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Archive for November, 2007

Celestial Mechanics

 

Have you ever wondered how Copernicus, and Galileo and Isaac Newton and that Kepler guy figured out that the earth revolves around the sun, and not the other way around? When I go out at night and look up at the heavens, I have no idea what I’m looking at. Oh, I can find the North Star and the dippers as well as the next guy, and the moon is easy, but if you asked me to point out where any of the planets are, I’d be literally lost in space. It’s a shame too, because we’re supposed to be so much smarter than the ancients. But if I had to swear to it, I’d have to admit I can’t prove Copernicus was right, or disprove Ptolemy.

Well, not any more, because the other day I found a wonderful book on the internet that teaches all about celestial mechanics. You can download it for free in PDF format at Celestial Mechanics Link. It will take a minute or so to download depending on your connection speed, because it is about 45 MB in size, but it is well worth the read.

“An Introduction to Celestial Mechanics” by Forest Ray Moulton is not an easy book, and not a dumbed-down treatment of the subject. Written in 1902, it was intended for Senior college students and graduate students. But we’ve come a long way since then, haven’t we? I’m not completely through the book yet, so I haven’t gotten into the more advanced topics, such as perturbations but I’d say that the book requires an understanding of mathematics, especially the concepts of Calculus and Differential Equations, and a basic course in Physics for sure. For those who didn’t take those subjects in school, please read on because I think it’s important, even if you don’t read the book, to understand the process of how our present day understanding came about, and how we do the math today to track the planets, the moon, the asteroids and comets, and even other star systems.

We know from experiments that two bodies of mass attract each other with a force that is proportional to the product of their masses, and inversely proportional to the square of the distance between their centers. That means simply, the larger the masses, the stronger the force; and the farther apart the masses are, the weaker it gets. That’s only common sense. We word it in the more quantitative way, so we can take accurate measurements though. From these basic considerations, scientists formulate a number of equations of motion – 3 equations for each of the two masses; one is for the x direction, one for the y direction, and one for z – 6 equations. These equations are written in the form of 2nd order differential equations, because the force in each direction is found to be the second derivative of the momentum in that direction – and differentiation, in Calculus means finding the derivative. The derivative is simply the limiting value of a very small change in one quantity divided by a very small change in a second quantity. In this case we want to know how much the motion changes in each direction x, y, or z in a very small time, which we call the velocity, and we would denote each by the expression dx/dt, dy/dt, and dz/dt. If we take it one step further, we get the rate of change of the rate of change in each direction (acceleration) – our second order derivatives and multiplied by the mass, they are equal to the force. To solve these equations we need to go backwards and find x, y, and z, and to do this we integrate each equation twice. Integration is the inverse of operation of differentiation, which you also learn in Calculus.

We can simplify the problem immensely, by working with the coordinates of the center of mass of the two bodies, and in the process we find that the motions of one of the two bodies with respect to the other will always be in a plane passing through the center of the other.

Now that the we know the motion is planar, we can switch to a simpler coordinate system based in that plane. Instead of three coordinates, it now only takes two to locate the two bodies relative to one another, an angle and a distance. Solving these equations we end up with an equation for the distance, which by inspection we find to be that for a conic section, i.e. a parabola, hyperbola or ellipse with the origin at one focus. By comparing this equation to the standard mathematical equation for a conic section, we derive 6 constants or elements that determine the orbit. For example, one of these elements designated by the small letter “e” determines that the orbit is elliptical when e-squared is less than 1.

There’s somewhat more to it than that, more constants that we derive when we solve what is known as Kepler’s equation. Kepler’s equation allows us to find the exact position on the ellipse by measuring the time elapsed and the difference between the eccentric anomaly, an angle which can be solved for, and the angular distance it would have travelled at a uniform rate. These values designated E and M can be calculated as precisely as needed, by representing them in the form of an infinite series of terms, and just using the terms we need and discarding the rest.

That’s about the gist of it, except we do need to convert these values to our earth system of coordinates. When we have done that, one can then view the body in the sky by its right ascension or direction angle and its declination or angle of elevation. If the object is where we predict, we can assume our equations and our basic understanding of planetary motion is correct.

