超大質量ブラックホールへのガス降着の鍵は超新星爆発？

超大質量ブラックホールへのガス降着の鍵は超新星爆発？

銀河中心の超大質量ブラックホールの周囲に広がる高密度分子ガス円盤が、ブラックホールへの質量の供給源として重要であることが初めて示された。円盤で発生する超新星爆発がブラックホールの成長を駆動するという理論予測を観測的に支持するもので、超大質量ブラックホールの起源の解明につながる観測結果である。

【2016年8月15日 東京大学／アルマ望遠鏡】

多くの銀河の中心には、太陽の100万倍以上もの質量を持つ超大質量ブラックホールが普遍的に存在している。しかし、その形成過程は謎に包まれており、現代天文学が解決すべき最重要テーマの一つとなっている。また、中心部での星形成が活発な銀河ほど銀河中心ブラックホールへのガス質量降着率も大きい（ブラックホールに落ち込むガスの量が多い）ことが知られているが、この2つの現象を結びつける物理機構はわかっていない。

東京大学大学院理学系研究科の泉拓磨さん、河野孝太郎さん、呉工業高等専門学校の川勝望さんの研究チームは、アルマ望遠鏡などで得た高解像度の電波観測データを解析し、近傍宇宙に存在する銀河10個の中心に潜む超大質量ブラックホールについて、その周囲数百光年にわたって広がる低温・高密度の分子ガス円盤を調べた。

（左）NGC 7469の可視光線画像、（右）NGC 7469の中心領域におけるシアン化水素分子輝線の強度分布図（擬似カラー）
（左）今回調査された銀河の一つ、ぺガスス座に位置するNGC 7469の可視光線画像、（右）NGC 7469の中心領域におけるシアン化水素分子輝線の強度分布図（擬似カラー）。中心の十字が超大質量ブラックホールの位置（提供：（左）NASA, ESA, Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration, and A.Evans、（右）東京大学、アルマ望遠鏡）

解析から、シアン化水素分子輝線の電波強度から見積もった「高密度分子ガス円盤の質量」と、別の観測で知られている「超大質量ブラックホールへのガス質量降着率」に、強い正の相関があることが明らかになった。また、別に見積もった「銀河全体の高密度ガスの総量」と「ブラックホールへのガス質量降着率」には相関が見られなかった。超大質量ブラックホール成長に重要なガス質量の供給源として、中心部の小さな高密度分子ガス円盤こそが重要な機能を果たしていることを示す結果である。

さらに、「高密度分子ガス円盤内で形成された大質量星が超新星爆発を起こし、ガス中に強い乱流が発生することで、さらに内側へのガス供給が促進される」という理論モデルと、実際の観測データをもとに、高密度分子ガス円盤からさらに内側に流入するガス質量を計算した。この値を超大質量ブラックホール近傍で実際に消費されているガスの総量（ブラックホールの成長に使われる質量とブラックホール近傍で発生する強い放射で外部に吹き飛ばされてしまう質量の合計）と比較したところ、これら2つの量は一致した。

この理論モデルでは、超新星爆発（星形成と関連する現象）がガス質量降着のかぎとなっており、「中心部での星形成が活発な銀河ほど銀河中心ブラックホールへのガス質量降着率も大きい」ことを自然と説明できる。今回の結果は、観測からこのモデル予測を裏付けるものだ。

銀河中心部で起こっている超大質量ブラックホールへのガス質量降着過程の想像図
銀河中心部で起こっている超大質量ブラックホールへのガス質量降着過程の想像図。高密度分子ガス円盤中で発生した超新星爆発が周囲に強い乱流を引き起こし、安定な運動を妨げられたガスが中心に向かって流入する様子を表している（出典：東京大学プレスリリース）

超大質量ブラックホールと星成分の研究に低温高密度分子ガスの観測を加えることで、銀河中心数百光年以内の小さい領域におけるガス質量の流入・流出の収支が初めて整合的に説明された。今後、アルマ望遠鏡等を用いた遠方ブラックホールの詳細な観測から、宇宙の古今にわたるブラックホールの成長の包括的な理解が進むと期待される。

