1
0:00:02 --> 00:00:08
The speed of sound is
340 meters per second--

2
00:00:12 --> 00:00:18
it depends a little bit
on the temperature--

3
00:00:14 --> 00:00:20
about 770 miles per hour.

4
00:00:17 --> 00:00:23
When I speak to you, my sound
reaches you with that speed.

5
00:00:20 --> 00:00:26
I produce
a certain frequency here,

6
00:00:24 --> 00:00:30
a certain number
of oscillations per second.

7
00:00:27 --> 00:00:33
They reach you,

8
00:00:28 --> 00:00:34
your eardrum starts to oscillate
with the same frequency

9
00:00:30 --> 00:00:36
and you hear that tone.

10
00:00:32 --> 00:00:38
I have here a tuning fork

11
00:00:34 --> 00:00:40
which oscillates
440 times per second.

12
00:00:37 --> 00:00:43
(tuning fork produces
medium-pitched tone )

13
00:00:38 --> 00:00:44
Your eardrum oscillates
440 times per second--

14
00:00:44 --> 00:00:50
you hear this tone.

15
00:00:46 --> 00:00:52
Here I have
256 oscillations per second.

16
00:00:50 --> 00:00:56
(metal rod emits lower tone )

17
00:00:53 --> 00:00:59
Your eardrum is now shaking,

18
00:00:56 --> 00:01:02
going back and forth
256 times per second.

19
00:00:59 --> 00:01:05
If you stay where you are
and you don't move

20
00:01:02 --> 00:01:08
and I move these tuning forks,

21
00:01:04 --> 00:01:10
you will hear
a different frequency

22
00:01:06 --> 00:01:12
and that's what we call
Doppler effect.

23
00:01:09 --> 00:01:15
If my sound source
approaches you,

24
00:01:15 --> 00:01:21
you will hear a frequency
f prime which is larger

25
00:01:18 --> 00:01:24
than the frequency
of the tuning fork.

26
00:01:21 --> 00:01:27
If it moves away from you,
which I will call receding,

27
00:01:27 --> 00:01:33
then f prime equals
lower... lower frequency.

28
00:01:33 --> 00:01:39
For instance, I move to you
a sound source--

29
00:01:37 --> 00:01:43
I call that a transmitter--

30
00:01:39 --> 00:01:45
with a speed of
about one meters per second.

31
00:01:46 --> 00:01:52
Transmitter is
the sound transmitter.

32
00:01:49 --> 00:01:55
Then if it approaches you here,
you will hear f prime,

33
00:01:55 --> 00:02:01
which is 1.003 times f.

34
00:01:58 --> 00:02:04
This three here is
the one part out of 340

35
00:02:03 --> 00:02:09
that you get an increase
in frequency.

36
00:02:06 --> 00:02:12
If I move it away from you,
then f prime would be

37
00:02:09 --> 00:02:15
0.997 times the frequency
of the source itself.

38
00:02:17 --> 00:02:23
You stay where you are.

39
00:02:19 --> 00:02:25
I have here a tuning fork
which generates 4,000 hertz,

40
00:02:24 --> 00:02:30
a very high frequency.

41
00:02:26 --> 00:02:32
If I move it to you with the
speed of one meter per second,

42
00:02:29 --> 00:02:35
which I can do, then you get
an increase in pitch of 0.3%.

43
00:02:33 --> 00:02:39
That makes it 4,012 hertz.

44
00:02:35 --> 00:02:41
And when I move it away from you
there is a decrease of 0.3%.

45
00:02:38 --> 00:02:44
And you can clearly hear
that difference.

46
00:02:40 --> 00:02:46
I will first make you listen
to the 4,000 hertz

47
00:02:43 --> 00:02:49
without my moving.

48
00:02:45 --> 00:02:51
(tuning fork produces
very high pitched tone )

49
00:02:47 --> 00:02:53
Can you hear it?
Very high frequency.

50
00:02:49 --> 00:02:55
Is it painful, really?

51
00:02:51 --> 00:02:57
High frequency.

52
00:02:53 --> 00:02:59
Most of you are young enough

53
00:02:54 --> 00:03:00
you should be able
to hear 4,000 hertz.

54
00:02:56 --> 00:03:02
Okay, now I am going to move it
to you one meter per second

55
00:02:59 --> 00:03:05
and away from you.

56
00:03:00 --> 00:03:06
(high tone goes up and down
slightly in pitch )

57
00:03:05 --> 00:03:11
Did you hear it? Once more.

58
00:03:08 --> 00:03:14
(tone goes up and down
quickly again )

59
00:03:12 --> 00:03:18
(class laughs )

60
00:03:13 --> 00:03:19
When it comes to you, it's clear
that the frequency goes up,

61
00:03:17 --> 00:03:23
and when it moves away from you,
the frequency is down.

62
00:03:21 --> 00:03:27
Now imagine that I'm going
to rotate the sound source

63
00:03:26 --> 00:03:32
around in a circle.

64
00:03:27 --> 00:03:33
Now the sound that you receive,
the frequency that you receive

65
00:03:31 --> 00:03:37
will change
in a sinusoidal fashion.

66
00:03:35 --> 00:03:41
If this is that circle, and this
is the radius of that circle,

67
00:03:41 --> 00:03:47
and if you are here
in the plane of the circle,

68
00:03:48 --> 00:03:54
then when the source comes
straight to you

69
00:03:52 --> 00:03:58
with the velocity v-- let's say
it's a uniform circular motion--

70
00:03:56 --> 00:04:02
f prime will be larger than f

71
00:03:58 --> 00:04:04
and it will, in this case,
reach a maximum.

72
00:04:02 --> 00:04:08
When it is at 90 degrees
relative to you--

73
00:04:05 --> 00:04:11
I don't have to give it
a vector notation--

74
00:04:09 --> 00:04:15
f prime equals f.

75
00:04:10 --> 00:04:16
When it moves away from you,
f prime is smaller than f,

76
00:04:15 --> 00:04:21
you hear a minimum.

77
00:04:17 --> 00:04:23
And when it is here again--

78
00:04:19 --> 00:04:25
when the angle between the
velocity and your direction

79
00:04:22 --> 00:04:28
is again 90 degrees--
then f prime equals f again.

80
00:04:27 --> 00:04:33
And so this phenomenon
is called the Doppler effect.

81
00:04:35 --> 00:04:41
So if I twirl it around,

82
00:04:37 --> 00:04:43
you will hear a sinusoidal
fluctuation in f prime.

83
00:04:43 --> 00:04:49
Suppose I plot,
as a function of time, f prime

84
00:04:50 --> 00:04:56
the way you will receive it--
you sit still,

85
00:04:52 --> 00:04:58
but I'm going to move the
sound source around like this.

86
00:04:56 --> 00:05:02
Then you will have a curve
that looks something like this:

87
00:04:59 --> 00:05:05
some sinusoidal-cosinusoidal
fluctuation of f prime.

88
00:05:05 --> 00:05:11
This will be the value f

89
00:05:09 --> 00:05:15
produced by
the sound source itself.

90
00:05:11 --> 00:05:17
This will be f prime maximum and
this will be f prime minimum.

91
00:05:24 --> 00:05:30
If you could record this, there
is an amazing number of things

92
00:05:27 --> 00:05:33
that you can deduce
from this curve.

93
00:05:31 --> 00:05:37
First of all, you can take...

94
00:05:34 --> 00:05:40
You can measure
f prime max divided by F,

95
00:05:39 --> 00:05:45
because you see this curve,
so you know what f is here,

96
00:05:41 --> 00:05:47
you see what f prime max is,

97
00:05:43 --> 00:05:49
and that should allow you
to retrieve immediately

98
00:05:46 --> 00:05:52
v velocity
of the transmitter.

99
00:05:48 --> 00:05:54
If that number were 1.003,
then you know

100
00:05:53 --> 00:05:59
that the speed in the orbit
was one meter per second.

101
00:05:56 --> 00:06:02
So this ratio immediately gives
you the transmitter velocity.

102
00:06:00 --> 00:06:06
This time separation
gives you immediately

103
00:06:05 --> 00:06:11
the period of rotation,

104
00:06:07 --> 00:06:13
but since two pi R--
if R is the radius

105
00:06:13 --> 00:06:19
divided by the velocity
of the transmitter--

106
00:06:16 --> 00:06:22
since that is the... oh, I can
reverse it, it doesn't matter.

107
00:06:20 --> 00:06:26
Two pi R divided
by the time to go around

108
00:06:24 --> 00:06:30
is the velocity
of the transmitter.

109
00:06:26 --> 00:06:32
Since you know the velocity of
the transmitter from this ratio

110
00:06:30 --> 00:06:36
since you know the period,
which is this,

111
00:06:32 --> 00:06:38
you now also find the radius R.

112
00:06:35 --> 00:06:41
So from that curve--
and keep that with you,

113
00:06:37 --> 00:06:43
because it's going to be
important in what follows--

114
00:06:40 --> 00:06:46
we can derive three things:

115
00:06:42 --> 00:06:48
the radius,
the period of rotation

116
00:06:46 --> 00:06:52
and the speed of the object
as I twirl it around.

117
00:06:52 --> 00:06:58
I have here what we call
a wind organ.