So if your kids ever ask you, how do you know the planets orbit around the sun, just tell them, “it’s really complicated and that’s why you need to study math,” and point them to this book.

Most of us don’t know why we study math, and what it is good for. I wish somebody had explained this to me, because I never could see how we had figured it out. Even after 3 years of college Physics, I knew about Newton’s laws and Kepler’s laws, but wondered why NASA used orbital elements to track the satellites and space shuttles.

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Fun With Photoshop

 

I ordered two of my favorite movies the other day from Walmart’s online website. I could have rented them from Netflix, I suppose, but I wanted to have my own personal copies in case there is ever a run on them. I waited such a long time for “Sweet Bird of Youth” to be released. I don’t think I could tell you why it has always been one of my favorites, not until I see the movie again and think about it some more,  since I’ve only really seen the movie once, and another time as a boy I fell asleep right after the opening credits. Something about it just struck me as memorable. Nor do I know why it took so long to be released; I’ve had all my other favorite Paul Newman’s for many years now.

The other movie I ordered was “Two for the Road,” starring Audrey Hepburn and Albert Finney, a fine British actor. I’ve seen that one many times, but never seem to get enough of it. I love movies about relationships, and movies about travelling, and relationships that travel, and this one is all over the place. It’s not very believable, I suppose – what man in his right mind could ever be unfaithful to Audrey Hepburn? She was Holly Golightly, for heaven’s sake.

“Breakfast at Tiffany’s” is probably my very favorite movie of all-time. I’ve said “that” all over the place. And if ever I’ve had a dream-girl, she would be Holly. I think there’s quite a few girls like her in New York City right at this very moment. They don’t know it, of course, they’re too wrapped up in their careers. I saw one just the other day in fact.

 I think the problem is there’s too few “Fred, baby-s.”

Oh yeah, the Photoshop thingy. Well I found a black and white photo on the internet, that was a much better photo than the color screen captures that were out there, and thought I might try adding some soft colors to it. I didn’t think Audrey would mind. The result is the first two pictures. The second image was colorized only, and turned out better than I had anticipated, so much so that for the top one I tried to get a little artsy with an artistic filter effect and later added a paper texture in Fractal Painter. Even tagged on it.

Here’s the original black and white photo that I used:

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A Fine Line

First, a man at a Florida University forum wouldn’t stop speaking and leave the podium.

Then another man driving in a car with his pregnant wife and baby wouldn’t sign a ticket in Utah after he was pulled over by a police officer for doing 50 in a 40 MPH zone, seemingly way out in the middle of nowhere, I might add.

As this clip from Monty Python and the Holy Grail demonstrates, it’s a fine line that separates comedy from tragedy.

I wonder how many people laughed when they saw the first two videos for the first time. I know I laughed at the first one when I saw it on the evening news.

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Oh yeah, Grease is the word!!

** Or High School Musical. Whatever. The VW probably wasn’t green, but I painted it green to look like my first car.

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My Poetry

Every once in a while I remember a fragment or two from the poems that I wrote in college for a Literature class. I never liked or understood poetry before that, but since then I’ve learned that it’s a language unto itself, a very powerful language that appeals directly to all seven of the senses, seven if you include the heart and the intellect along with the usual suspects.

I’ve tinked with this poem for a long time. For some reason it just keeps popping up and biting me in the ass, metaphorically speaking.

I don’t mind the wait for love
Nor do dismal days that hum, and flee
Like pin-prick planes, bother me.

I don’t mind my company, the moon above
And night’s playful populace
Who mock my every sleepless dream.

In darkness on a crumpled map
I seek to mark my way to love
Tap, again tap, side to boundless side.

I don’t mind waiting for love,
Yet, lest love become its own pawn
Hasten love, and soothe my bright,
My dreary day.

In case you didn’t pick up on the imagery right away, the crumpled map is my computer keyboard. But I was also using it as a larger symbol for the brain, you know, how it’s all crumpled up looking and the tap, tap is that of a blindman tapping his way along and his map is all crumpled up anyway. And its reaaaally dark outside.

Poetry can get you in a lot of trouble too, and it can be very embarrassing. Women claim to love poetry, but I think they’d rather get handed a big bug, from someone they like than the most heart-felt poem from someone that they don’t care about, or don’t consider much. I sent this poem to a girl once, and that’s how I know:

I found this flower in the trash
And thought to myself,
To myself I thought
I’ll send it to So-and-so with some chocolate.

