PROS Bambang S. Solar Physics Long Term Full text
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
SOLAR PHYSICS LONG TERM RESEARCH RESULT:
THE BUTTERFLY DIAGRAM OF ACTIVE REGIONS
Bambang Setiahadi
Indonesian National Institute of Aeronautics and Space (LAPAN)
Watukosek, Gempol P.O. Box 04, Pasuruan 67155
bambangsetiahadi@rocketmail.com; bsetiapx@gmail.com
ABSTRACT
Solar activity daily monitoring at Watukosek Solar Observing Site at Watukosek, East Java, has resulted
important solar activity regions mapping. The mapping is coded internationally as Solar Butterfly Diagram. This
result demonstrated a capability of Watukosek in conducted international-coordinated solar activity observations.
The results may be extracted by our ASCII data base and flexibly might be used for multiyear researches.
Keywords: Daily observations, ASCII code, Butterfly Diagram
sky condition is abundant compare to fourseason observation, and consequently the data is
huge. Even we make only one report in a day;
the data we compile in a year is around 365 or
366 reports. This is about two times larger than
four seasons report’s observatory.
1. Introduction
Scientific results from solar physics research
have proven its utility to human life. It is not
surprisingly that more research institutions and
universities are gradually being involved to solar
physics researches and observations. Various
hardware or software-based solar physics
technology, as solar active battery cells and solar
pond generators for small scale electricity, is
proved to be more environment friendly
technology.
Nearly everyday we have a report as an output
from our sunspot telescope. Beside conventional
sunspot reports in a piece of a paper, we
consider to construct an ASCII data format and
compile the year’s data in very compact manner.
Scientist, who needs to look out the data from
certain day of observations, may have it
immediately without look out the original data.
Our original data is opened when we consider it
is necessary to look the original configuration of
sunspot and its heliographic position.
The solar research needs continual and
consistent observational activity, based-on
everyday coordinated solar international
standard observation procedure with other solar
observatory through-out the world [4]. It is
worth than after decades international
coordinated observations that our star, the Sun,
hides many plasma physics interactions as
periodic phenomena, ranging from minutes to
hundred year periodicities.
2. Observations
Solar observation is basically has the same
limitation as other sky observation. Clear and
stable sky is preferential for the solar
observations, except the solar has abundant light
that makes it different with other weak light sky
observations. But solar observation may pose a
danger to human eyes or to the other optical
sensors, because of the danger to over expose to
Since the observatory is located near equator at
about 07034 S and 112 0 40! E , observation
only face the tropical dry and rainy seasons in a
year. The chances to observe the sun under clear
317
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
sun light rays and radiation. Protective filter as
well as operation procedure has to be innovated
to make safe solar observations.
starts and ends time coordinate. The lastly
procedure to fulfill is East-West direction on
sunspot sketch. This direction is a line across
sunspot sketch paper to serve as fundamental
line to solar actual equator and North-South
direction of true solar pole.
In case white light sunspot digital observations,
with direct image method, the telescope has to
apply a neutral filter to reduce sun light intensity
[2]. Practically one has to reduce the sun light
intensity as low as 10-6 times to original
intensity. More over, to reduce stray lights, a
green filter has to be added in front of a digital
sensor. Those two apparatuses generally comply
to convey safe observation to both over-all
hardware’s and with observer.
3. Method and Processing
Soon after the international procedure produces
a sunspot sketch data, further sequential
procedure to convert the data to be more
scientifically valuable. The sequential procedure
has final results at least two kinds of data, a
conventional paper sunspot sketch data and
ASCII code. The paper sketch provides
international data compiling as a continuation of
400 years international solar observations. While
the ASCII sunspot sketch data provides quick
access and may be further used for statistical or
long term solar physics researches [3].
Our solar observations activity has international
collaborations and consequently international
procedure must be adopted. There are several
parameters in conjunction with sunspot whitelight observations. The very first is it conveys
international sunspot-sketch using B4 size
white-paper with good quality and withstand
against local humidity for more than 50 years
ahead. Good filing system with only special staff
could reach and accessed original data [5].