118
00:06:55 --> 00:07:01
When I twirl this around,
it produces a particular tone.

119
00:06:59 --> 00:07:05
We will talk later about 801

120
00:07:01 --> 00:07:07
why it produces
a particular tone.

121
00:07:03 --> 00:07:09
Sometimes you hear two tones.

122
00:07:05 --> 00:07:11
I'll try to make you hear
only one.

123
00:07:07 --> 00:07:13
And as I swirl it around,
the sound is coming...

124
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the sound source, the
transmitter is coming to you.

125
00:07:15 --> 00:07:21
This, when it goes like this,
it's 90-degree angle

126
00:07:17 --> 00:07:23
so you should not hear
any Doppler shift.

127
00:07:20 --> 00:07:26
When it is here,
it's moved away from you

128
00:07:23 --> 00:07:29
and so you hear
a sinusoidal change in f prime.

129
00:07:27 --> 00:07:33
Try to hear that.

130
00:07:28 --> 00:07:34
(wind organ producing tone
that changes pitch )

131
00:07:36 --> 00:07:42
Can you hear, when it's coming
to you, that it's higher-pitched

132
00:07:38 --> 00:07:44
than when it's going
away from you?

133
00:07:40 --> 00:07:46
Can you hear that?

134
00:07:41 --> 00:07:47
Just say no
if you don't hear it.

135
00:07:43 --> 00:07:49
Not very clear.

136
00:07:44 --> 00:07:50
(wind organ again producing
varying tone )

137
00:07:51 --> 00:07:57
For me, it's impossible to hear

138
00:07:52 --> 00:07:58
because I'm standing
right under it, of course.

139
00:07:54 --> 00:08:00
Well, I tried.

140
00:07:57 --> 00:08:03

141
00:08:00 --> 00:08:06
I now want to change
to electromagnetic waves.

142
00:08:03 --> 00:08:09
Electromagnetic waves travel
with the speed of light,

143
00:08:07 --> 00:08:13
which is 300,000 kilometers
per second.

144
00:08:09 --> 00:08:15
And if you want to treat
that correctly,

145
00:08:11 --> 00:08:17
you would have to use
special relativity.

146
00:08:15 --> 00:08:21
In the case of sound,
I stressed repeatedly

147
00:08:19 --> 00:08:25
that you in the audience
should not move

148
00:08:22 --> 00:08:28
but that the sound source
is moving.

149
00:08:25 --> 00:08:31
In the case
of electromagnetic radiation

150
00:08:27 --> 00:08:33
when you deal
with the speed of light,

151
00:08:29 --> 00:08:35
you don't have to ask
that question.

152
00:08:31 --> 00:08:37
It is a meaningless question
in special relativity.

153
00:08:33 --> 00:08:39
To ask whether you are moving
relative to me

154
00:08:36 --> 00:08:42
or whether I am moving relative
to you, it doesn't matter.

155
00:08:39 --> 00:08:45
All that matters
in special relativity

156
00:08:41 --> 00:08:47
is the relative motion,

157
00:08:42 --> 00:08:48
so you can always think
of yourself as standing still

158
00:08:46 --> 00:08:52
and make the source
of electromagnetic radiation

159
00:08:49 --> 00:08:55
move to you, or away from you,
relative to you.

160
00:08:53 --> 00:08:59
Electromagnetic radiation
is optical light, infrared,

161
00:08:56 --> 00:09:02
ultraviolet, radio,
x-rays, gamma rays.

162
00:08:59 --> 00:09:05
All of that is
electromagnetic radiation.

163
00:09:03 --> 00:09:09
If the velocity of the source
of electromagnetic radiation--

164
00:09:08 --> 00:09:14
the transmitter--

165
00:09:09 --> 00:09:15
if that is way, way smaller
than the speed of light,

166
00:09:13 --> 00:09:19
then it is very easy to predict

167
00:09:16 --> 00:09:22
the change in frequency
due to Doppler shift.

168
00:09:20 --> 00:09:26
Let this be the transmitter
which produces frequency f,

169
00:09:26 --> 00:09:32
and here is the receiver which
receives the frequency f prime.

170
00:09:33 --> 00:09:39
And let the velocity

171
00:09:37 --> 00:09:43
of the source of
electromagnetic radiation be v--

172
00:09:43 --> 00:09:49
I could put transmitter here,
but we can drop that index--

173
00:09:47 --> 00:09:53
and let this angle be theta.

174
00:09:49 --> 00:09:55
Then this is the component
in your direction--

175
00:09:54 --> 00:10:00
we call that
the radial component--

176
00:09:56 --> 00:10:02
which is v cosine theta.

177
00:09:59 --> 00:10:05
So I delete the tr.

178
00:10:00 --> 00:10:06
This is just the velocity
of the source

179
00:10:04 --> 00:10:10
relative to you at that angle.

180
00:10:06 --> 00:10:12

181
00:10:09 --> 00:10:15
If now we want to know
what f prime is,

182
00:10:14 --> 00:10:20
then f prime equals f
times one plus v over c

183
00:10:20 --> 00:10:26
times the cosine of theta.

184
00:10:23 --> 00:10:29
What matters is only the radial
component of the velocity.

185
00:10:30 --> 00:10:36
This is the radial component.

186
00:10:32 --> 00:10:38
If theta is 90 degrees,
just like we had with sound,

187
00:10:36 --> 00:10:42
then f prime equals f.

188
00:10:37 --> 00:10:43
So 90 degrees, the cosine
of theta is zero,

189
00:10:41 --> 00:10:47
f prime equals f.

190
00:10:43 --> 00:10:49
If theta is smaller
than 90 degrees,

191
00:10:47 --> 00:10:53
then it's coming towards you,

192
00:10:51 --> 00:10:57
then f prime equals
larger than f.

193
00:10:55 --> 00:11:01
If theta equals larger
than 90 degrees,

194
00:10:58 --> 00:11:04
it's going away from you,

195
00:11:01 --> 00:11:07
then f prime equals
smaller than f.

196
00:11:05 --> 00:11:11

197
00:11:07 --> 00:11:13
You would get
a similar equation for sound

198
00:11:10 --> 00:11:16
by replacing this c
by the speed of sound.

199
00:11:14 --> 00:11:20
But I want to stress

200
00:11:15 --> 00:11:21
that this only holds for
electromagnetic radiation

201
00:11:19 --> 00:11:25
if v over c is much,
much smaller than one.

202
00:11:23 --> 00:11:29

203
00:11:29 --> 00:11:35
Now, when we deal with sound,

204
00:11:31 --> 00:11:37
there is something
mechanically oscillating.

205
00:11:33 --> 00:11:39
Something is vibrating.

206
00:11:36 --> 00:11:42
With electromagnetic radiation,
charges are vibrating.

207
00:11:39 --> 00:11:45
Electrons are vibrating,

208
00:11:41 --> 00:11:47
and they are vibrating
with a certain frequency,

209
00:11:44 --> 00:11:50
and that means

210
00:11:46 --> 00:11:52
there is a certain period
of one oscillation.

211
00:11:49 --> 00:11:55
And that period of one
oscillation is, of course,

212
00:11:54 --> 00:12:00
one over the frequency.

213
00:11:57 --> 00:12:03
I can ask myself now
the question,

214
00:11:59 --> 00:12:05
how far does
electromagnetic radiation,

215
00:12:01 --> 00:12:07
how far does light travel in the
time of one period capital T?

216
00:12:07 --> 00:12:13
Well, it goes
with the speed of light,

217
00:12:08 --> 00:12:14
so in T seconds,
it moves a distance cT.

218
00:12:14 --> 00:12:20
And that distance we call

219
00:12:16 --> 00:12:22
the wavelength
of electromagnetic radiation,

220
00:12:19 --> 00:12:25
lambda equals cT, for which you
can also write c divided by F.

221
00:12:24 --> 00:12:30
So this is the wavelength of
the electromagnetic radiation--

222
00:12:27 --> 00:12:33
the speed of light,
300,000 kilometers per second--

223
00:12:30 --> 00:12:36
the period of one oscillation,
say, of the electrons,

224
00:12:33 --> 00:12:39
and this is the frequency,
which you can give in hertz.

225
00:12:38 --> 00:12:44
I could give you
a specific example.

226
00:12:42 --> 00:12:48
I, for instance,
can take a period T

227
00:12:45 --> 00:12:51
of two times
ten to the minus 15 seconds.

228
00:12:53 --> 00:12:59
That would give me a wavelength

229
00:12:55 --> 00:13:01
of about six times
ten to the minus seven meters--

230
00:12:59 --> 00:13:05
six times ten
to the minus seven meters--

231
00:13:02 --> 00:13:08
and that you would experience
as red light.

232
00:13:06 --> 00:13:12
If I make the period shorter--

233
00:13:09 --> 00:13:15
say, 1.3 times
ten to the minus 15 seconds--

234
00:13:14 --> 00:13:20
I get a shorter wavelength.

235
00:13:16 --> 00:13:22
I get four times
ten to the minus seven meters,

236
00:13:21 --> 00:13:27
and you would experience that
as blue light.

237
00:13:26 --> 00:13:32

238
00:13:29 --> 00:13:35
In astronomy,
in optical astronomy

239
00:13:31 --> 00:13:37
we cannot measure the period or
the frequency of optical light.

