This is a very special word in mathematics, and it has an entangled state in a macro sense.
There may be nothing behind this word, or there may be a lot of content filling the page.
At the same time, even if the page is filled with content, the final result is likely to be the same as nothing.
In addition, it has nothing to do with the appearance or stationery of the solver.
Of course.
As the initiator of this observation, Xu Yun naturally would not be the former.
So after writing down a solution, he continued to draw the initial calculations.
As for the initial entry point for calculation...
Of course it is the Titius-Bode rule.
As we all know.
As an important branch of the history of civilization, the history of human science can be said to be as bright as a galaxy.
These talented people are basically geniuses, but there are also rising stars who have become superstars with their incredible and shocking conjectures.
For example, Faraday, or Erdar Arikan, who only wrote the 5G standard channel coding at the age of 51.
Another example is a German high school teacher named Johann Titius.
John Tydeus lived in the 18th century. At that time, people knew that there were six planets in the solar system.
That is Mercury, Venus, Earth, Mars, Jupiter, and Saturn.
Tidius was an astronomy enthusiast. After long-term observations, he wrote such a sequence in 1766:
a=0.4 0.3X2^k.
The a in it refers to the average distance from the planet to the sun, which is 150 million kilometers.
Among them, k=0,1,2,4,8,16..., the numbers after 0 are 2 raised to the nth power.
If the distance between the sun and the earth...that is, 150 million kilometers is taken as an astronomical unit, then the ratios of the distances from the six planets to the sun are:
0.4,0.7,1.0,1.6,5.2,10.0.
The actual value is:
0.39,0.71,1.0,1.52,5.2,9.8.
Are you surprised?
That's right.
In the reference system of the starry sky, the two results can be said to be infinitely close to the same.
In 1781, Herschel discovered Uranus at a position close to 19.6 (the eighth item in the sequence).
Since then, people have firmly believed in this rule.
According to this certain rule.
The fifth item in the sequence...that is, the position of 2.8 should also correspond to a planet or asteroid, but it had not been discovered at that time.
So many astronomers and astronomy enthusiasts embarked on a journey to find this new planet with great enthusiasm.
This asteroid is Ceres, and its discoverer was none other than Gauss at the scene.
Later, this rule was summarized by Bode, director of the Berlin Observatory, and summarized into an empirical formula called the Titius-Bode rule.
Having said this, it’s time to whip up Zhidu Encyclopedia again.
If you search for Titius-Bode rule on Baidu, you will see a sentence in the detailed introduction:
[Because Neptune discovered in 1846 and Pluto discovered in 1930 deviated greatly from this formula, many people still hold a negative attitude."]
Among them, the estimated data of Neptune given by the encyclopedia is 38.8 astronomical units, and the actual distance is 30.2 astronomical units.
The estimated data of Pluto is 77.2 astronomical units, and the actual distance is 39.6 astronomical units.
Yes, seeing this, students majoring in astronomy should have discovered a problem:
A certain editor calculated Pluto's data to be 77.2 - this is the distance to the inner boundary of the solar system...
Actually.
During the calculation process, due to the existence of k degree polynomials, Pluto and Neptune are calculated using n=8.
Therefore, calculated according to the Titius-Bode rule, the error rate of Pluto is 2%, not 200%.
This is something that will be clearly marked in the textbooks in the second semester of astrophysics and astrometry. As an encyclopedia column, it is quite helpless to make such a mistake...
In Xu Yun's previous life, there happened to be a certain plot where Tidius-Bode's rule was used. When harassing... ahem, when consulting a friend who worked at the Phoenix Mountain Observation Station, the other party once expressed some extreme remarks to the encyclopedia.
Kind regards and blessings.
Of course.
A large part of the reason for this situation can be attributed to the lack of popularity of knowledge. Tidius-Bode's rule itself is a niche knowledge, let alone Pluto, a niche within a niche.
all in all.
Later generations basically have no opinion on the numerical value of Titius-Bode's rule in mathematical calculations.
Its main controversy is that its physical meaning is vague and it is a purely empirical formula that is difficult to explain in principle.
Other measurement methods such as an 1: an = β are basically mathematically accurate but have unclear physical meaning.
Xu Yun then wrote down two more formulas, which are the function of k degree polynomial and the minimum error value:
f(x)≈g(x)=a0 a1x a2x2 a3x3 ... akxk.
loss=i=0∑10(g(i)?f(i))2.
Thus.
As long as the appropriate coefficient is found, the error value can be minimized.
And while Xu Yun was optimizing the function.