The important slot to determine solar surface
phenomena such as the sunspots, is carefully
determined the orientation of solar globe seen
relative from an observer on earth [2]. This
provide by carefully determine the equatorial
angle of North-South and East-West direction.
Tensorial heliographic coordinate system and
procedure developed in Watukosek Solar
Observatory uses the previously mention
procedure as initial data to guarantee the
uniqueness one-to-one correspondence of
sunspot coordinate on solar surface.
The second is the international time keeping that
keeps a clock at a solar observatory ticking until
accuracy within 10-1 second. This is the only
parameter connect among one observatory to
other international observatory, that is why
every observer has deep concern with time
coordinate.
International
coordinated
observation applies the Greenwich Mean Time
(or GMT) as international time coordinate. Other
so-called is Universal Time Coordinate (UTC).
The Watukosek Solar Observatory has 7+UTC
in its time coordinate.
The system and procedure maps not just the
sunspots positions uniquely, but every point on
solar surface as well. This is accomplished by
considering all aspect of solar terrestrial and
earth dynamic geometry in one year orbit on the
ecliptical plane. The sun makes an angel of ~70.0
relative to the ecliptical plane; in our ASCII
code is coded as B. While the earth during its
revolution around the sun makes an angel at
about ~230.5; enter to our ASCII code as P. Such
the angel is manifestation of four seasons during
a year. While the sun it self rotates on its axis of
The third is carefully notes environmental and
local weather every time an observation is done.
The parameters are related to atmospheric
transparency that could affect the sunspot
counting, there are seeing, scintillation, and
image stability. After all those parameters are
noted carefully, observation of sunspot is
starting and ending with precise UTC time
recorded at the start and ending. The presume
time coordinate at the observation is half of
318
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
observer’s name, number of sheet, and sunspot
counting number R, f, and g. One may read-out
ASCII data and makes suitable programming
according to one’s capability and scientific
objectives. One of that is the famous solar
Butterfly Diagram.
rotation in a period of 27 days along its equator;
coded as L.
The ASCII code of our sunspot sketch data
includes all parameters give necessary
information of every day report on UTC time of
observation, date-month-year of observation,
weather condition, celestial angel P, B, and L;
Figure 3.1: Output of ASCII-coded for four seasons extreme in March, June, September, and
December 2012. The important notes can be read on the left side of the report, and the orthographic
two dimensional projection of solar disk with sunspots and active regions positions and numbering
systems are obvious.
is the reason why all material of the sun should
be in sub-atomic and dynamical state. From
spectral observations the sun consist of most
abundance electrons, followed by protons or
hydrogen nucleus, and ions.
4. Result
Modern observations reveal important and
interesting reality that the sun rotates on its axis
exhibits non-massive celestial body as its
planets. Instead, the sun rotates differentially
that proven all material building up the solar
globe is fluid rather than solid. The density of
the fluid is about the same order with water
density.
The global physical state of the sun drawn
consequences to every solar-surface observable
phenomena. One this is the sunspot appearance.
The sunspots are supposed to appear on the
surface by local buoyancy force. Solar
differential rotation makes sunspots demonstrate
non-zero systematical flow in the direction of
The sun keeps its high temperature body, at least
on the surface is no less than 60000 Kelvin. This
319
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
dynamo generates from non-linear dynamo
processes below observable solar surface.
longitudinal and at the same time latitudinal
flow components. On the average the sunspots
have resultant speed on solar surface no less than
10km/day.
Inner study from solar deeper layers might open
the discovery of very initial process of solar
sunspots instability and solar surface energetic
energy liberations. The effect of this may
propagates through interplanetary space, and
may affect satellite communications and satellite
navigation, or to other orbital technologies.
Relatively small speed will not be seen with
relatively short period solar observation. It must
be observed years to be calculated and resulting
significant number of kilometers per day. The
systematical daily observation with good
observation procedure and data compiling
system makes long-term phenomena of sunspot
is easy to access. The Watukosek ASCII code
accommodates not just the phenomena, it may
be used to various scientific goals and generates
international scientific collaborations.