240
00:13:36 --> 00:13:42
All we can measure
is the wavelength.

241
00:13:39 --> 00:13:45
And so if I want
to use this equation,

242
00:13:42 --> 00:13:48
then I have to replace f prime
by c divided by lambda prime

243
00:13:49 --> 00:13:55
and f I have to replace
by c divided by lambda.

244
00:13:55 --> 00:14:01
And when I do that,
I get the following result.

245
00:13:58 --> 00:14:04
I get lambda prime equals lambda

246
00:14:02 --> 00:14:08
times one minus
v over c cosine theta.

247
00:14:10 --> 00:14:16
If this is a plus,
this is a minus.

248
00:14:12 --> 00:14:18
Check that for yourself.

249
00:14:14 --> 00:14:20
You have to use
the small number approximation,

250
00:14:17 --> 00:14:23
the Taylor expansion,

251
00:14:19 --> 00:14:25
namely that v over c is
much, much smaller than one.

252
00:14:24 --> 00:14:30
So you can see now
if the object comes to you,

253
00:14:30 --> 00:14:36
in other words, if
f prime is larger than f,

254
00:14:34 --> 00:14:40
if the frequency is higher, then
the wavelength will be smaller.

255
00:14:41 --> 00:14:47
And so let me write that down.

256
00:14:43 --> 00:14:49
When the cosine of theta...
so the object is coming to you--

257
00:14:47 --> 00:14:53
when the cosine of theta
is larger than zero,

258
00:14:50 --> 00:14:56
the object is approaching you,

259
00:14:54 --> 00:15:00
then the wavelength lambda prime
will be less than lambda.

260
00:15:00 --> 00:15:06
And that has a name--
we call that blue shift.

261
00:15:04 --> 00:15:10
And the reason why we call
that blue shift is

262
00:15:06 --> 00:15:12
that if the wavelengths
become shorter,

263
00:15:08 --> 00:15:14
it moves towards
the blue end of the spectrum,

264
00:15:11 --> 00:15:17
because blue has
a lower wavelength than red.

265
00:15:15 --> 00:15:21
If cosine theta is negative,

266
00:15:20 --> 00:15:26
then the object
is receding from you,

267
00:15:25 --> 00:15:31
then lambda prime is larger
than lambda,

268
00:15:29 --> 00:15:35
and we call that red shift.

269
00:15:33 --> 00:15:39
These are the terms that
astronomers use all the time.

270
00:15:39 --> 00:15:45
When you make a spectrum
of a star--

271
00:15:42 --> 00:15:48
you can do that using prisms
or by other means--

272
00:15:47 --> 00:15:53
and you look
at the light intensity

273
00:15:49 --> 00:15:55
as a function of wavelength--

274
00:15:51 --> 00:15:57
so here is the light intensity
as a function of wavelength--

275
00:15:58 --> 00:16:04
then you may expect to see
some kind of a continuum.

276
00:16:00 --> 00:16:06
But, in fact,
what you do see is...

277
00:16:03 --> 00:16:09
Superimposed on a continuum

278
00:16:05 --> 00:16:11
you see sometimes
very sharp absorption lines--

279
00:16:09 --> 00:16:15
black, missing light,
called absorption lines

280
00:16:13 --> 00:16:19
And these absorption lines
correspond

281
00:16:16 --> 00:16:22
to elements in the atmosphere
of the star.

282
00:16:20 --> 00:16:26
In fact, if you see the
absorption lines, you can tell

283
00:16:23 --> 00:16:29
what kind of elements
are present in the star.

284
00:16:27 --> 00:16:33
Some are very characteristic
absorption lines:

285
00:16:29 --> 00:16:35
some for hydrogen, some
for calcium, some for silicon,

286
00:16:31 --> 00:16:37
some for magnesium and so on.

287
00:16:33 --> 00:16:39
It's actually interesting

288
00:16:35 --> 00:16:41
that when you look
at the spectrum of the sun--

289
00:16:38 --> 00:16:44
when people did that first,

290
00:16:40 --> 00:16:46
when they had the means
of doing that,

291
00:16:43 --> 00:16:49
they found absorption lines
in the spectrum of the sun

292
00:16:46 --> 00:16:52
which could not be identified.

293
00:16:48 --> 00:16:54
They had never been seen here
on Earth, these lines,

294
00:16:50 --> 00:16:56
and so they called these lines
after the sun.

295
00:16:53 --> 00:16:59
The sun is Helios,
and so they called it helium.

296
00:16:56 --> 00:17:02
So helium was
first discovered on the sun

297
00:16:58 --> 00:17:04
before it was
later discovered on Earth

298
00:17:01 --> 00:17:07
by looking at the absorption
lines of the solar spectrum.

299
00:17:07 --> 00:17:13
If a star moves to you,
then all the lines--

300
00:17:11 --> 00:17:17
every single line--
will be blue-shifted.

301
00:17:13 --> 00:17:19
And if the star moves
away from you,

302
00:17:15 --> 00:17:21
all the lines will be
red-shifted.

303
00:17:18 --> 00:17:24
If you take an example:

304
00:17:19 --> 00:17:25
With lambda prime
divided by lambda

305
00:17:21 --> 00:17:27
and you pick
any one of those lines--

306
00:17:24 --> 00:17:30
it doesn't matter
which you pick

307
00:17:25 --> 00:17:31
because they will all do
exactly the same.

308
00:17:27 --> 00:17:33
If this were,
for instance, 1.00333--

309
00:17:32 --> 00:17:38
I just pick a very nice number;

310
00:17:34 --> 00:17:40
that means lambda prime, as
you see, is larger than lambda;

311
00:17:39 --> 00:17:45
the wavelengths get longer,
so we have a red shift--

312
00:17:43 --> 00:17:49
and you substitute that
in that equation,

313
00:17:46 --> 00:17:52
then you'll find
that the velocity

314
00:17:48 --> 00:17:54
at which that star is moving
relative to you--

315
00:17:51 --> 00:17:57
that gives you
immediately the answer there--

316
00:17:54 --> 00:18:00
equals minus 0.00333
times the speed of light, c,

317
00:18:03 --> 00:18:09
and that is
minus 100 kilometers per second.

318
00:18:08 --> 00:18:14
And the minus, then, reminds you

319
00:18:11 --> 00:18:17
that the object
is receding from you.

320
00:18:14 --> 00:18:20
So that gives you a red shift.

321
00:18:16 --> 00:18:22

322
00:18:17 --> 00:18:23
I just wrote down

323
00:18:19 --> 00:18:25
that the velocity v is
minus 100 kilometers per second.

324
00:18:22 --> 00:18:28
It's, of course, v cosine theta

325
00:18:24 --> 00:18:30
that is minus 100
kilometers per second.

326
00:18:27 --> 00:18:33
It's theradial velocity--
that's all you can measure.

327
00:18:31 --> 00:18:37
You have
no information on theta.

328
00:18:33 --> 00:18:39
So it is this component,
v cosine theta--

329
00:18:37 --> 00:18:43
which we call
the radial velocity--

330
00:18:40 --> 00:18:46
that is minus 100 kilometers
per second.

331
00:18:43 --> 00:18:49

332
00:18:44 --> 00:18:50
Half of all the stars
in the sky are binaries,

333
00:18:48 --> 00:18:54
and so when you look
at the spectra,

334
00:18:51 --> 00:18:57
you will see them go
around each other.

335
00:18:54 --> 00:19:00
And so you, in principle,
can measure the red shifts

336
00:18:58 --> 00:19:04
and the blue shifts
as they go around each other.

337
00:19:01 --> 00:19:07
You see Doppler effect.

338
00:19:02 --> 00:19:08
If they come to you,
you see blue shift.

339
00:19:04 --> 00:19:10
If they go away from you,
you see red shift.

340
00:19:07 --> 00:19:13
So, in principle,

341
00:19:08 --> 00:19:14
you can determine
for each one of those stars

342
00:19:11 --> 00:19:17
the velocity in orbit,
the radius of their orbit

343
00:19:14 --> 00:19:20
and, of course, the period
of the binary system.

344
00:19:18 --> 00:19:24
So it's an extremely
powerful tool in astronomy

345
00:19:21 --> 00:19:27
if you have a binary system,
when the stars exactly do this,

346
00:19:24 --> 00:19:30
to determine

347
00:19:26 --> 00:19:32
all these quantities
that you would like to know.

348
00:19:30 --> 00:19:36
I first would like to show you
now some slides.

349
00:19:34 --> 00:19:40

350
00:19:41 --> 00:19:47
The first slide... oh, I have
to lower the screen, by the way.

351
00:19:47 --> 00:19:53
That would help, wouldn't it?

352
00:19:48 --> 00:19:54
(chuckles )

353
00:19:50 --> 00:19:56
The first slide is a spectrum
made in the laboratory

354
00:19:56 --> 00:20:02
of hydrogen, helium
and calcium and sodium.

355
00:20:01 --> 00:20:07
It shows you emission lines,
no absorption lines.

356
00:20:05 --> 00:20:11
These lines are produced
by lamps,

357
00:20:07 --> 00:20:13
and the frequencies
are very well known.

358
00:20:11 --> 00:20:17
Here you see
the famous sodium yellow lines.

