Images of Jupiter and Saturn and some related info

After a close look  at planet  Mars in a previous post , here are images and info related to Jupiter and Saturn.

Jupiter is the biggest planet in the solar system , fifth from the Sun between Mars and Saturn.It
was named after the Roman king of the gods.

This first image of Jupiter shows the planet with the (multicolored) ring system surrounding it.The Grest Red Spot is also visible (image made with Universe Sandbox).

Jupiter rings and Great Red Spot

The faint diffuse ring system  of Jupiter was  identified by the Voyager 1 spacecraft in 1979.The
ring system consists mostly of dust particles , and comprises three main parts: the halo closest to
Jupiter, the main ring ,and the gossamer ring outside the main ring.

The Great Red Spot (GRS) of Jupiter is a huge high pressure anti-cyclonic storm , similar to a big
hurricane.Three planets each having the same size of the Earth could comfortably fit in it.
The image below gives a detailed view of the GRS and its surroundings (made with Starry Night).

Great Red Spot

Next is a picture of Saturn with its ( multicolored ) system of rings .Planet Earth is shown to the right of Saturn. Image made with Universe Sandbox.

Saturn rings and Earth

Saturn is the second largest planet in the solar system after Jupiter.
Saturn has bright rings made of lots of small particles having sizes from a centimeter to a few
meters to more than a kilometer.The particles of the rings are composed mainly of aggregated
water ice pieces and some rocks.The rings extend away from Saturn and have a very large
diameter ,but they are very thin and have a thickness of less than one kilometer.Galileo first
observed the rings in 1610 ,but he thought they were large moons on both sides of planet Saturn.
A few decades later in 1655 Christian Huygens explained that Saturn was in fact surrounded by
rings.These rings were divided into many sections by astronomers and scientists.The D ring is
closest to Saturn ,the F, G ,E rings and the Phoebe ring are the outermost rings .

The final image below is of Saturn and its rings in March 2015 (made with Starry Night and a touch of Photoshop;the lighting is an added effect).

Saturn and rings

The rings of Saturn lie within the Roche limit.Inside this borderline distance (approximately 2.44
Saturn radii ) a celestial body or moon disintegrates due to the stronger tidal forces of Saturn and
rings are formed  , while outside it a body or disk of orbiting material is expected to accrete and
coalesce.

A general formula for calculating the Roche limit is:

The Roche limit

The Roche limit varies for rigid bodies and for fluid satellites.

Additional reference work related to this post and the Roche limit :
Planets, Stars and Stellar Systems , Volume 3:Solar and Stellar Planetary Systems ( edited by Terry D. Oswalt ,
Linda M. French and Paul Kalas).

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Images of planet Mars and related input

It was by studying the orbit of planet Mars and the obesrvational data collected by Tycho Brahe that
Johannes Kepler was able to formulate his laws of planetary motion in the early 1600s ,
concluding that the orbits of the planets are ellipses, with the Sun at one focus of the ellipse.
The Martian atmosphere is very thin , composed mainly of carbon dioxide (between 95% and
96%) ,and  contains nitrogen , oxygen ,argon and traces of water vapor.
Mars appears to have a red or red-orange color.Its surface , rocks and soil contain dust composed
mainly of iron which reacted with oxygen , giving iron oxide or rust.This red colored dust has been
carried by storms into the atmosphere and has covered most of the Martian surface and  landscape.

The following images of Mars were made with Redshift and Starry Night.

We begin with an image of Mars from a position behind the moon Deimos.The date for the image is February 2015.

Deimos Mars and Phobos

Phobos and Deimos were discovered in 1877 by Asaph Hall.Both have irregular shapes.Phobos (with a diameter of 22.2 km across or 13.8 miles) is larger than Deimos (with a diameter of 12.6 km or 7.8 miles) and is closest to Mars.

Next is an image of Mars showing also Phobos , Deimos and some planets of the solar system.

Mars Phobos Deimoes and planets

The third image features Phobos and Mars with Planum Boreum , the North polar cap of Mars , consisting mostly of water ice.