References
[1]. Adipranata, R., Setia Budhi, G., Setiahadi,
B. (2013), Automatic Classification of
Sunspot Groups for space Weather
Analysis,
International
Journal
of
Multimedia and Ubiquitous Engineering,
Vol. 8, No. 3, p. 41.
[2]. Bray, R.J., Loughhead, R.E. (1964),
Sunspots, Dover Publ. Inc., New York.
[3]. Cooke, D., Craven, A.H., Clarke, G.M.
(1982), Basic Statistical Computing,
Edward-Arnold Publ. Co., Sussex.
[4]. Setiahadi, B. (2007a), Daily Observational
Sunspot ASCII Data for Long-Term Solar
Magnetohydrodynamo Studies at LAPAN
Watukosek, University of
Indonesia,
National Conference
on
Computer
Science and Information Technology,
Jakarta, (ISSN 0126-2866), p. 137.
[5]. Setiahadi, B. (2007b), Solar Observation
Procedures to Obtain the Watukosek White
Light Solar ASCII Data for Statistical
Space Warnings, ITB Research Center on
ICT (Information & Communication
Technology), Bandung, (ISBN 978-97915509-1-8), p. 101.
Figure 4.1: The solar butterfly diagram resulted
from daily observations at Watukosek. The
diagram is resulted from reading our ASCII code
data compiled for many years systematical
procedure according to international sunspot
sketch procedure.
5. Discussions
Appendix
Further work may be directed to many ways
depending on scientific focus or goal. Study for
systematical motion of sunspots in years
generate solar sunspot statistical study [1]. The
results of the study provide solar differential
curve which can be applied into solar energy
Our heliographic coordinate system to provide
orthographic projection for solar disk dynamical
orientation during a year are presented below
with the celestial angels P, B, and L. The
position of sunspots or every points on solar disk
are guaranteed uniquely and satisfy the one-to320
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
one correspondence algebraic set theory, as well
as unique transformation.
Repeated upper and lower indexes are applied
the Einstein-ian summation rule that is usually
implemented in special or general relativity
physics for celestial body, such as the Sun,
Planets, Stars, and Galaxies
( 0) y +
(0) z ⎤ ⎡ x ⎤
⎡ xˆ ⎤ ⎡ (1) x +
⎢ yˆ ⎥ = ⎢(0) x + (cos P) y + (− sin P) z ⎥ ⎢ y ⎥
⎢ ⎥ ⎢
⎥⎢ ⎥
⎢⎣ zˆ ⎥⎦ ⎢⎣(0) x + (sin P) y + (cos P ) z ⎥⎦ ⎢⎣ z ⎥⎦
⇔
Xˆ α = Pβα X β
⎡ xˆ ⎤ ⎡ (cos B) x + (0) y + (sin B) z ⎤ ⎡ x ⎤
⎢ yˆ ⎥ = ⎢ (0) x +
(1) y +
(0) z ⎥⎥ ⎢⎢ y ⎥⎥
⎢ ⎥ ⎢
⎢⎣ zˆ ⎥⎦ ⎢⎣(− sin B ) x + (0) y + (cos B) z ⎥⎦ ⎢⎣ z ⎥⎦
⇔
Xˆ α = Bβα X β
⎡ xˆ ⎤ ⎡ (cos L) x + (sin L) y + (0) z ⎤ ⎡ x ⎤
⎢ yˆ ⎥ = ⎢(− sin L) x + (cos L) y + (0) z ⎥ ⎢ y ⎥
⎢ ⎥ ⎢
⎥⎢ ⎥
⎢⎣ zˆ ⎥⎦ ⎢⎣ (0) x +
( 0) y +
(1) z ⎥⎦ ⎢⎣ z ⎥⎦
⇔
Xˆ α = Lαβ X β
Xˆ α = Pβα Bγβ Lγδ X δ
321
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
SOLAR PHYSICS LONG TERM RESEARCH RESULT:
THE BUTTERFLY DIAGRAM OF ACTIVE REGIONS
Bambang Setiahadi
Indonesian National Institute of Aeronautics and Space (LAPAN)
Watukosek, Gempol P.O. Box 04, Pasuruan 67155
bambangsetiahadi@rocketmail.com; bsetiapx@gmail.com
ABSTRACT
Solar activity daily monitoring at Watukosek Solar Observing Site at Watukosek, East Java, has resulted
important solar activity regions mapping. The mapping is coded internationally as Solar Butterfly Diagram. This
result demonstrated a capability of Watukosek in conducted international-coordinated solar activity observations.