359
00:20:13 --> 00:20:19
So here is the red part
of the spectrum

360
00:20:15 --> 00:20:21
and there is the blue part
of the spectrum.

361
00:20:17 --> 00:20:23

362
00:20:19 --> 00:20:25
So we know these frequencies,

363
00:20:21 --> 00:20:27
we know these wavelengths
very well.

364
00:20:23 --> 00:20:29
And here you see
the spectrum of the sun

365
00:20:26 --> 00:20:32
with all these absorption lines
that I mentioned to you.

366
00:20:29 --> 00:20:35
It's plastered
with absorption lines,

367
00:20:31 --> 00:20:37
and each of them
can be identified.

368
00:20:34 --> 00:20:40
These are due to calcium,
iron, hydrogen and so on.

369
00:20:37 --> 00:20:43
Here is the blue part
of the spectrum,

370
00:20:39 --> 00:20:45
here is the green part,
the green part,

371
00:20:42 --> 00:20:48
and here is the red part
of the spectrum.

372
00:20:44 --> 00:20:50

373
00:20:46 --> 00:20:52
And here you see the basic idea
behind a binary system.

374
00:20:53 --> 00:20:59
Suppose you have a binary system

375
00:20:55 --> 00:21:01
that only one star is visible
and the other one is invisible

376
00:20:58 --> 00:21:04
and the one star shows you
three clear absorption lines.

377
00:21:02 --> 00:21:08
Then as the star moves
around the center of mass,

378
00:21:05 --> 00:21:11
you see that all these lines
drift in unison.

379
00:21:08 --> 00:21:14
And out of this information

380
00:21:09 --> 00:21:15
you get the radius,
the velocity and the period,

381
00:21:12 --> 00:21:18
assuming that you are on Earth

382
00:21:15 --> 00:21:21
in the plane of the orbit
of the stars.

383
00:21:17 --> 00:21:23
If you have a binary system
whereby both stars are visible

384
00:21:22 --> 00:21:28
so you get
the spectrum of both stars,

385
00:21:25 --> 00:21:31
then you see the Doppler shift
of both stars in the spectrum.

386
00:21:29 --> 00:21:35
Here we have a simple case

387
00:21:30 --> 00:21:36
that we only have two absorption
lines, not to confuse the issue,

388
00:21:34 --> 00:21:40
and so in one...
in the case of one star,

389
00:21:36 --> 00:21:42
the shift will be towards
the left of the two lines,

390
00:21:39 --> 00:21:45
but the other star, the shift
will be to the right,

391
00:21:42 --> 00:21:48
because if you have
a binary system

392
00:21:44 --> 00:21:50
when one star comes to you,

393
00:21:45 --> 00:21:51
the other star goes away
from you, and vice versa.

394
00:21:48 --> 00:21:54
So now you are very lucky,
now you have an ideal situation

395
00:21:52 --> 00:21:58
that you can find for both stars
the radius of the orbit,

396
00:21:56 --> 00:22:02
the velocity in orbit
and the period for each star,

397
00:22:00 --> 00:22:06
which, of course,
is the same for both.

398
00:22:03 --> 00:22:09

399
00:22:05 --> 00:22:11
And here you see real data.

400
00:22:07 --> 00:22:13
You see here, first of all,

401
00:22:08 --> 00:22:14
the emission lines which are
measured in the laboratory

402
00:22:12 --> 00:22:18
that I just showed you.

403
00:22:13 --> 00:22:19
They are always done
simultaneously

404
00:22:15 --> 00:22:21
with the measurements.

405
00:22:16 --> 00:22:22
You always must be sure

406
00:22:18 --> 00:22:24
that you have a good calibration
of your wavelength.

407
00:22:22 --> 00:22:28
And this spectrum a,
the top spectrum, is of a star,

408
00:22:25 --> 00:22:31
a binary system, that has
a period of 20.5 days.

409
00:22:28 --> 00:22:34
And you see here single lines,
if you have good eyes.

410
00:22:32 --> 00:22:38
That means at this very moment

411
00:22:34 --> 00:22:40
both stars move relative to you
at angles of 90 degrees,

412
00:22:37 --> 00:22:43
so you don't see
any Doppler shift.

413
00:22:39 --> 00:22:45
But now look here.

414
00:22:41 --> 00:22:47
Later in time, you see that
this line has split in two lines

415
00:22:45 --> 00:22:51
and this one has also split
in two lines.

416
00:22:47 --> 00:22:53
Clearly, one component
is coming to you

417
00:22:50 --> 00:22:56
and the other component
is moving away from you.

418
00:22:53 --> 00:22:59
And so you get all this
useful information in astronomy

419
00:22:58 --> 00:23:04
by making the Doppler shift
measurements of binary systems.

420
00:23:04 --> 00:23:10

421
00:23:11 --> 00:23:17
I want to pursue the idea
of binary stars.

422
00:23:16 --> 00:23:22
They give us

423
00:23:18 --> 00:23:24
not only the information that
we want regarding the orbits,

424
00:23:23 --> 00:23:29
but there is even more
that we can get out of it

425
00:23:27 --> 00:23:33
which is even more exciting.

426
00:23:29 --> 00:23:35
So I will remind you what
a binary system looks like.

427
00:23:33 --> 00:23:39

428
00:23:38 --> 00:23:44
Remember the second exam.

429
00:23:42 --> 00:23:48
I'm sure you will never forget
that second exam

430
00:23:45 --> 00:23:51
and maybe never forgive me
for that.

431
00:23:47 --> 00:23:53

432
00:23:49 --> 00:23:55
Binary system: star one,
radius r1, mass m1, velocity v1,

433
00:24:02 --> 00:24:08
and star two-- going about
their common center of mass--

434
00:24:12 --> 00:24:18
mass m2, radius r2
and velocity v2.

435
00:24:22 --> 00:24:28
m1 r1 equals m2 r2.

436
00:24:27 --> 00:24:33
That's the way the center
of mass is defined.

437
00:24:30 --> 00:24:36
Imagine that you as an observer

438
00:24:33 --> 00:24:39
are somewhere in the plane of
this orbit, and you are here.

439
00:24:37 --> 00:24:43
And you are observing
the system going around.

440
00:24:42 --> 00:24:48
Kepler's Third Law,
which you derived on your exam

441
00:24:48 --> 00:24:54
as well as on an assignment:
the period squared equals

442
00:24:54 --> 00:25:00
four pi squared times r1 plus r2
to the power three.

443
00:25:00 --> 00:25:06
divided by G times m1 plus m2.

444
00:25:07 --> 00:25:13
Let me check that to make sure
I have that right.

445
00:25:08 --> 00:25:14
Yes, that is correct.

446
00:25:10 --> 00:25:16
Imagine now you can make

447
00:25:12 --> 00:25:18
the Doppler shift measurements
of both stars.

448
00:25:15 --> 00:25:21
You make the Doppler shift
measurement of star number one,

449
00:25:18 --> 00:25:24
so you measure lambda one prime
as a function of time.

450
00:25:22 --> 00:25:28
Out of that pops immediately
the period of rotation.

451
00:25:26 --> 00:25:32
Out of that pops the velocity,
as we discussed.

452
00:25:29 --> 00:25:35
Out of that pops the radius, r1.

453
00:25:32 --> 00:25:38
And now you measure

454
00:25:33 --> 00:25:39
the Doppler shift of star two
as a function of time.

455
00:25:38 --> 00:25:44
Out of that pops the period

456
00:25:40 --> 00:25:46
which, of course,
better be the same.

457
00:25:42 --> 00:25:48
Out of that pops
its velocity in orbit

458
00:25:44 --> 00:25:50
and out of it pops its radius.

459
00:25:47 --> 00:25:53
All these things come out of
the Doppler shift measurements,

460
00:25:50 --> 00:25:56
but if you know r1
and you know r2,

461
00:25:53 --> 00:25:59
then you also know r1 plus r2,

462
00:25:56 --> 00:26:02
so you know this part
in Kepler's Third Law.

463
00:26:01 --> 00:26:07
Since you also know the periods,
you can find what m1 plus m2 is.

464
00:26:06 --> 00:26:12
So now you get an extra bonus.

465
00:26:08 --> 00:26:14
You know now what the sum
of the mass of the two stars is

466
00:26:13 --> 00:26:19
in the binary system, but you
also know that m1 r1 is m2 r2.

467
00:26:18 --> 00:26:24
So now you have two equations.

468
00:26:20 --> 00:26:26
You know what m1 plus m2 is

469
00:26:22 --> 00:26:28
and you know this equation,
and you can solve for m1 and m2,

470
00:26:25 --> 00:26:31
which is an amazing thing
when you come to think of it.

471
00:26:28 --> 00:26:34
So we finally end up

472
00:26:30 --> 00:26:36
with the mass of star one
and the mass of star two.

473
00:26:33 --> 00:26:39
All this comes out
of Doppler shift measurements:

474
00:26:36 --> 00:26:42
the velocities,
the radii, the periods

475
00:26:39 --> 00:26:45
and even the masses
of these objects.

476
00:26:43 --> 00:26:49
Now, if you
as an observer on Earth

477
00:26:46 --> 00:26:52
are not exactly
in the plane of the orbit,

478
00:26:48 --> 00:26:54
then the situation is
a little bit more complicated,

479
00:26:51 --> 00:26:57
and I will not discuss
that here today

480
00:26:53 --> 00:26:59
because, in principle,

481
00:26:54 --> 00:27:00
it doesn't affect the idea
behind Doppler shift.