Sweets for the sweet,
Hot damn! Oh, fudge
Sweet would just spoil a complex so fair
And So-and-so is fairly complex

So I’ll just send the rose and a kiss
And God blex.

I wrote another poem for a girl once titled “So-and-so, whom the gods would destroy.” Apparently I was in my pantheonic period then, because it was all about the Greek gods. I only remember a small fragment from that poem:

Or cool woman, a gown of sighs
Pulled smooth across her frame
From burning eyes
She watches her sister’s robin’s breast arise
And fall, on tender sticks of symmetry
Her jealousy, sweet Sappho could but surmise.

Right after she read it, So-and-so introduced me to her boyfriend, who worked as the bouncer in the bar. He was real nice, but I’ll never write a poem for a cocktail waitress ever again. Or a pole-dancer.

Nowadays you don’t even have to write a full poem, just wax a little poetic and women will cut you right off at the knees. The other day I sent an email to a woman, telling her that her eyes could melt the polar caps, and she right after that wrote in her blog that her boyfriend reads her email.

But some women have poetic souls, I suppose. Emily Dickinson wrote some great poems, and this very brief one is perhaps my favorite:

To make a prairie it takes a clover and one bee,
One clover, and a bee,
And revery.
The revery alone will do,
If bees are few.

I think Emily had it right; it doesn’t take a whole lot to make a person happy or fulfilled, or in love for that matter. In return, I wrote a poem for Emily:

Jogging days,
And Disco nights.
I like Emily’s advice –
Little girls must have pudding.

I also wrote a poem about a bee, called Bee-Lines. It’s not as good as Emily’s bee poem.

Came I but by the money
To fetch for you a pleasure
I’d entrust it to no bumbling bee
To post it at his leisure.

He’d breach a spider’s web
For wrap, at high velocity
Then slow, while slender, silver butterflies
Slide scissors down the rainbow.

Past sprinkling with nectar
Fresh from the vernal vines
He’d whisk it off to you dear
And right his tie, and learn his lines.

And of course there was Elizabeth Barrett Browning. She’s always seemed a little high-brow to me, but I see where she’s coming from:

How Do I Love Thee?

How do I love thee? Let me count the ways.
I love thee to the depth and breadth and height
My soul can reach, when feeling out of sight
For the ends of Being and ideal Grace.
I love thee to the level of every day’s
Most quiet need, by sun and candlelight.
I love thee freely, as men strive for Right;
I love thee purely, as they turn from Praise.
I love with a passion put to use
In my old griefs, and with my childhood’s faith.
I love thee with a love I seemed to lose
With my lost saints, I love thee with the breath,
Smiles, tears, of all my life! and, if God choose,
I shall but love thee better after death.

You’d be dead-meat, if you ever sent that poem to a girl.

My favorite poems were the spiritual ones. I’ve always felt that God was poetical too.

I, convinced that I could,
Strove on tip-toes and a stack of books
To touch the place of God.
Til, on my knees
I learned to count infinities
On my two, little hands.

I’ve always thought of God as the nicest neighbor you could imagine. His yard is verdant and bursting with colorful flora, the lawn is always neatly mowed, His bushes are manicured, and He knows just the right flowers to plant for each season. And I always wonder, how splendid must be the inside of his home too.

Speaking of that, I wrote about the universe a lot too, like in this fragment:

Like the light that long after sun has set
Lingers to congeal into a billion shining parts.
I give of these to thee a shining star…

Where the cool blue diamonds of desert night
Glide along the taut fabric and go straight for my soul…

Sweet So-and-so…these sacred sounds bite my soul awake.
Like a stick-match struck on the cheek of god.

I wanted the poem to conjure up the image of a quasar on the very edge of the universe. It used to be one of my favorites. I wish I could remember more of it.

One time when I was baby-sitting my little nephew, I watched him walk off to bed with his two teddy bears, and thought of this:

Bears in Arms, Bears in Arms,
Pottied, and in pod-ed P.J.s
But light the light, just in case.