The results may be extracted by our ASCII data base and flexibly might be used for multiyear researches.
Keywords: Daily observations, ASCII code, Butterfly Diagram
sky condition is abundant compare to fourseason observation, and consequently the data is
huge. Even we make only one report in a day;
the data we compile in a year is around 365 or
366 reports. This is about two times larger than
four seasons report’s observatory.
1. Introduction
Scientific results from solar physics research
have proven its utility to human life. It is not
surprisingly that more research institutions and
universities are gradually being involved to solar
physics researches and observations. Various
hardware or software-based solar physics
technology, as solar active battery cells and solar
pond generators for small scale electricity, is
proved to be more environment friendly
technology.
Nearly everyday we have a report as an output
from our sunspot telescope. Beside conventional
sunspot reports in a piece of a paper, we
consider to construct an ASCII data format and
compile the year’s data in very compact manner.
Scientist, who needs to look out the data from
certain day of observations, may have it
immediately without look out the original data.
Our original data is opened when we consider it
is necessary to look the original configuration of
sunspot and its heliographic position.
The solar research needs continual and
consistent observational activity, based-on
everyday coordinated solar international
standard observation procedure with other solar
observatory through-out the world [4]. It is
worth than after decades international
coordinated observations that our star, the Sun,
hides many plasma physics interactions as
periodic phenomena, ranging from minutes to
hundred year periodicities.
2. Observations
Solar observation is basically has the same
limitation as other sky observation. Clear and
stable sky is preferential for the solar
observations, except the solar has abundant light
that makes it different with other weak light sky
observations. But solar observation may pose a
danger to human eyes or to the other optical
sensors, because of the danger to over expose to
Since the observatory is located near equator at
about 07034 S and 112 0 40! E , observation
only face the tropical dry and rainy seasons in a
year. The chances to observe the sun under clear
317
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
sun light rays and radiation. Protective filter as
well as operation procedure has to be innovated
to make safe solar observations.
starts and ends time coordinate. The lastly
procedure to fulfill is East-West direction on
sunspot sketch. This direction is a line across
sunspot sketch paper to serve as fundamental
line to solar actual equator and North-South
direction of true solar pole.
In case white light sunspot digital observations,
with direct image method, the telescope has to
apply a neutral filter to reduce sun light intensity
[2]. Practically one has to reduce the sun light
intensity as low as 10-6 times to original
intensity. More over, to reduce stray lights, a
green filter has to be added in front of a digital
sensor. Those two apparatuses generally comply
to convey safe observation to both over-all
hardware’s and with observer.
3. Method and Processing
Soon after the international procedure produces
a sunspot sketch data, further sequential
procedure to convert the data to be more
scientifically valuable. The sequential procedure
has final results at least two kinds of data, a
conventional paper sunspot sketch data and
ASCII code. The paper sketch provides
international data compiling as a continuation of
400 years international solar observations. While
the ASCII sunspot sketch data provides quick
access and may be further used for statistical or
long term solar physics researches [3].
Our solar observations activity has international
collaborations and consequently international
procedure must be adopted. There are several
parameters in conjunction with sunspot whitelight observations. The very first is it conveys
international sunspot-sketch using B4 size
white-paper with good quality and withstand
against local humidity for more than 50 years
ahead. Good filing system with only special staff
could reach and accessed original data [5].