482
00:26:57 --> 00:27:03
But for astronomers,
it is very important.

483
00:26:59 --> 00:27:05
It's really a nuisance,

484
00:27:01 --> 00:27:07
but I will not discuss that
in any detail.

485
00:27:04 --> 00:27:10
I want to discuss
a fascinating application

486
00:27:06 --> 00:27:12
that we have in x-ray astronomy.

487
00:27:10 --> 00:27:16
Namely, we have x-ray binaries.

488
00:27:13 --> 00:27:19

489
00:27:16 --> 00:27:22
What is an x-ray binary?

490
00:27:18 --> 00:27:24
Well, it is a binary system.

491
00:27:21 --> 00:27:27
This is a star
not unlike our sun.

492
00:27:23 --> 00:27:29
It has as certain mass,
has a certain radius,

493
00:27:25 --> 00:27:31
and it is in orbit, let's say,
with a neutron star,

494
00:27:28 --> 00:27:34
even though it could be
a black hole.

495
00:27:30 --> 00:27:36
But for now, let's just assume
it is a neutron star.

496
00:27:33 --> 00:27:39
And if these two masses
are the same,

497
00:27:35 --> 00:27:41
which I only use
for the sake of simplicity--

498
00:27:38 --> 00:27:44
in practice,
they could be very different--

499
00:27:40 --> 00:27:46
then there is a point between
these two, right in the middle,

500
00:27:45 --> 00:27:51
whereby the gravitational pull
in one direction is the same

501
00:27:48 --> 00:27:54
as the gravitational pull
in the other direction.

502
00:27:51 --> 00:27:57
And we call that
the inner Lagrangian point.

503
00:27:54 --> 00:28:00
In other words,
if you were there,

504
00:27:56 --> 00:28:02
the neutron star would pull
at you

505
00:27:58 --> 00:28:04
with exactly the same force
as the other star.

506
00:28:01 --> 00:28:07
So you wouldn't know
where to go.

507
00:28:03 --> 00:28:09
If this inner
Legrangian point lies

508
00:28:07 --> 00:28:13
below the surface of this star,
that means

509
00:28:10 --> 00:28:16
if the stars are a little closer
than I have drawn them here,

510
00:28:13 --> 00:28:19
then the matter of this star
will fall

511
00:28:16 --> 00:28:22
towards the neutron star,

512
00:28:18 --> 00:28:24
because the pull in this
direction is, then, larger

513
00:28:21 --> 00:28:27
than the pull in this direction.

514
00:28:23 --> 00:28:29
Now, of course,
this system is a binary system.

515
00:28:26 --> 00:28:32
They go around in the plane
of the blackboard, say,

516
00:28:30 --> 00:28:36
and so this matter
cannot fall radially in

517
00:28:32 --> 00:28:38
but it will fall in
and spiral in

518
00:28:35 --> 00:28:41
and forms what we call
an accretion disk

519
00:28:40 --> 00:28:46
around the neutron star.

520
00:28:44 --> 00:28:50
This is called the accretor
and this is called the donor.

521
00:28:48 --> 00:28:54
There is mass transfer from
the donor to the neutron star.

522
00:28:52 --> 00:28:58
Oops, I just noticed I
misspelled the word "accretion."

523
00:28:56 --> 00:29:02
There is an "r" in "accretion."

524
00:28:58 --> 00:29:04
And as that occurs,

525
00:28:59 --> 00:29:05
there is a tremendous amount
of energy that is released.

526
00:29:03 --> 00:29:09
I want to blow up
the neutron star.

527
00:29:05 --> 00:29:11
Very simple
801 considerations, now.

528
00:29:10 --> 00:29:16
What comes is
extremely pedestrian.

529
00:29:13 --> 00:29:19
This is the mass
of the neutron star

530
00:29:16 --> 00:29:22
and this is the radius
of the neutron star.

531
00:29:19 --> 00:29:25
And I take a little bit
of matter m,

532
00:29:21 --> 00:29:27
and I drop it from a large
distance onto the neutron star.

533
00:29:25 --> 00:29:31
At what speed

534
00:29:27 --> 00:29:33
will that little piece of matter
reach the neutron star?

535
00:29:30 --> 00:29:36
You should almost be able
to close your eyes

536
00:29:33 --> 00:29:39
and give me that answer
right now.

537
00:29:35 --> 00:29:41
The kinetic energy when
it reaches the neutron star

538
00:29:38 --> 00:29:44
equals one-half m v squared.

539
00:29:41 --> 00:29:47
That is the speed

540
00:29:42 --> 00:29:48
at which it will crash
onto the neutron star,

541
00:29:45 --> 00:29:51
and that must be

542
00:29:47 --> 00:29:53
mM neutron star G divided by
the radius of the neutron star.

543
00:29:52 --> 00:29:58
You always lose your m,
and so you find

544
00:29:55 --> 00:30:01
that the speed at which
it reaches the neutron star

545
00:29:58 --> 00:30:04
is the square root of two
M neutron star

546
00:30:02 --> 00:30:08
times G divided
by R neutron star.

547
00:30:04 --> 00:30:10
You should remember
this equation.

548
00:30:07 --> 00:30:13
This was the equation
that we had for escape velocity.

549
00:30:10 --> 00:30:16
If you were here,
and you go back to infinity,

550
00:30:13 --> 00:30:19
you reach exactly that speed,
so if you fall in from infinity

551
00:30:17 --> 00:30:23
that is exactly the speed

552
00:30:18 --> 00:30:24
at which you reach
the neutron star.

553
00:30:19 --> 00:30:25
It should obviously
be the same number.

554
00:30:21 --> 00:30:27
And you don't really have to be
infinitely far away;

555
00:30:23 --> 00:30:29
you just have to be
much further away

556
00:30:25 --> 00:30:31
than the radius
of the neutron star.

557
00:30:29 --> 00:30:35
When this matter crashes
onto the neutron star,

558
00:30:33 --> 00:30:39
the kinetic energy that is
released is one-half mv squared.

559
00:30:37 --> 00:30:43
It is converted to heat,
and to give you some feeling

560
00:30:42 --> 00:30:48
for the incredible power
of a neutron star,

561
00:30:46 --> 00:30:52
if you make this little m
as little as ten grams--

562
00:30:50 --> 00:30:56
think of it as a pretty
full-sized marshmallow--

563
00:30:54 --> 00:31:00
and you throw a marshmallow

564
00:30:56 --> 00:31:02
from a large distance
onto a neutron star,

565
00:30:58 --> 00:31:04
the energy that is released
is comparable

566
00:31:01 --> 00:31:07
to the atomic bomb
that was used on Hiroshima.

567
00:31:04 --> 00:31:10
A ten-gram object thrown
onto a neutron star--

568
00:31:07 --> 00:31:13
the reason being
that this velocity

569
00:31:10 --> 00:31:16
becomes enormously high.

570
00:31:12 --> 00:31:18
If you put in
for the neutron star

571
00:31:15 --> 00:31:21
a mass of about three times
ten to the 30th kilograms,

572
00:31:20 --> 00:31:26
and you take for the radius
of the neutron star

573
00:31:23 --> 00:31:29
about ten kilometers,

574
00:31:25 --> 00:31:31
you will find
that that velocity becomes

575
00:31:28 --> 00:31:34
about two times ten
to the eighth meters per second,

576
00:31:33 --> 00:31:39
which is about 70%
of the speed of light.

577
00:31:38 --> 00:31:44
And because
of thisenormous speed--

578
00:31:40 --> 00:31:46
one-half mv squared
is horrendously high--

579
00:31:43 --> 00:31:49
it is a conversion

580
00:31:44 --> 00:31:50
of gravitational potential
energy to kinetic energy

581
00:31:47 --> 00:31:53
and then ultimately to heat.

582
00:31:51 --> 00:31:57
Now, nature is transferring mass
at an extraordinarily high rate

583
00:32:01 --> 00:32:07
in many of these binary systems.

584
00:32:04 --> 00:32:10
There are at least
some hundred or so that we know

585
00:32:08 --> 00:32:14
in our own galaxy.

586
00:32:11 --> 00:32:17
The mass transfer rate,
which I call dm/dt--

587
00:32:15 --> 00:32:21
so that is the transfer rate

588
00:32:17 --> 00:32:23
from the donor
onto the neutron star--

589
00:32:20 --> 00:32:26
that transfer rate

590
00:32:22 --> 00:32:28
is roughly ten to the 14th
kilograms per second.

591
00:32:29 --> 00:32:35
It is ahorrendous
mass transfer rate.

592
00:32:32 --> 00:32:38
You can calculate--

593
00:32:33 --> 00:32:39
by multiplying it
with one-half v squared--

594
00:32:37 --> 00:32:43
how many joules per second
are released

595
00:32:39 --> 00:32:45
in the form of kinetic energy.

596
00:32:41 --> 00:32:47
That means in the form of heat.

597
00:32:43 --> 00:32:49
And I call that
the power of that neutron star,

598
00:32:45 --> 00:32:51
and that, then,
for this mass transfer rate,

599
00:32:49 --> 00:32:55
that's about two times ten
to the 30th joules per second,

600
00:32:54 --> 00:33:00
which is watts.