Mars and PlanumBoreum

Below is a detailed image of Planum Boreum and its surroundings , including Vastitas Borealis , the largest lowland region of Mars.

 details of Planum Boreum

Click here to view an enlarged and more detailed image of Planum Boreum.

Here is a simple way to find the surface gravity g for Mars.The same equation used to determine
the value of g on Earth’s surface can also be used to determine the acceleration of gravity or
surface gravity on the surface of other planets.
We equate the force of gravity at the surface of a planet,or the force on an object in a gravitational
field F=mg (also called weight) with the force of gravity between objects in space given by the
Universal Law of gravitation F =\frac {G M m} {r^2} .
Hence:

g-mars-calculation
The value of g obtained here may be very slightly different from values cited in textbooks because
we used specific  and more detailed or accurate values of the constants.
The ratio of the surface gravity of Mars compared to Earth is 3.72761/9.80665 or 0.38011,which
means the surface gravity of Mars is about 38% that of Earth.

Mars is close to Earth and the fourth planet from the Sun , and it has become famous in the past
decades or years  because people and humans have been planning and wanting to go and set foot on the
red planet , some people intending to go there in an unprepared  and unrealistic way.
I think it must be taken into account that a manned mission to Mars and the first mission/trip ( or even trips) to
Mars should be undertaken as part of an international enterprise with international cooperation and an international crew , consisting of people very well prepared ,well trained ,well versed in science ,engineering , technology , astronautics and aviation (preferably having pilot skills) , and having planned everything to the tiniest detail in order
for the crew to go land on Mars , stay there for a short determined period of time , conduct
experiments , establish a base for future trips , and come back to Earth safely.
Not to offend anyone ,but this is not a game or a one-way voyage with uncertain or harmful results and consequences.This will be a very important event in the history of humankind , and not everyone is ready,
prepared or able to make the journey to Mars .

Just an image about symmetry

I was thinking  about making some sort of image related to symmetry , either for general usage or  for my Twitter account , so I experimented a little with Photoshop to draw something about symmetry , which is an essential and important property in mathematics , physics , science and knowledge . Symmetry is also present in nature , art , architecture , etc

I came up first with this image :

symmetry image one

 

Then I noticed it wasn’t “symmetrical” enough , and I changed it till I got this  :

symmetry image two

I found this last image more satisfactory.

Wherever there is symmetry  , there is mathematics , science , precision , and all the other nice things mentioned/written/carved in the image.

Then finally I added a (symmetric) curve and “wrapped” it around the words .

Both y = \pm\frac {1} {x} and y = \pm\frac {1} {x^2} are symmetric with respect to the origin of the coordinate axes and either one of them can be used . The final image is the following one :

symmetry curve

A few calculus results related to the sinc function

I’ve tried to sink my teeth into the sinc function and obtained the following calculus solutions , mostly by tinkering with Mathematica.

The sinc function is generally defined by:

\text {sinc} (x) = \frac {\sin (x)} {x}

with sinc(0) = 1.
The sinc function is sometimes called the filtering or interpolation function and is often used in digital signal processing and in engineering . Sometimes a distinction is made between the unnormalized sinc function and the normalized sinc function sin(πx)/(πx) , but I’m going to consider mostly the unnormalized function.

Graph of the sinc function (done with Mathematica):

sinc function graph

Here is a table of the jth derivative of sinc(x) for j between 1 and 6:

sinc-deriv-1-6

And for j between 7 and 10:

sinc-deriv-7-10

Another table of derivatives for sinc of x with x to the nth power:

calculus-sinc-1

 

Also :

sinc integral set 4

Si(x) is the sine integral function .

Here is an indefinite integral of sinc(f(x)) with

 f(x)=x^n

sinc-properties-int-1

 

For the definite integral we get:

sinc definite integral

Another indefinite integral :

sinc integral 2

Ei(x) is the exponential integral function .

And below is a table of values for a definite integral of sinc(x) to the jth power:

sinc integral 3