Parallax -

Japanese: 視差 - しさ

The difference in direction when a celestial object is viewed from two points. When viewing one point P from two points A and B, ∠APB is called parallax ( Figure A ). Parallax is an important quantity (a numerical value/measurement, the size of an angle is expressed in degrees) for determining distances AP and BP when they cannot be measured directly. Measuring parallax is the most basic method for determining the distance of a celestial object, except when it can be measured directly with radar or laser. There are various possible parallaxes, depending on how observation points A and B are located.

[Naoaki Owaki]

Geocentric parallax

In Figure B (1), if A and B are two points on the Earth, the parallax is ∠APB. In this case, it is convenient to calculate the parallax as seen from the center of the Earth, C, instead of B, and the parallax thus calculated, ∠APC, is called geocentric parallax. More conveniently, geocentric parallax (horizontal parallax) is used when a celestial body is seen on the horizon from the Earth's surface, and furthermore, horizontal parallax as seen by an observer on the equator (equatorial-horizontal parallax). Geocentric parallax is used to measure the distance of celestial bodies in the solar system. That is, in Figure B (2), if the distance between the center of the Earth, C, and celestial body P is r and the distance between observation point A and the center of the Earth, C, is a , then,

It becomes.

[Naoaki Owaki]

Annual parallax and triangular parallax

In Figure C , the direction of star S as seen from Sun C is CS, i.e. the position of the star on the celestial sphere is C', but when seen from Earth A on its orbit, S appears to be located at A' on the celestial sphere. This deviation between C' and A', i.e. ∠ASC, is the parallax, which changes with a yearly cycle as the Earth revolves around the Sun, so it is called the annual parallax. At positions P and Q on the orbital plane that are perpendicular to CS, the annual parallax reaches its maximum. This maximum value is called the trigonometric parallax p (it is often referred to as the annual parallax). Now, if the average distance between the Earth and the Sun is a and the distance to the star is r , then,

From this relationship, r can be calculated by measuring p . This is the most basic method for calculating the distance to a star. However, in reality, stars are extremely far away, and the trigonometric parallax is very small. Therefore, this method can only be applied to nearby stars (with trigonometric parallax of a few tenths of an arcsecond or more). Note that 1 second is 1/1,296,000th of a circumference (360 degrees), and a distance with a trigonometric parallax of 1 arcsecond is called 1 parsec, which is equivalent to approximately 200,000 astronomical units, approximately 31 trillion kilometers, or 3.26 light years.

[Naoaki Owaki]

Secular parallax

The sun moves among the stars in the direction of Hercules at a speed of about 19.5 kilometers per second. Therefore, if the sun is C and the star is S ( Figure D ), at a certain point the star will appear in the direction of CS, but after a few years it will appear in the direction of C'S. The angle C'SC at this time is called secular parallax. In reality, S is also moving, so the distance of S cannot be determined immediately from this, but if we assume that the specific motion of each of the many stars is zero on average, we can statistically estimate the distance of the star from the magnitude of the deviation in direction.

[Naoaki Owaki]

Stellar Parallax

Some star clusters are known to move in the same direction relative to the Sun (e.g., the Hyades). Star S in these clusters moves parallel to the Sun C, so they appear to converge to a point K on the celestial sphere ( Figure E ). In this case, the distance to S can be determined by observing ∠SCK, the annual change in the direction of S (i.e., the proper motion of S), and the radial velocity of S relative to the Sun. This method of determining distance is called the stellar parallax method.

The above is a method of calculating parallax (i.e. distance) by geometric or kinematic methods, and so the original word "parallax" is used, but distance can also be estimated by other methods, such as the brightness of stars. The word "parallax" is also used for distances obtained by such methods, and is sometimes called photometric parallax.

[Naoaki Owaki]

"Experimental Astronomy Workbook" by Roger B. Culver, translated by Toshio Hasegawa (1988, Koseisha Kouseikaku)" ▽ "Astronomical Calculation Classroom" by Hiroshi Saida, new edition (1998, Chijin Shokan)" ▽ "Sunrise and Sunset Calculations - How to Find the Times of Rising and Setting of Celestial Bodies" by Takumi Nagasawa (1999, Chijin Shokan)"

Parallax (Figure A)
©Shogakukan ">

Parallax (Figure A)

Geocentric parallax (Figure B)
(1)∠APB＝parallax when P is seen from A and B (2)＝horizontal parallax ©Shogakukan ">

Geocentric parallax (Figure B)

Annual parallax/triangular parallax [Figure C]
∠ASC＝Annual parallax＝Trigonometric parallax (This is sometimes called annual parallax) ©Shogakukan ">

Annual parallax/triangular parallax [Figure C]

Secular parallax (Fig. D)
C: The position of the sun in a certain year C': The position of the sun several years later = secular parallax C'. If these are known, the distance to S can be calculated. ©Shogakukan ">

Secular parallax (Fig. D)

Stellar Parallax (Figure E)
From C, the stars at S appear to converge to a single point K on the celestial sphere. If , λ, and μ are known from observations, the distance to the cluster can be calculated from μ = tan λ .