601
00:32:56 --> 00:33:02
And that is
about 5,000 times larger

602
00:32:58 --> 00:33:04
than the power of our own sun.

603
00:33:02 --> 00:33:08
But the temperature
of this neutron star--

604
00:33:04 --> 00:33:10
because of this enormous
amount of energy released,

605
00:33:07 --> 00:33:13
the temperature
would reach values

606
00:33:09 --> 00:33:15
of about
ten million degrees Kelvin,

607
00:33:11 --> 00:33:17
and at that high temperature,

608
00:33:13 --> 00:33:19
the neutron star would emit
almost exclusively x-rays.

609
00:33:16 --> 00:33:22
You and I are very cold bodies,
only 300 degrees Kelvin.

610
00:33:20 --> 00:33:26
We radiate
electromagnetic radiation

611
00:33:23 --> 00:33:29
in the infrared part
of the spectrum.

612
00:33:25 --> 00:33:31
We have warm bodies.

613
00:33:26 --> 00:33:32
When you hold someone
in your arms, you can feel that.

614
00:33:29 --> 00:33:35
If I would heat you up
to 3,000 degrees Kelvin,

615
00:33:32 --> 00:33:38
you would become red-hot.

616
00:33:34 --> 00:33:40
And you actually...

617
00:33:35 --> 00:33:41
we could turn the light off
and I would see you.

618
00:33:38 --> 00:33:44
You're just emitting red light.

619
00:33:40 --> 00:33:46
If I would heat you up
to three million degrees,

620
00:33:43 --> 00:33:49
you would start to begin
to radiate in x-rays.

621
00:33:46 --> 00:33:52
You may not like it,
but that's a detail, of course.

622
00:33:48 --> 00:33:54
So I want you to appreciate
the fact that the...

623
00:33:52 --> 00:33:58
the kind of radiation that
you get depends strongly

624
00:33:55 --> 00:34:01
on the temperature,
and at ten million degrees

625
00:33:57 --> 00:34:03
you're dealing almost
exclusively with x-rays.

626
00:34:01 --> 00:34:07

627
00:34:05 --> 00:34:11
So these binary systems are
very potent sources of x-rays.

628
00:34:08 --> 00:34:14
The neutron stars rotate around,
we discussed that earlier--

629
00:34:12 --> 00:34:18
conservation
of angular momentum--

630
00:34:14 --> 00:34:20
and they have
strong magnetic fields.

631
00:34:17 --> 00:34:23
The matter that falls
onto the neutron star

632
00:34:20 --> 00:34:26
already heats up
during the infall

633
00:34:23 --> 00:34:29
because there is gravitational
potential energy released,

634
00:34:26 --> 00:34:32
and so the matter is so hot

635
00:34:27 --> 00:34:33
that, in general,
it's highly ionized.

636
00:34:30 --> 00:34:36
And highly ionized material
cannot reach

637
00:34:33 --> 00:34:39
a magnetic neutron star

638
00:34:35 --> 00:34:41
in all locations
that it prefers to do so.

639
00:34:38 --> 00:34:44
In 802, you will learn
why that's the case.

640
00:34:41 --> 00:34:47
However, the matter can reach

641
00:34:42 --> 00:34:48
the neutron star
at the magnetic poles.

642
00:34:46 --> 00:34:52
And so what you're going
to see now is

643
00:34:47 --> 00:34:53
you're going to have a neutron
star with magnetic poles,

644
00:34:51 --> 00:34:57
so the matter streams in
onto the magnetic poles,

645
00:34:53 --> 00:34:59
which gives you two hot spots.

646
00:34:55 --> 00:35:01
And if the axis of rotation
doesn't coincide

647
00:34:58 --> 00:35:04
with the line
through the two hot spots,

648
00:35:01 --> 00:35:07
if the neutron star rotates,

649
00:35:02 --> 00:35:08
you're going to see
x-ray pulsations.

650
00:35:05 --> 00:35:11
When the hot spot is here,
you will see x-rays,

651
00:35:08 --> 00:35:14
and when the hot spot is here,
you will not see x-rays.

652
00:35:11 --> 00:35:17
And so we observe from these
systems x-ray pulsations.

653
00:35:16 --> 00:35:22
Now think of the following.

654
00:35:18 --> 00:35:24
The x-ray pulsations
are a clock.

655
00:35:21 --> 00:35:27
It is the clock
of the rotating neutron star.

656
00:35:23 --> 00:35:29
If the neutron star
in a binary system--

657
00:35:25 --> 00:35:31
because all of these are
in a binary system,

658
00:35:27 --> 00:35:33
these x-ray binaries--

659
00:35:28 --> 00:35:34
if it's coming to you,
you see Doppler shift.

660
00:35:30 --> 00:35:36
The ticks of the clock come
a little closer together.

661
00:35:33 --> 00:35:39
If the neutron star moves
away from you,

662
00:35:35 --> 00:35:41
the ticks of the clocks are
a little bit further apart.

663
00:35:38 --> 00:35:44
That is exactly what
Doppler shift is all about.

664
00:35:40 --> 00:35:46
So by timing the pulses
of the x-rays,

665
00:35:43 --> 00:35:49
you can get a handle

666
00:35:45 --> 00:35:51
on the Doppler shift
of the neutron star.

667
00:35:47 --> 00:35:53
That means you can get
the speed of the neutron star,

668
00:35:50 --> 00:35:56
you can get the radius
of the orbit,

669
00:35:52 --> 00:35:58
and you can get the period,
just like we discussed before.

670
00:35:55 --> 00:36:01
But now you take an optical...

671
00:35:56 --> 00:36:02
The x-ray observations,
by the way, have to be made

672
00:35:58 --> 00:36:04
from outside
the Earth's atmosphere,

673
00:36:00 --> 00:36:06
because x-rays are absorbed
by the Earth's atmosphere.

674
00:36:02 --> 00:36:08
Now you take
an optical telescope

675
00:36:04 --> 00:36:10
and you look from the ground,

676
00:36:05 --> 00:36:11
and now you see the optical
spectrum of the donor.

677
00:36:09 --> 00:36:15
And what do you see
in the donor?

678
00:36:11 --> 00:36:17
You see these absorption lines.

679
00:36:12 --> 00:36:18
And as the donor moves
around the center of mass,

680
00:36:15 --> 00:36:21
these absorption lines move
back and forth.

681
00:36:17 --> 00:36:23
The Doppler shift of the donor.

682
00:36:19 --> 00:36:25
So you know the velocity
of the donor,

683
00:36:21 --> 00:36:27
you know the radius
of the donor...

684
00:36:23 --> 00:36:29
not the radius of the donor--

685
00:36:24 --> 00:36:30
you know the radius of the
orbit, and you know the period.

686
00:36:27 --> 00:36:33
So now we have a situation
that I just described earlier

687
00:36:31 --> 00:36:37
that you have the Doppler shift
of both objects,

688
00:36:35 --> 00:36:41
and remember, I told you
that you also get the masses.

689
00:36:39 --> 00:36:45
You get the mass of the donor
and the mass of the accretor.

690
00:36:43 --> 00:36:49

691
00:36:45 --> 00:36:51
Before I go ahead,
let me show you some slides.

692
00:36:49 --> 00:36:55
So we have to lower this again,
if that's possible-- yep.

693
00:36:54 --> 00:37:00

694
00:36:59 --> 00:37:05
I want to show you

695
00:37:01 --> 00:37:07
an artist's conception
of such a binary system.

696
00:37:05 --> 00:37:11

697
00:37:15 --> 00:37:21
So this is
what it may look like.

698
00:37:18 --> 00:37:24
You see the donor there

699
00:37:20 --> 00:37:26
and you see the neutron star
right here,

700
00:37:24 --> 00:37:30
so small, of course,
that it's invisible.

701
00:37:26 --> 00:37:32
And this is the accretion disk.

702
00:37:28 --> 00:37:34
Swirls in, the matter
ends up on the neutron star.

703
00:37:31 --> 00:37:37
And this is another view that
gives you an idea of the donor

704
00:37:38 --> 00:37:44
and then the swirl of matter,

705
00:37:40 --> 00:37:46
and then it swirls in and ends
up here on the neutron star.

706
00:37:45 --> 00:37:51
And here you see data
that were obtained in 1971.

707
00:37:51 --> 00:37:57
It's clear evidence
for the existence

708
00:37:53 --> 00:37:59
of these rotating neutron stars
with these x-ray hot spots.

709
00:37:57 --> 00:38:03
You see here the observed x-ray
intensity as a function of time.

710
00:38:01 --> 00:38:07
And the actual data are
these very thin lines.

711
00:38:05 --> 00:38:11
And this bold line was drawn
over by the authors

712
00:38:08 --> 00:38:14
to convince you that you see a
signal which is highly periodic.

713
00:38:12 --> 00:38:18
The time from here to here
is 1.24 seconds.

714
00:38:15 --> 00:38:21
This object was called
Hercules X-1.

715
00:38:17 --> 00:38:23
So this is one
of the magnetic poles

716
00:38:19 --> 00:38:25
and this is
the other magnetic pole.