The important slot to determine solar surface
phenomena such as the sunspots, is carefully
determined the orientation of solar globe seen
relative from an observer on earth [2]. This
provide by carefully determine the equatorial
angle of North-South and East-West direction.
Tensorial heliographic coordinate system and
procedure developed in Watukosek Solar
Observatory uses the previously mention
procedure as initial data to guarantee the
uniqueness one-to-one correspondence of
sunspot coordinate on solar surface.
The second is the international time keeping that
keeps a clock at a solar observatory ticking until
accuracy within 10-1 second. This is the only
parameter connect among one observatory to
other international observatory, that is why
every observer has deep concern with time
coordinate.
International
coordinated
observation applies the Greenwich Mean Time
(or GMT) as international time coordinate. Other
so-called is Universal Time Coordinate (UTC).
The Watukosek Solar Observatory has 7+UTC
in its time coordinate.
The system and procedure maps not just the
sunspots positions uniquely, but every point on
solar surface as well. This is accomplished by
considering all aspect of solar terrestrial and
earth dynamic geometry in one year orbit on the
ecliptical plane. The sun makes an angel of ~70.0
relative to the ecliptical plane; in our ASCII
code is coded as B. While the earth during its
revolution around the sun makes an angel at
about ~230.5; enter to our ASCII code as P. Such
the angel is manifestation of four seasons during
a year. While the sun it self rotates on its axis of
The third is carefully notes environmental and
local weather every time an observation is done.
The parameters are related to atmospheric
transparency that could affect the sunspot
counting, there are seeing, scintillation, and
image stability. After all those parameters are
noted carefully, observation of sunspot is
starting and ending with precise UTC time
recorded at the start and ending. The presume
time coordinate at the observation is half of
318
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
observer’s name, number of sheet, and sunspot
counting number R, f, and g. One may read-out
ASCII data and makes suitable programming
according to one’s capability and scientific
objectives. One of that is the famous solar
Butterfly Diagram.
rotation in a period of 27 days along its equator;
coded as L.
The ASCII code of our sunspot sketch data
includes all parameters give necessary
information of every day report on UTC time of
observation, date-month-year of observation,
weather condition, celestial angel P, B, and L;
Figure 3.1: Output of ASCII-coded for four seasons extreme in March, June, September, and
December 2012. The important notes can be read on the left side of the report, and the orthographic
two dimensional projection of solar disk with sunspots and active regions positions and numbering
systems are obvious.
is the reason why all material of the sun should
be in sub-atomic and dynamical state. From
spectral observations the sun consist of most
abundance electrons, followed by protons or
hydrogen nucleus, and ions.
4. Result
Modern observations reveal important and
interesting reality that the sun rotates on its axis
exhibits non-massive celestial body as its
planets. Instead, the sun rotates differentially
that proven all material building up the solar
globe is fluid rather than solid. The density of
the fluid is about the same order with water
density.
The global physical state of the sun drawn
consequences to every solar-surface observable
phenomena. One this is the sunspot appearance.
The sunspots are supposed to appear on the
surface by local buoyancy force. Solar
differential rotation makes sunspots demonstrate
non-zero systematical flow in the direction of
The sun keeps its high temperature body, at least
on the surface is no less than 60000 Kelvin. This
319
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
dynamo generates from non-linear dynamo
processes below observable solar surface.
longitudinal and at the same time latitudinal
flow components. On the average the sunspots
have resultant speed on solar surface no less than
10km/day.
Inner study from solar deeper layers might open
the discovery of very initial process of solar
sunspots instability and solar surface energetic
energy liberations. The effect of this may
propagates through interplanetary space, and
may affect satellite communications and satellite
navigation, or to other orbital technologies.
Relatively small speed will not be seen with
relatively short period solar observation. It must
be observed years to be calculated and resulting
significant number of kilometers per day. The
systematical daily observation with good
observation procedure and data compiling
system makes long-term phenomena of sunspot
is easy to access. The Watukosek ASCII code
accommodates not just the phenomena, it may
be used to various scientific goals and generates
international scientific collaborations.