Others were not idle either, and were acting according to their predetermined plans.
For example, Lao Tang was taking today's star map with technicians from the Greenwich Observatory, and Gauss was sorting out the unique observation records left by the Bradley family:
If you need to calculate planetary orbit data only through mathematics, then Kepler's three laws of planets must be used:
First law:
Each planet orbits the sun along its own elliptical orbit, with the sun at one focus of the ellipse.
This chapter is not finished yet, please click on the next page to continue reading the exciting content! The second law:
In equal periods of time, the lines connecting the sun and the moving planets sweep out equal areas.
That is Sab=Scd.
The third law is:
The square of the orbital period of each planet around the sun is proportional to the cube of the semi-major axis of their elliptical orbit.
That is, T2/a3=K, T is the planetary period, and K is a constant.
In addition, the elliptic curve in the Cartesian coordinate system needs to be used, namely:
Ax2 Bxy Cy2 Dx Ey F=0.
With these, you only need to add a certain tool to perform calculations.
With the advanced technology of later generations, the tool for calculating orbits is generally numpy, and the results can be calculated in a few seconds.
Although there is no assistance from numpy at the moment, the calculation logic of this thing is actually the least squares method.
The inventor of the least squares method is none other than Gauss...
"The next group is 0.31468531...0.21538462....0.12960373...."
"0.05337995....0.01724942....0.32307692...." (Note: All data comes from NASA's open database, not fabricated)
About ten minutes passed.
Riemann, who was responsible for the final calculation, wiped the sweat from his forehead and wrote a number on the paper:
0.4857342657342658.
Although it is still not possible to know the specific location of Pluto, let alone its weight.
But it has been mentioned before.
After deducting Neptune's gravity, Uranus's orbit is still somewhat abnormal.
This abnormal data is the entry point for calculation, which is the number calculated by Riemann and others.
Gauss took the piece of paper, glanced at it a few times, and shook his head.
The observation records they compiled this time can be traced back to 1012, with nearly 32,000 hand-drawn drawings and about 2,700 black-and-white photos.
Faced with these data, the results calculated by cubic polynomials obviously cannot be accurately fitted.
However, this situation was already expected by Gauss and Xu Yun, and the cubic polynomial was just a low-cost test.
If the accuracy of the results obtained is high enough, a lot of effort can be saved. If the accuracy is low, it will only take a little time.
Gauss's expression did not change at all, he turned to Riemann and said:
"Bornhard, open up the higher power."
Riemann nodded, hesitated for a moment, and asked:
"Teacher, should we still use Huang Jing?"
Gauss thought for a while, waved his hand and said:
"Continue to use Huang Jing, go to the...eighth power!"
When he heard the word eighth power, Riemann's expression suddenly became serious:
"clear!"
My classmates who are rarely seen in this life probably don’t know.
In planetary orbit calculations.
x’ is the true position of the planet and x is the mean position.
The orbital longitude is γN NX', and these two angles are on two different orbits.
Draw a celestial longitude line vertically through the true position of the planet x' and intersect at x" on the ecliptic, then γx" is the celestial longitude L.
Then Gauss looked at Sylvester aside and asked:
"James, have you calculated your time?"
Sylvester swallowed when he heard this, frowned and said:
"The results have been calculated and are in the third round of verification. They will be ready soon!"
Previously, Xu Yun divided the entire team into several modules, and Sylvester was responsible for time correction.
This is also a very critical link - because there is an error in the Julian number of days and thousands of years.
Assume that the given time JDE is a standard Julian day and t is a thousand years.
Then the expression of t is t=(JDE - 2451545.0)/365250.
In today's calculations of this magnitude, even one decimal place can make a huge difference.
five minutes later.
Sylvester suddenly raised his head and said to Gauss:
"The verification is correct, t is 0.00834422!"
Gauss turned his head and said to Riemann:
"Bornhard, did you remember it?"
Riemann quickly filled in the numbers, and even had time to say "hmm".
After the calculation has reached this step, the next thing is very simple, only calculation is left.
The entire formula is L=(L0 L1*t L2*t^2 L3*t^3 L4*t^4... L8*t^8....)/10^8.
L'= L - 1°.397*T - 0.00031*T^2.
Correction value of ΔL=-0.09033 0.03916*( cos(L') sin(L'))*tan(B).
Correction value of ΔB = 0.03916*( cos(L')- sin(L')).
Brush brush brush——
The scene where hundreds of people gathered was silent at this time, and everyone's eyes were focused on the 43 mathematical tool people.
