Presented at the GHOU 2018 conference in Vienna, Austria from 13-15 August. http://handsonuniverse.org/ghou2018/program/

1. DEVELOPMENT AND IMPLEMENTATION
OF STUDENT ACTIVITY TO
FIND MASS OF BLACK HOLE
Naoki Matsumoto
Keio Senior High School : JAPAN

2. SURPRISING AT MICHELLE'S PRACTICE
Almost same (^o^)
I'm happy to meet
teachers who make new
teaching materials using
new result of observations.
There are some
differences, so please
listen. m(_ _)m

3. BACKGROUND 1
In the high school curriculum of Japan,
astronomy is learned by a part of earth
science.
Earth science includes geophysics, earth
history, weather, ocean, contents are diverse.
The earth science teacher does not spend
too much time on astronomy.
However, many students like cosmos,
astronomy and black holes.

4. Survey of interest in science terms for university first
graders. (M.Kato, et al. 2004)
Humanities students(#409)
Science students(#415)Black Hole

5. BACKGROUND 2
FINDING MASS PRACTICE
USING M106
NGC4258（M106）
Determine the velocity of the gas
cloud from radio observation using
H2O maser(22GHz) .
The mass of the black hole is
derived from the velocity v and the
distance from the center of this
galaxy r.
(H.Shinohara, et al. 2015)

6. BACKGROUND 2
FINDING MASS PRACTICE
USING M106
NGC4258（M106）
This student practice designed for
physic class.
Difficult (little bit).
I thought that to develop a more
simple practice using our galaxy
for earth science class.

7. KEPLER’S LAW
The Japanese national curriculum
of earth science contains Kepler 's
law.
Using Kepler's third law, we can
find the mass of a celestial body.
Object used in this practice is S2
star which was observed with
infrared at European Southern
Observatory.
(Gillessen, et al.2009)

8. Revolution period Ｐ
Orbit radius ａ
KEPLER'S 3RD LAW
a
一定2
3
P
a

9. 2018/9/6
軌道半径
（天文単位）
公転周期
（年）
水星 0.39 0.24
金星 0.72 0.62
地球 1.00 1.00
火星 1.52 1.88
木星 5.20 11.86
土星 9.55 29.46
天王星 19.22 84.02
海王星 30.11 164.77
冥王星 39.54 247.80
a^3/P^2
0.9995
0.9998
1.0000
0.9999
1.0008
1.0052
1.0055
1.0055
1.0068
天文単位(Astronomical Unit：au ) 地球･太陽距離を １ とする

10. CONSTANT ORBIT →
GRAVITATION AND CENTRIFUGAL FORCE ARE BALANCED
Gravitation force
Centrifugal force
a
Mm
万有引力
F
2
a
Mm
GF 
遠心力
F’
2
22
P
ma4
a
mv
'F


Planet mass m
Solar mass M
Gravity constant G
Revolution speed v=2πa／P

11. WORKSHEET PRACTICE
Word files of this worksheet can be downloaded
from this QR code ⇗.
Lower part of the first page
2πa
P
𝑚∙4π2 𝑎2
𝑎𝑃2 =
4π2
𝑚𝑎
𝑃2

12. NEWTON'S KEPLER'S THIRD LAW EVOLVED FORM
2
2
2
P
ma4
a
Mm
G


22
P
a4
a
M
G


Gravity = Centrifugal force
2
32
P
a4
GM


22
3
4
GM
P
a


a : Orbital radius
P : Revolution Period
G: Gravity constant
M: Mass of central
object
=Const.