References
[1]. Adipranata, R., Setia Budhi, G., Setiahadi,
B. (2013), Automatic Classification of
Sunspot Groups for space Weather
Analysis,
International
Journal
of
Multimedia and Ubiquitous Engineering,
Vol. 8, No. 3, p. 41.
[2]. Bray, R.J., Loughhead, R.E. (1964),
Sunspots, Dover Publ. Inc., New York.
[3]. Cooke, D., Craven, A.H., Clarke, G.M.
(1982), Basic Statistical Computing,
Edward-Arnold Publ. Co., Sussex.
[4]. Setiahadi, B. (2007a), Daily Observational
Sunspot ASCII Data for Long-Term Solar
Magnetohydrodynamo Studies at LAPAN
Watukosek, University of
Indonesia,
National Conference
on
Computer
Science and Information Technology,
Jakarta, (ISSN 0126-2866), p. 137.
[5]. Setiahadi, B. (2007b), Solar Observation
Procedures to Obtain the Watukosek White
Light Solar ASCII Data for Statistical
Space Warnings, ITB Research Center on
ICT (Information & Communication
Technology), Bandung, (ISBN 978-97915509-1-8), p. 101.
Figure 4.1: The solar butterfly diagram resulted
from daily observations at Watukosek. The
diagram is resulted from reading our ASCII code
data compiled for many years systematical
procedure according to international sunspot
sketch procedure.
5. Discussions
Appendix
Further work may be directed to many ways
depending on scientific focus or goal. Study for
systematical motion of sunspots in years
generate solar sunspot statistical study [1]. The
results of the study provide solar differential
curve which can be applied into solar energy
Our heliographic coordinate system to provide
orthographic projection for solar disk dynamical
orientation during a year are presented below
with the celestial angels P, B, and L. The
position of sunspots or every points on solar disk
are guaranteed uniquely and satisfy the one-to320
Prosiding Seminar Nasional Sains dan Pendidikan Sains VIII, Fakultas Sains dan Matematika, UKSW
Salatiga, 15 Juni 2013, Vol 4, No.1, ISSN:2087-0922
one correspondence algebraic set theory, as well
as unique transformation.
Repeated upper and lower indexes are applied
the Einstein-ian summation rule that is usually
implemented in special or general relativity
physics for celestial body, such as the Sun,
Planets, Stars, and Galaxies
( 0) y +
(0) z ⎤ ⎡ x ⎤
⎡ xˆ ⎤ ⎡ (1) x +
⎢ yˆ ⎥ = ⎢(0) x + (cos P) y + (− sin P) z ⎥ ⎢ y ⎥
⎢ ⎥ ⎢
⎥⎢ ⎥
⎢⎣ zˆ ⎥⎦ ⎢⎣(0) x + (sin P) y + (cos P ) z ⎥⎦ ⎢⎣ z ⎥⎦
⇔
Xˆ α = Pβα X β
⎡ xˆ ⎤ ⎡ (cos B) x + (0) y + (sin B) z ⎤ ⎡ x ⎤
⎢ yˆ ⎥ = ⎢ (0) x +
(1) y +
(0) z ⎥⎥ ⎢⎢ y ⎥⎥
⎢ ⎥ ⎢
⎢⎣ zˆ ⎥⎦ ⎢⎣(− sin B ) x + (0) y + (cos B) z ⎥⎦ ⎢⎣ z ⎥⎦
⇔
Xˆ α = Bβα X β
⎡ xˆ ⎤ ⎡ (cos L) x + (sin L) y + (0) z ⎤ ⎡ x ⎤
⎢ yˆ ⎥ = ⎢(− sin L) x + (cos L) y + (0) z ⎥ ⎢ y ⎥
⎢ ⎥ ⎢
⎥⎢ ⎥
⎢⎣ zˆ ⎥⎦ ⎢⎣ (0) x +
( 0) y +
(1) z ⎥⎦ ⎢⎣ z ⎥⎦
⇔
Xˆ α = Lαβ X β
Xˆ α = Pβα Bγβ Lγδ X δ
321