Xu Yun took this opportunity to walk to the other side of the shed.
He first glanced at Xiaomai, who was calculating their respective tasks, and then said to a yellow-skinned young man beside Xiaomai who was assisting in the calculation:
"Brother Haosuo, how do you feel?"
"Oh, it's Brother Luo Feng."
Tian Haosuo was frowning and thinking about how to write. Hearing this, he quickly raised his head and shook his head with a wry smile:
"It's a bit difficult, but I can barely keep up with the train of thought. I have to say that there are people outside the world, and God outside the world..."
Tian Haosuo's expression was somewhat emotional. This was the first time he had been exposed to such a high-level computing activity.
Xu Yun smiled and patted his shoulder, comforting him:
"It's okay, we mainly want to broaden our horizons, and we don't necessarily pursue results."
"I've watched you all the way and your performance is already better than many sophomores."
This chapter is not finished yet, please click on the next page to continue reading the exciting content! Tian Haosuo is one of the members of the computing power that Xu Yun invited to join yesterday. After all, this Oriental is also a student in the Department of Mathematics.
However, Xu Yun did not give him specific tasks. He mainly hoped to improve his vision and thinking pattern.
Anyway, this approach has no cost, and it is not likely to cause any harm. If it is not guaranteed, what surprises can be gained in the future?
Then Xu Yun and Tian Hao separated, came to Lao Tang in the center of the venue, and asked him in a low voice:
"Mr. Thomson, how is the visibility tonight?"
Lao Tang glanced around a few times and said in a low voice:
"God bless, the visibility is very good, almost all of the Hevelius star chart is visible."
Xu Yun breathed a sigh of relief and nodded.
Black-and-white photographs were invented in 1839. Before that, all planetary observation records relied on text or star charts.
For example, the positioning method of the Big Dipper in the Chinese "Historical Records·Tianguan Shu" is also known as the Star Bridge method:
The ladle carries the dragon's horn, weighs the Yin Nan Dou, and the Kui pillow holds the ginseng head.
What does it mean?
It consists of the four stars from the right among the seven stars that form the mouth of the spoon, which is called "Kui".
The three stars with relatively straight lines in the middle form the longer straight handle of the spoon, which is the "weight".
The angle of the connecting line between the two on the left is deflected, forming the part of the handle of the spoon that is held by the hand, which is what Sima Qian called the "ladle".
"Diao carries dragon horns" means that the line connecting two stars (diats) points directly to a very bright star.
The ancients believed that it was the horn of the Eastern Green Dragon in the sky, which was also the Arcturus star in later generations.
"Heng Yin Nan Dou" refers to the connection of the long handle part represented by "Heng", which points directly to the Nan Dou constellation among the twenty-eight constellations.
The last word "Kui pillow and ginseng head" means that the "Kui" representing the mouth of the spoon is facing the zodiac sign among the twenty-eight constellations.
In the Han Dynasty, Gansu and Rigel were added together to form a tiger.
Xiu represents the head of a tiger, so "Shenshou" is "Qisu".
In addition, in Su Shi's "Red Cliff Ode", "the moon rises above the east mountain and wanders among the bullfights", which is also a positioning method in poetry.
Besides the text, the rest is the star map.
The most famous star map in ancient China is the Suzhou stone astronomical map, which was drawn by Huang Shang, the teacher of Ningzong Zhao Kuo who taught him astronomy when he was the prince.
This star map has the North Pole as the center, and the three concentric circles represent the permanent manifest circle, the equatorial circle and the permanent hidden circle respectively.
As the name suggests.
The stars within the ever-present circle never set at all times; while those outside the ever-present circle were invisible to the active range of the ancients.
This star map was later engraved on a stone tablet 2.16 meters high and 1.06 meters wide, which is currently preserved in Changshu.
In addition, there are the Dunhuang star map and the Su Song star map drawn by Lao Su, etc. The star map drawn by Lao Su is the star map with the most celestial bodies among all ancient civilizations.
As for the more famous one in Europe, the Hevelius star chart is extremely vivid in shape and has high artistic value. (If you are interested, you can search it. It is indeed very beautiful.)
These days, the Hevelius chart is also used to determine visibility, which is a default method.
The more objects in the Hevelius chart that are observed, the better the observation environment is.
be honest.
It was indeed not an easy thing to encounter such a nice night near London in 1850.
And while Xu Yun was chatting with Thomson.
Riemann in the shed whispered a few words to the people around him, and then raised his head happily:
"The eighth root has been found, and the parameter of the deviation is 0.001273499338486!"