13. 2
32
GP
a4
M

22
3
4
GM
P
a


Planet's movement →
mass of the sun
Satellite movement →
mass of the planet
Stellar motion around BH →
mass of BH

14. WORKSHEET PRACTICE
Lower part of the first page
2πa
P
𝑚∙4π2 𝑎2
𝑎𝑃2 =
4π2
𝑚𝑎
𝑃2
4𝜋2 𝑎3
𝐺𝑃2

15. PRACTICE -- MASS OF THE SUN FROM EARTH’S ORBIT
2018/9/6
2
32
GP
a4
M


Second page
Earth’s A≒1.5×1011(m)
P≒3×107(sec)
G≒7×10-11 π2≒9
𝑀 =
4 ∙ 9 ∙
3
2
3
× 1033
7 × 10−11 ∙ 9 × 10143
30
=
4 ∙ 33 × 1030
23 ∙ 7
=
27
14
× 1030
≒ 2 × 1030
(kg)

16. BLACK HOLE IN THE CENTER OF THE OUR GALAXY
Sgr A*
Stars
revolve.
2018/9/6

17. BLACK HOLE IN THE CENTER OF THE GALAXY
Attention to the
star S2.
Revolution period
+
Orbit radius
→Celestial mass
Let's find it.
2018/9/6

18. IMAGE CORRECTION
Orbital inclination angle
as viewed from the Earth
of S2 is 45°.
This image has the
orbital inclination angle
corrected to 90°.
Therefore, it is
determined only by
measuring the orbital
radius with a ruler.

19. FIND ORBITAL RADIUS
Just measure it with
a ruler.
scale
8.9
0.195×1014m→1.2cm
1.45×1014

20. FIND REVORUTION PERIOD
The rule of thumb OK!
Apastoron → Periastron
is half period.
Periastron of BH
Apastron of BH
About 1:2(2002.33-2002.25)/3*2
=0.05
A: 2002.25+0.05
2002.30
About 1:3
1994.62
15.4 4.8×108
A
B

21. FIND MASS OF THE BLACK HOLE
2018/9/6
2
32
GP
a4
M


S2’s a≒1.45×1014(m)
P≒4.8×108(sec)
G≒7×10-11 π2≒9
𝑀 =
4 ∙ 9 ∙ 1.453 × 1014×3
7 × 10−11 ∙ 4.82 × 108×2
= 0.68 × 1037
= 6.8 × 1036
(kg)
=
4 ∙ 9 ∙ 1.453
7 ∙ 4.82
× 1037

22. Literature value : 4.1× 106M☉
Same order → Very good! in astronomy.
BH mass led with easy task and simple
calculation without science calculator.
FIND MASS OF THE BLACK HOLE
6.8 × 1036
3.4 × 106

23. EDUCATIONAL IMPLEMENTATION
At Yokohama Science
Frontier High School.
Teacher: Mitsuhiro Ishida.
July 20, 2016; 29students,
12th grade, physics class.
September 12, 2016;
18students, 12th grade,
physics class.
Using 1 school hour.
YSF

24. QUESTIONNAIRE RESULTS
WAS THIS ACTIVITY INTERESTING?

25. DID YOU UNDERSTAND THIS CONTENTS?

26. HAVE YOU GROWN INTERESTED BY
DEALING WITH THE BLACK HOLE?

27. DID YOU GET INTERESTED BY HANDLING THE
ACTUAL DATA USED FOR ASTRONOMICAL
RESEARCH?

28. QUESTIONNAIRE RESULTS
FREE DESCRIPTION
Was funny.
While listening to the explanation that the
teacher seemed to be having fun, I thought.
I felt a little less time.
The value is big.
I wanted a little more explanation.
Negative Answer 18%
 More much time, more easy.

29. USING SCIENTIFIC CALCULATOR
Initially, the practice was designed to use a scientific calculator.
These are slides explaining how to use a scientific calculator.

30. STUDENT PRACTICE TO
FIND MASS OF BLACK HOLE
This student practice has been simplified based on
reflection of implementation.
Please modify this worksheet and use it.
I expect that my worksheet will be held somewhere
in the world.
I'd like to finish by thanking you for listening.
Worksheet Lecture Slides This Presentation