717
00:38:21 --> 00:38:27
One magnetic pole
and the other magnetic pole.

718
00:38:23 --> 00:38:29
So you see here unmistakably
the rotation of the neutron star

719
00:38:26 --> 00:38:32
and the x-ray pulsations.

720
00:38:27 --> 00:38:33
Here you see data
from the same object,

721
00:38:30 --> 00:38:36
but now the time scale is
very different.

722
00:38:32 --> 00:38:38
From here to here is one day.

723
00:38:34 --> 00:38:40
This is two days.

724
00:38:36 --> 00:38:42
And when you look at this data
alone, forget this for now,

725
00:38:40 --> 00:38:46
notice that you see
the source is active in x-rays--

726
00:38:43 --> 00:38:49
the 1.42-second oscillations

727
00:38:45 --> 00:38:51
you cannot see,
of course, anymore

728
00:38:47 --> 00:38:53
because the time scale
is different,

729
00:38:49 --> 00:38:55
but notice here
there are no x-rays at all:

730
00:38:51 --> 00:38:57
1.7 days later,
no x-rays at all.

731
00:38:54 --> 00:39:00
1.7 days later, no
x-rays at all.

732
00:38:57 --> 00:39:03
And so what
you're looking at here

733
00:38:59 --> 00:39:05
are what we call x-ray eclipses.

734
00:39:01 --> 00:39:07
When the neutron star moves
behind the donor star,

735
00:39:04 --> 00:39:10
all the x-rays are absorbed
by the donor star

736
00:39:06 --> 00:39:12
and you get x-ray eclipses.

737
00:39:08 --> 00:39:14
In other words, you get...

738
00:39:10 --> 00:39:16
Independently
from the Doppler shift

739
00:39:12 --> 00:39:18
you also get the period of
the orbit by the x-ray eclipses.

740
00:39:19 --> 00:39:25
And this really changed
our whole concept of astronomy,

741
00:39:26 --> 00:39:32
the existence of these
neutron star binaries.

742
00:39:30 --> 00:39:36
And now comes a part, what are
the masses of these objects?

743
00:39:36 --> 00:39:42
I already alluded you
to the idea

744
00:39:38 --> 00:39:44
of the possibility
that there may be black holes.

745
00:39:42 --> 00:39:48
All the mass measurements
that have been done to date

746
00:39:45 --> 00:39:51
of these neutron stars
where you see the pulsations...

747
00:39:49 --> 00:39:55
all of them are very close
to 1.4 solar mass.

748
00:39:55 --> 00:40:01
And there's a good reason for
that-- that's not an accident.

749
00:39:58 --> 00:40:04
In 1930, the physicist
Chandrasekhar predicted

750
00:40:03 --> 00:40:09
that white dwarfs
could not exist

751
00:40:06 --> 00:40:12
if their mass is larger
than 1.4 solar mass.

752
00:40:09 --> 00:40:15
It was a quantum
mechanical calculation

753
00:40:11 --> 00:40:17
for which he received
in 1983 the Nobel Prize.

754
00:40:14 --> 00:40:20
Remember we discussed
white dwarfs earlier.

755
00:40:16 --> 00:40:22
A white dwarf has about
a radius of 10,000 kilometers,

756
00:40:19 --> 00:40:25
about the same as the Earth.

757
00:40:21 --> 00:40:27
And imagine
that you have a white dwarf,

758
00:40:23 --> 00:40:29
and you add matter
to the white dwarf

759
00:40:26 --> 00:40:32
and you pass
the 1.4-solar-mass mark.

760
00:40:28 --> 00:40:34
Then the white dwarf
will collapse

761
00:40:30 --> 00:40:36
and becomes a neutron star.

762
00:40:32 --> 00:40:38
And so when we measure
the masses of neutron stars,

763
00:40:35 --> 00:40:41
it turns out,
maybe somewhat by surprise,

764
00:40:37 --> 00:40:43
that they're all
very close to 1.4.

765
00:40:41 --> 00:40:47
If you could add more matter
to the neutron star

766
00:40:44 --> 00:40:50
by accreting
more and more matter

767
00:40:47 --> 00:40:53
and you reach the point
that the neutron star becomes

768
00:40:50 --> 00:40:56
as massive as three times
the mass of the sun,

769
00:40:53 --> 00:40:59
we believe that the neutron star
can no longer support itself

770
00:40:58 --> 00:41:04
and becomes a black hole.

771
00:40:59 --> 00:41:05
And so now comes the question,
what is a black hole?

772
00:41:03 --> 00:41:09
A black hole is the most bizarre
object that you can imagine,

773
00:41:07 --> 00:41:13
and it is something

774
00:41:09 --> 00:41:15
that you want
to stay away from, too.

775
00:41:12 --> 00:41:18
A black hole has no size,
unlike a neutron star.

776
00:41:18 --> 00:41:24
It has no size, but it does have
mass, and it has a lot of mass--

777
00:41:24 --> 00:41:30
three times the mass of the sun,
ten times the mass of the sun,

778
00:41:27 --> 00:41:33
a hundred times the mass
of the sun.

779
00:41:29 --> 00:41:35
So it has mass,
but it has no size.

780
00:41:34 --> 00:41:40
We identify around the black
hole a sphere with radius R

781
00:41:40 --> 00:41:46
which we call the event horizon.

782
00:41:43 --> 00:41:49

783
00:41:46 --> 00:41:52
Imagine you are
at the event horizon

784
00:41:48 --> 00:41:54
and you want to get away
from the black hole.

785
00:41:51 --> 00:41:57
What kind of speed do you need?

786
00:41:54 --> 00:42:00
You should be able to give me
that answer immediately.

787
00:41:57 --> 00:42:03
The escape velocity must be

788
00:42:00 --> 00:42:06
2 MG divided by the radius
of that event horizon.

789
00:42:05 --> 00:42:11
In other words, the radius
of the event horizon itself

790
00:42:09 --> 00:42:15
equals 2 MG
divided by c squared.

791
00:42:13 --> 00:42:19
If you tell me what m is,

792
00:42:15 --> 00:42:21
I will tell you what the radius
of the event horizon is.

793
00:42:18 --> 00:42:24

794
00:42:19 --> 00:42:25
I went a little fast here.

795
00:42:21 --> 00:42:27
I skipped an important step.

796
00:42:23 --> 00:42:29
v is the escape velocity
from the event horizon,

797
00:42:26 --> 00:42:32
which is at a distance capital R
from the mass M.

798
00:42:29 --> 00:42:35
So we see that here.

799
00:42:30 --> 00:42:36
Now, this escape velocity
can never be larger

800
00:42:33 --> 00:42:39
than the speed of light, so
the maximum value possible is c.

801
00:42:38 --> 00:42:44
And if now you look
at this part of this equation

802
00:42:41 --> 00:42:47
and you take the radius
on one side,

803
00:42:43 --> 00:42:49
you'll get that the radius
of the event horizon

804
00:42:46 --> 00:42:52
equals 2 MG divided
by c squared,

805
00:42:53 --> 00:42:59
and that's how I found
that equation.

806
00:42:55 --> 00:43:01
Sorry that I went
a little too fast.

807
00:42:57 --> 00:43:03

808
00:42:58 --> 00:43:04
If M is the mass of the Earth,

809
00:43:00 --> 00:43:06
the radius of the event horizon
is one centimeter.

810
00:43:03 --> 00:43:09
If M is the mass of the sun,

811
00:43:05 --> 00:43:11
the radius of the event horizon
is three kilometers.

812
00:43:08 --> 00:43:14
If M is three times
the mass of the sun,

813
00:43:11 --> 00:43:17
the radius of the event horizon
would become ten kilometers.

814
00:43:15 --> 00:43:21
It scales linearly
with the mass.

815
00:43:18 --> 00:43:24
If you were inside
the event horizon,

816
00:43:20 --> 00:43:26
you could never escape
the black hole

817
00:43:22 --> 00:43:28
because you would need

818
00:43:24 --> 00:43:30
a speed which is larger
than the speed of light.

819
00:43:28 --> 00:43:34
Therefore, you can never escape
from inside the event horizon.

820
00:43:31 --> 00:43:37
Nothing can get out of it,

821
00:43:33 --> 00:43:39
not x-rays, no radio emission,
no light, nothing.

822
00:43:36 --> 00:43:42
Once you're inside the event
horizon, you've had it.

823
00:43:39 --> 00:43:45
You cannot escape it.

824
00:43:40 --> 00:43:46
And so the question now
that comes up:

825
00:43:43 --> 00:43:49
Can we see x-rays
from a black hole?

826
00:43:45 --> 00:43:51
Because if nothing can come
out of a black hole,

827
00:43:48 --> 00:43:54
how can we see x-rays?

828
00:43:49 --> 00:43:55
And the answer is yes, we can,

829
00:43:51 --> 00:43:57
because as long as the matter
that swirls in

830
00:43:54 --> 00:44:00
is outside the event horizon,
it would still be very hot.

831
00:43:58 --> 00:44:04
Because gravitational
potential energy

832
00:44:00 --> 00:44:06
would already have been
released, it would be very hot

833
00:44:04 --> 00:44:10
and it would emit x-rays.

834
00:44:06 --> 00:44:12
So we can see x-rays
outside a black hole.

