the Interstellar Asteroid.” arXiv preprint arXiv:1711.03155 (2017).
Accessed on 19 December 2018.
Meech, Karen J., et
al. “A brief visit from a red and extremely elongated interstellar
asteroid.” Nature 552.7685 (2017): 378.
Accessed on 19 December 2018.
Micheli, Marco, et
al. “Non-gravitational acceleration in the trajectory of 1I/2017 U1 (‘ʻOumuamua).” Nature 559.7713