Stellar Parallax (Figure E)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

ある天体を2地点から見たときの方向の差。2点A、Bから1点Pを見るときに∠APBを視差という（図A）。視差は、距離APやBPを直接測定できないとき、これらを求めるのに重要な量（数値・計測値で、角の大きさは角度で表す）である。天体の距離を求めるのに、レーダーやレーザーなどで直接に測定できる場合は別にして、視差を測るのがもっとも基本的な方法である。視差は観測点A、Bのとり方などにより、いろいろ考えられる。

［大脇直明］

地心視差

図Bの(1)において、A、Bを地球上の2点とすると、視差は∠APBとなる。このときBのかわりに地球中心Cから見たような視差を計算しておくと便利であり、こうして求めた視差∠APCを地心視差という。また、より便利なものとして、地表で天体を地平線に見たときの地心視差（地平視差）、さらに赤道上の観測者が見た地平視差（赤道地平視差）が用いられる。地心視差は太陽系天体の距離測定に用いられる。すなわち、図Bの(2)において、地心Cと天体Pとの距離をr、観測点Aと地心Cとの距離をaとすると、

となる。

［大脇直明］

年周視差・三角視差

図Cにおいて、太陽Cから見た恒星Sの方向はCS、すなわち天球上の星の位置はC'であるが、公転軌道上の地球Aから見るとSは天球上のA'に位置して見える。このC'とA'のずれ、すなわち∠ASCが視差で、地球の公転運動に伴い1年周期で変化するので年周視差とよばれる。軌道面上でCSに垂直な位置P、Qでは年周視差が極大となる。この極大値を三角視差pという（しばしばこれを年周視差ということがある）。いま、地球・太陽間の平均距離をa、恒星までの距離をrとすると、

という関係から、pを測ればrが求められる。これが恒星までの距離を求めるもっとも基本的な方法である。しかし現実には恒星はきわめて遠方にあり、三角視差は非常に小さい。したがって、この方法は、近い恒星（三角視差が数十分の1秒角以上）にしか適用できない。なお、1秒は円周（360度）の129万6000分の1の角度で、三角視差が1秒角の距離を1パーセクといい、これは約20万天文単位、約31兆キロメートル、3.26光年に相当する。

［大脇直明］

永年視差

太陽は恒星の間をヘルクレス座の方向に秒速約19.5キロメートルの速度で運動している。したがって、太陽をC、恒星をSとすると（図D）、ある時点には星がCSの方向に見えるが、何年かたつとC'Sの方向に見える。このときの∠C'SCを永年視差という。実際はSも動いているので、これからただちにSの距離は求められないが、多数の星それぞれの特有の運動が平均的にみてゼロと仮定すると、統計的に方向のずれの大小で星の距離を推定することができる。

［大脇直明］

星流視差

星団などのなかには、太陽に対して同じ方向に運動していることが知られているものがある（例、ヒヤデス星団）。これらの星団に属する星Sは太陽Cに対して平行運動をしているので、天球上の1点Kに収束して動いているように見える（図E）。このとき∠SCK、Sの方向の年間変化（すなわちSの固有運動）およびSの太陽に対する視線速度を観測すると、Sまでの距離がわかる。このような距離の求め方を星流視差の方法という。

　なお、以上は幾何学的ないし運動学的方法によって視差（すなわち距離）を求めるもので、視差という本来のことばが用いられているのであるが、ほかの方法、たとえば恒星の明るさからも距離を推定することができる。このような方法で得られた距離に対しても視差ということばを用い、たとえば測光学的視差という場合もある。

［大脇直明］

『ロジャー・B・カルバー著、長谷川俊雄訳『実験天文学ワークブック』（1988・恒星社厚生閣）』▽『斉田博著『天文の計算教室』新装版（1998・地人書館）』▽『長沢工著『日の出・日の入りの計算――天体の出没時刻の求め方』（1999・地人書館）』