0.001273499338486.
Compared with the previous 0.4857342657342658, it is hundreds of times more accurate!
After all, one is to the third power and the other is to the eighth power. The difficulty and accuracy are the same.
But then again.
This value is almost the upper limit of human speed calculation.
The result of the 17-a-side speed calculation competition organized by Oxford University in 1937 was about 8% lower than this figure.
This parameter represents the correction coefficient of Uranus, which is the gravitational effect of Pluto on it.
With this coefficient, the next step is very clear.
As mentioned before, there are only two macroscopic feedbacks from Pluto's gravitational effect on Uranus.
One is the orbit of Uranus.
The second is the ecliptic angle of Uranus.
Huang Jing L has been calculated before, so there is only one task left for the calculation team:
Compare the difference in orbit offset.
What does it mean?
Suppose a magnet A moves on a horizontal plane. In the absence of other external forces, its trajectory is straight.
If another weak heteropolar magnet B is added during its movement - for example, placed ten meters to the left of A, then the movement trajectory of A will appear slightly while maintaining the original direction of movement.
offset.
Uranus is magnet A and Pluto is magnet B.
The movement trajectory of magnet A after deflection is the trajectory of Uranus that is observed and recorded by the naked eye.
After deducting the correction coefficient calculated by Riemann and others, what is obtained is its theoretical original trajectory - that is, the trajectory without being attracted by Pluto, that is, the "straight line".
This way.
There will be a coordinate difference between the two trajectories.
It's like a tourist who was supposed to go to Magic City today, but ended up in Jinmen.
Regardless of what happened in the middle, at least the geographical differences in longitude and latitude can be determined.
Then compare those observation records and find out a large number of coordinate differences at different times and locations, and you can use multivariate equations to calculate the position of Pluto - because according to the Titius-Bode rule, the distance of Pluto can be roughly determined
of.
In other words.
The so-called ‘difference in comparison of orbital offsets’, to put it bluntly, is...
Compare observation records!
Precisely.
It is a comparison of tens of thousands of observation records.
Of course.
Due to the existence of perihelion and aphelion, and the reference significance of some early images is greater than the actual significance, the data that really need to be identified is not so exaggerated.
This chapter is not finished yet, please click on the next page to continue reading the exciting content! According to rough statistics, there are about 4,000 copies in total.
Subsequently, the counting members on the scene began to form pairs.
One person reports the coordinates, and the other begins to calculate the deviation.
Among them, the ability of tool workers who report coordinates is slightly lower, and they are mainly students from the Department of Mathematics.
The computing power is provided by big names such as Riemann, Jacobi, and Weierstrass.
On average, each person needs to calculate more than two hundred observation records.
The calculation and comparison of a record takes about one minute. After all, there are only two coordinates to set the formula, so it takes about four hours in total.
Xu Yun and Lao Tang were not idle either, and took the initiative to take on part of the calculation tasks.
"4.6692568......6283.07585...."
"462.61....12.5661517...."
"2.0371......529.691......"
"2.92..........0.067...."
Soon, the coordinate system parameters of different specifications were reported one by one.
Some statistics come from the Bradley family, and data that has been hidden for many years has appeared in front of the world for the first time.
Many of the data even exceed similar documents from the Greenwich Observatory in terms of accuracy.
For example, Daniel Bradley's father, Condon Bradley, recorded the trajectory of Makemake twenty years ago.
Although it is only a record of the trajectory rather than an accurate discovery, it is already very scary in nature - because according to historical development, this thing will not be discovered until 2005.
2005 and 1830.
From the perspective of the accuracy of observation equipment, it is basically two eras...
It can be seen from this that the Bradley family has been holding back a lot of energy in order to reverse the case of their ancestors...
Maybe it was because I was touched by the atmosphere at the scene.
after awhile.
Several mathematics students actually walked out of the crowd and took the initiative to take over the work of the mathematicians who reported the numbers, allowing them to fully utilize their abilities in the calculation process.
According to Lao Tang, one of them was also a follower of Frederick Agar Ellis.
Looking at the ugly Earl of Eisley not far away, Xu Yun felt inexplicably emotional.
Perhaps this is the charm of science.
Many times, its appeal is invisible.
Then he thought of something again, raised his head, and looked around.
750 years ago.
He once worked with a group of Chinese sages to conquer the sky day and night.
750 years later.
It was also a night without snow.
Xu Yun worked together with another group of European mathematicians and looked across the sky to the vast starry sky.