835
00:44:08 --> 00:44:14
However, you will
never see pulsations,

836
00:44:11 --> 00:44:17
because a black hole has no
surface, like a neutron star.

837
00:44:14 --> 00:44:20
So there's no such thing

838
00:44:16 --> 00:44:22
as two hot spots
which rotate around.

839
00:44:18 --> 00:44:24
And so now comes the problem
for astronomers:

840
00:44:21 --> 00:44:27
How can you determine
the mass of the accretor

841
00:44:24 --> 00:44:30
if the accretor is
not a pulsating neutron star

842
00:44:28 --> 00:44:34
but if the accretor
is a black hole?

843
00:44:30 --> 00:44:36
Well, you can only now measure
the Doppler shift of the donor,

844
00:44:35 --> 00:44:41
because the donor, in general,
is quite well visible.

845
00:44:39 --> 00:44:45
It's an optical star.

846
00:44:41 --> 00:44:47
But you will not be able
to measure

847
00:44:43 --> 00:44:49
the Doppler shift of the
black hole-- no pulsations.

848
00:44:47 --> 00:44:53
If, however, an astronomer
can make an estimate

849
00:44:51 --> 00:44:57
of the mass of that donor,

850
00:44:53 --> 00:44:59
then you will find
the mass of the accretor.

851
00:44:56 --> 00:45:02
In other words,
instead of having

852
00:44:59 --> 00:45:05
the Doppler shift measurements
of both stars--

853
00:45:02 --> 00:45:08
the neutron star
and the donor star,

854
00:45:05 --> 00:45:11
which gives you the mass
of two stars--

855
00:45:07 --> 00:45:13
now you have to settle for the
Doppler shift of only the donor

856
00:45:11 --> 00:45:17
and the mass
of the donor itself.

857
00:45:13 --> 00:45:19
And if you have a reasonable
idea of what that mass will be,

858
00:45:17 --> 00:45:23
then you can find the mass
of the accretor.

859
00:45:20 --> 00:45:26
And there is a very famous case

860
00:45:22 --> 00:45:28
that was the first one
discovered in the early '70s,

861
00:45:26 --> 00:45:32
which is called Cygnus X-1.

862
00:45:29 --> 00:45:35
Cygnus X-1 is an x-ray binary

863
00:45:31 --> 00:45:37
which has an orbital period
of 5.6 days.

864
00:45:35 --> 00:45:41
The Doppler shift measurements
of the donor were made,

865
00:45:39 --> 00:45:45
and astronomers simply looking
at the spectrum--

866
00:45:42 --> 00:45:48
at the absorption lines

867
00:45:43 --> 00:45:49
and the structure
of the absorption lines

868
00:45:45 --> 00:45:51
and the kind
of absorption lines--

869
00:45:48 --> 00:45:54
were able to say, "Yeah,
the mass of the donor

870
00:45:50 --> 00:45:56
is probably
approximately 30 solar masses."

871
00:45:58 --> 00:46:04
And with that information
and with the Doppler shift,

872
00:46:03 --> 00:46:09
you can now arrive
at the mass of the accretor,

873
00:46:09 --> 00:46:15
and that is, in this case...

874
00:46:12 --> 00:46:18
oh, by the way

875
00:46:13 --> 00:46:19
there is an r missing
in the word "accretion" there--

876
00:46:16 --> 00:46:22
that mass turns out to be
about 15 solar masses.

877
00:46:19 --> 00:46:25

878
00:46:22 --> 00:46:28
Now, when this was found
in the early '70s,

879
00:46:25 --> 00:46:31
most people concluded
this has to be a black hole.

880
00:46:28 --> 00:46:34
It is a very compact object.

881
00:46:29 --> 00:46:35
Otherwise it wouldn't emit
x-rays in the first place.

882
00:46:32 --> 00:46:38
And clearly, if the mass
of that compact object

883
00:46:36 --> 00:46:42
is way larger
than three solar masses,

884
00:46:39 --> 00:46:45
then there is no doubt in our
minds that this is a black hole.

885
00:46:42 --> 00:46:48
Since that time,

886
00:46:44 --> 00:46:50
many black hole x-ray binaries
have been discovered.

887
00:46:47 --> 00:46:53
So, if I summarize,
the amazing thing is

888
00:46:50 --> 00:46:56
from studying the Doppler shift

889
00:46:53 --> 00:46:59
of binary systems
like x-ray binaries,

890
00:46:56 --> 00:47:02
you can derive the orbital
parameters, orbital radius,

891
00:47:01 --> 00:47:07
orbital periods, the speed
of the stars in orbit,

892
00:47:06 --> 00:47:12
but you can also find
the masses.

893
00:47:09 --> 00:47:15
And whenever you make
a measurement of the mass

894
00:47:12 --> 00:47:18
when it is a neutron star

895
00:47:13 --> 00:47:19
when you see
the x-ray pulsations,

896
00:47:15 --> 00:47:21
you almost always find

897
00:47:17 --> 00:47:23
that it is very close to 1.4
times the mass of the sun.

898
00:47:20 --> 00:47:26
But in a few cases,
you will find

899
00:47:22 --> 00:47:28
that the mass is
substantially larger.

900
00:47:25 --> 00:47:31
Admittedly you have to do
without the Doppler shift, then,

901
00:47:28 --> 00:47:34
of the accretor, but you have
to use some other information,

902
00:47:30 --> 00:47:36
and then you can conclude
in most cases

903
00:47:33 --> 00:47:39
with pretty good confidence

904
00:47:35 --> 00:47:41
that you're dealing
with something like...

905
00:47:37 --> 00:47:43
bizarre as a black hole,

906
00:47:38 --> 00:47:44
which you can only define
the event horizon...

907
00:47:41 --> 00:47:47
And you can never escape
a black hole

908
00:47:44 --> 00:47:50
when you're inside
the event horizon,

909
00:47:46 --> 00:47:52
because that is
when the escape velocity

910
00:47:49 --> 00:47:55
would be larger
than the speed of light.

911
00:47:51 --> 00:47:57
So this is the escape velocity.

912
00:47:53 --> 00:47:59
If you set that equal to c,

913
00:47:55 --> 00:48:01
then you can solve for
the radius of the event horizon,

914
00:47:58 --> 00:48:04
and out of it pops
this equation.

915
00:48:00 --> 00:48:06
I would like to show you now
a slide of Cygnus X-1,

916
00:48:05 --> 00:48:11
which is the oldest known
black hole x-ray binary.

917
00:48:09 --> 00:48:15

918
00:48:14 --> 00:48:20
I have to lower the screen.

919
00:48:17 --> 00:48:23

920
00:48:20 --> 00:48:26
And there it comes.

921
00:48:22 --> 00:48:28

922
00:48:25 --> 00:48:31
This was really a bombshell
when this was discovered.

923
00:48:28 --> 00:48:34
I still remember reading
that first publication.

924
00:48:31 --> 00:48:37
Two people discovered this
independently, by the way.

925
00:48:34 --> 00:48:40
They came independently
to the same conclusion.

926
00:48:37 --> 00:48:43
Tom Bolton
and it was Paul Merlin--

927
00:48:41 --> 00:48:47
two independent groups.

928
00:48:43 --> 00:48:49

929
00:48:50 --> 00:48:56
All right, here is an optical
picture-- it is a negative,

930
00:48:55 --> 00:49:01
so you see the stars dark
and you see the sky bright--

931
00:48:59 --> 00:49:05
and right here is the star
that is Cygnus X-1.

932
00:49:03 --> 00:49:09
It is the donor.

933
00:49:04 --> 00:49:10
It is a very large star,
a supergiant, huge radius,

934
00:49:08 --> 00:49:14
and it is believed to have

935
00:49:10 --> 00:49:16
a mass of 30 times
that of the sun.

936
00:49:12 --> 00:49:18
You see here the close-up.

937
00:49:14 --> 00:49:20
This is not the companion,
believe me.

938
00:49:16 --> 00:49:22
This is just an image
of that star.

939
00:49:19 --> 00:49:25
The position was... it was hard
to get an accurate position.

940
00:49:23 --> 00:49:29
Various groups made
a major contribution

941
00:49:26 --> 00:49:32
to finding the position.

942
00:49:28 --> 00:49:34
One of the rocket flights of MIT

943
00:49:31 --> 00:49:37
found a position
that is quite precise

944
00:49:33 --> 00:49:39
and there was no doubt later...

945
00:49:35 --> 00:49:41
When the orbital period
was found of 5.6 days,

946
00:49:38 --> 00:49:44
there was no doubt that
this was the x-ray source.

947
00:49:41 --> 00:49:47
And so this is a system

948
00:49:42 --> 00:49:48
whereby you can only see
the donor in the optical light.

949
00:49:46 --> 00:49:52
You can measure the
Doppler shift of the donor,

950
00:49:49 --> 00:49:55
and by looking at the spectrum
of this star alone,

951
00:49:52 --> 00:49:58
you come to the conclusion

952
00:49:54 --> 00:50:00
that the mass must be
about 30 solar masses,

953
00:49:57 --> 00:50:03
and then you can argue

954
00:49:59 --> 00:50:05
that the invisible x-ray source
must be a black hole.

955
00:50:02 --> 00:50:08

956
00:50:04 --> 00:50:10.000