That was supposed to sound like a call to arms, you know, to Arms, to Arms… Oh well, I guess I’m just not cut out to be a poet. 🙂

I don’t know why they even teach poetry in schools. It doesn’t get you anywhere. 

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I’ve been trying to broaden my horizons a bit lately, by reading some of the other Blog material on the internet. There’s a plethora of interesting articles and photo and video Post-its. That’s really the only way to describe them. Most of it is about as interesting and memorable as a trainwreck, and it can become a little overwhelming at times – if Virginia Woolf were still around, she’d probably be sticking a needle in her head, or snorting a line of something by now and Sylvia Plath would have her head inside a bell-jar again for sure.

I gather from reading a few of the blogs, and from viewing the following real video that one of the latest trends in city chic is white girls flashing all manner of gang-signs.

In real life the gang-signs mean something to the members, by signifying friend or foe, and often result in a drive-by shooting or at the very least a high-speed chase down the 605 freeway. I don’t know how or when this new phenomenon first caught on in the East, but it’s already gotten into the mainstream, and even been attempted in some of the more fashionable places in New York.

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A Tale of Two Kitties

It was the best of times, it was the worst of times. I was feeling anxious most of the times, so I decided one day to dig up the back yard again, and plant a vegetable garden. It was about that time that I adopted a stray cat. A very beautiful grey and brownish-black Persian cat had wandered into our backyard and had a litter of kittens of assorted colors, all beautiful in their own way, but none as unique as she. There was a snow white one, and two tan tabby-colored kittens, and several others. I let her and her brood stay for as long as she wanted, and each day I put food and milk out for her. She would lie there and nurse the kittens most of the day, and kind of stick her nose up real snooty-like at my proffer, but it was gone the next day, just the same.

I don’t know what happened to all the kittens. Some nights I’d hear a commotion out in the backyard, as only cats can make, and each day another kitten would be gone. Finally it was just her and one little tabby colored cat. I would watch her teaching him how to hunt birds and climb trees, it was quite an experience really watching the rigorous training that she put him through. One day she left too, but he stayed, and I kept putting out the food, and that’s when he began making up to me.

I didn’t want to domesticate “little pussycat” too much, he seemed to have quite a nice little life going. When I worked out in the yard, he would always come around and “show off” for me. He had great claws and he loved to climb our trees. And if ever there was any sign of danger, right up the tree he would go.

One day when I was digging out a section of the yard for my garden, he was playing with the shadows as the pick rose and fell, and finally when I came down with the heavy pick one time, he was right there and I struck him with it dead center. Well, the cat went flying out of the yard, and I finally found him about a half hour later hiding behind the lantana camara bush in the front yard. He was frightened of course, but not nearly as much as I was. I thought I had cut the little guy in two, but everything seemed to be still attached. I sat with him and comforted him, and after that he just became real friendly; he’d come to the screen-door every day about the same time, and I would let him in, and he’d jump up on me and sit on my lap while I was watching television. In the evening, when it was time for bed, he’d begin nosing around the front door, and I knew to let him out.

But evil was lurking only a yard away. I began seeing another cat snooping around that looked just like him. It was a bigger, older cat, but they really could have been twins. One day I saw the little girl next door holding him, so I assumed the “daddy” cat belonged to her. She was a cute little Hispanic girl, who loved animals. She had a big rabbit for a while too, and often she would see me outside and hit me up for carrots for her rabbit. I’d have to get them out of the refrigerator though, because her rabbit had eaten all the tops off my carrots out in the garden one morning not long after they had appeared. Before long this old cat was always up on my front porch, and I could tell he was after my little cat. They had already had several run-ins, so I started shooing him off when I saw him in the yard, but the little guy could really hold his own against him. His mom had taught him well, he was so agile and good with his claws.

One day after a real donny-brook the big cat left the territory, and I began seeing the little girl next door playing with my cat. And right after that she went away and took the cat with her. I don’t think it was a case of mistaken identity, but I felt it was a far better thing for me not to protest about it. Oddly enough, that wasn’t the last time I saw him. One day I was out in the front yard mowing the lawn, and it was very hot and for like a full 5 seconds I saw the little brown cat standing by the porch. He wasn’t moving, but just stood there transfixed, as I must have been. I was afraid to look away because I knew he wouldn’t be there when I looked again.

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