フレッシュアイペディアで調べるpowered http://www.astroarts.co.jp/news/2016/08/15smbh/index-j.shtml


documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 185　: The importance of the division by zero $z/0=0$}
\author{{\it Institute of Reproducing Kernels}\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall state the importance of the division by zero $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
{\bf Introduction}
\bigskip
%\label{sect1}
By {\bf a natural extension of the fractions}
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.　
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$}
\medskip
Furthermore, note that Hiroshi Michiwaki gave the important interpretation of the division by zero $z/0=0$ by the intuitive meaning of the division, independently of the concept of the product (see \cite{ann}).
We shall state the importance of the division by zero $z/0=0$.
\bigskip
\section{}
On AD 628, the zero was appeared in India, and the zero division $z/0=0$ was discovered on Feburary 2, 2014, definitely with the clear definition and motievation. The uniquess and the natural interpretation were given in \cite{taka, ttk,kmsy} and \cite{ann}, respectively. Several physical interpretations of the division by zero were given in \cite{kmsy}.
\bigskip
\section{}
By the introduction of the division by zero $z/0=0$, four arithmetic operations; that is,
addition, subtraction, multiplication, and division are always possible; note that for division, we were not able to divide by zero. There was one exceptional case for the division by zero.
\section{}
For the Euclidean (B.C. 3 Centuary ) geometry, two non-Euclidean geometries were appered about 2 hundred years ago, and in particular, in the elliptic type non-Euclidean geometry, the point at infinity was introduced by the stereoprojection of the Euclidean plane to the sphere and the concept is a standard one in complex analysis around over one hundered years. And then we have considered as $1/0= \infty$ (\cite{ahlfors}). However, surprisingly enough, the division by zero means that $1/0=0$.
\section{}
We shall recall the fundamental law by Newton:
\begin{equation}
F = G\frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, we obtain the important interpretation:
\begin{equation}
F = 0 = G \frac{m_1 m_2}{0}.
\end{equation}
Of course, here, we can consider the above interpretation for the mathematical formula (4.1) as the new interpretation (4.3). We can find many physical formulas with the division by zero.
\section{}
In complex analysis, linear fractional functions
$$
W = \frac{az + b}{cz + d}, \quad ad -bc \ne 0,
$$
map the extended complex plane onto the extended complex plane containing the point at infinity, one to one, conformally, beautifully. This beautiful property is changed as the beautiful formula that linear fractional functions map the whole complex plane onto the whole complex plane, one to one, however, at one point of the singular point, the linear fractional functions have strong discontinuity.
The division by zero excludes the infinity from the numbers.
\section{}
We did, essentially, not consider the division by zero, and so the property of the division by zero; that is, at the isolated singular points of analytic functions, to consider the analytic functions are new mathematics and new research topics, essentially.
\section{}
The impact to complex analysis is unclear, we, however, obtain a typical new theorem:
\medskip
{\bf Theorem :} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.} The definite value is given by the first coefficient of the regular part in the Laurent expansion around the isolated singular point.
\medskip
This will be the fundamental theorem on the division by zero in Complex Analysis and we have many applications for the Sato hyperfunction theory, generating functions theory and singular integral theory (\cite{mst}).
\section{}
In particular, the divison by zero gives new interpretations on the finite part of Hadamard
for singular integrals and the Cauchy's principal values. The division by zero will represent discontinuity properties on the universe.
\section{}
Even for middle high shool students, the division by zero may be accepted as the beautiful result with great pleasures:
For the elementary function
$$
y = f(x) = \frac{1}{x},　
$$
we have $f(0) = 0$; that is, $1/0=0$.　
\section{}
We can introduce the division by zero $100/0=0,0/0=0$ with the simple and natural definition for the division by the Hiroshi Michiwachi method (\cite{ann}) in the elementary school. The division by zero will request the change of all the related books and scientific books.
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$

The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
\section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12):　 $100/0=0, 0/0=0$ 　--　 by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division　--　The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9):　 Should be changed the education of the division by zero
\medskip
{\bf Announcement 179} (2014.10.22):　 Division by zero is clear as z/0=0 and it is fundamental in mathematics
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).

\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{mst}
H. Michiwaki, S. Saitoh, and M. Takagi,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields, Tokyo Journal of Mathematics (in press).
\bibitem{ann}
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics,
Institute of Reproducing Kernels, 2014.10.22.
\end{thebibliography}
\end{document}