-
Notifications
You must be signed in to change notification settings - Fork 2
/
orbits.html
195 lines (191 loc) · 9.92 KB
/
orbits.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
<!DOCTYPE html>
<html>
<head>
<title>Alexander Mead | Orbits</title>
<link rel="shortcut icon" href="images/favicon.ico" type="image/x-icon" />
<meta charset="UTF-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<link rel="stylesheet" type="text/css" href="css/reset.css" />
<link rel="stylesheet" type="text/css" href="css/main.css" />
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<script type="text/javascript" src="scripts/script.js"></script>
<script type="text/javascript" src="scripts/menu.js"></script>
<!--[if IE]>
<script src="//html5shiv.googlecode.com/svn/trunk/html5.js"></script>
<![endif]-->
</head>
<body>
<div class="page-wrapper">
<div class="nav-container clearfix" id="nav-home">
<div class="nav-logo">
<p class="logo"><a href="index.html">AM</a></p>
</div>
<nav class="nav-menu">
<ul>
<li><a href="index.html#about">About</a></li>
<li><a href="index.html#publications">Publications</a></li>
<li>
<a href="https://github.com/alexander-mead/HMcode">HMcode</a>
</li>
<li><a href="#" class="selected">Orbits</a></li>
<li><a href="gravity.html">Gravity</a></li>
<li><a href="universe.html">Universe</a></li>
<li><a href="covid.html">COVID-19</a></li>
<li><a href="maths.html">Maths</a></li>
<li><a href="mandelplot.html">Mandelplot</a></li>
<li><a href="index.html#contact">Contact</a></li>
</ul>
</nav>
</div>
<div class="intro">
<h3>3:2 spin-orbit resonance</h3>
</div>
<div class="subpage-wrapper">
<div class="content-wrapper spacer clearfix">
<div class="subpage-left">
<p>
This movie shows a blue planet orbiting a red star in a special
way that is called 'spin-orbit resonance', where the spin period
of the planet is connected to the orbital period by a simple
ratio. In this example, the planet spins exactly three times on
its axis for every two orbits of the star. The right-hand side of
the movie shows the orbit, while the left-hand side shows the flux
received and what the sun looks like from the point of view of an
observer on the equator of the planet at the location of the large
arrow. A 3:2 spin-orbit resonance is exactly the configuration of
the planet Mercury. For comparison, the Earth spins ~366 times per
orbit (not ~365, think about it!), and is not in a resonant
configuration. Tidal forces exerted by the Sun on Mercury are
thought to be responsible for driving it into this 3:2 spin-orbit
resonance, much in the same way that tidal forces from Earth have
driven the Moon into a 1:1 spin-orbit resonance, which is why you
only ever see one face of the moon.
</p>
</div>
<div class="subpage-right">
<img src="images/32so_e0p00_beta0p00.gif" />
</div>
</div>
<div class="content-wrapper spacer clearfix">
<div class="subpage-left">
<p>
It is thought that some eccentricity is required to drive planets
into resonances such as 3:2, and in the previous example the orbit
of the planet was circular. In this movie, the orbit has changed
to be mildly eccentric, with e = 0.15. The large arrow denotes the
'up' direction from the point of view of our observer: the sun is
directly overhead when this arrow is pointing towards the star and
this is then mid-day. The two smaller arrows indicate the horizon
and when one of these is pointing towards the star; it would be
either sunrise or sunset for our observer. Note that the passage
of the sun slows down in the sky around mid-day, which is a bit
strange. A quick look at the previous movies, combined with a
small amount of thinking, will convince you that one 'day' lasts
two 'years' (orbital periods) for our observer, and that the
system is exactly periodic over two orbital periods. Is this
interesting? Maybe a bit.
</p>
</div>
<div class="subpage-right">
<img src="images/32so_e0p15_beta0p00.gif" />
</div>
</div>
<div class="content-wrapper spacer clearfix">
<div class="subpage-left">
<p>
Things get more interesting when we consider higher eccentricities
for the same 3:2 system, such as in this movie where e = 0.25. The
eccentricity means two things for our observer: First, that the
distance to the sun changed appreciably over one orbit, hence the
sun in the sky appears larger or smaller depending on the orbital
phase. Second, the angular velocity changes significantly over the
orbit, with it being fastest when the planet is closest to the sun
and vice versa. Note that the sun actually moves backwards in the
sky here around mid-day! How the sun transitions across the sky
for our observer depends on a competition between the (constant)
angular-spin rate of the planet and the (not constant)
angular-orbit rate. For these higher eccentricities the increased
orbital angular speed around the point of orbital closest approach
means that actually the sun moves backwards in the sky for a short
time, prolonging the mid-day for our observer.
</p>
</div>
<div class="subpage-right">
<img src="images/32so_e0p25_beta0p00.gif" />
</div>
</div>
<div class="content-wrapper spacer clearfix">
<div class="subpage-left">
<p>
Things get stranger if we consider our observer to be at a
different longitude on the surface of the planet, such as in this
movie. In this case the 'day' is actually broken up into three
distinct phases of daylight and the sun will rise thrice and set
thrice throughout the 'day'. For this longitude, this manifests as
one longer period of daylight when the planet is furthest from the
sun and then two short bursts of light with the sun closer, but
low in the sky. Note that this also means that the total received
flux, and the rate at which this flux is deposited, varies as a
function of planetary longitude, as well as the standard
latitudinal dependence. This is different to Earth, where all
longitudes are equivalent in terms of the flux they receive over a
year.
</p>
</div>
<div class="subpage-right">
<img src="images/32so_e0p25_beta0p50.gif" />
</div>
</div>
<div class="content-wrapper spacer clearfix">
<div class="subpage-left">
<p>
As the eccentricity becomes more extreme so does the weird
behaviour. When e = 0.4, as in this example, the observer receives
more instantaneous flux from the short sunset-sunrise times than
they do during the longer day, so the best tanning time occurs
when the sun is low in the sky, but compensatingly close. There is
a critical value of the eccentricity at which the retrograde
motion of the sun and the corresponding strange behaviour becomes
possible. It turns out that this can be obtained as the solution
to a cubic equation, and the critical eccentricity is ~0.193. The
eccentricity of Mercury is ~0.205 so the retrograde motion of the
Sun would only be visible for a short time, and there are only
very specific longitudes on the surface of Mercury when you would
see a triple sunrise phenomenon over the course of one 'day',
which on Mercury lasts for ~176 Earth days.
</p>
</div>
<div class="subpage-right">
<img src="images/32so_e0p40_beta0p50.gif" />
</div>
</div>
<div class="content-wrapper">
<div class="subpage-center">
<p>
This may all seem like an academic exercise, but many planets have
been discovered around low-mass stars, and the habitable zone for
these stars is much closer to the star compared to that around our
Sun. The relative importance of tidal forces increases more
quickly than the gravitational force as you go near a star, it has
an r-cubed dependence rather than r-squared, so planets in the
habitable zone around low-mass stars are expected to be in
spin-orbit coupled configurations. These could be 1:1, like the
Moon-Earth resonance, but other situations such as 3:2, 2:1, 5:2,
... are also possible. So there could be funny aliens out there in
the cosmos who have a very different conception of what a day
means compared to that which we have on earth. These aliens would
also have to get used to the fact that the total received flux in
one 'day' varies as a function of longitude.
</p>
</div>
</div>
</div>
<footer>
<p>
© 2023 Alexander Mead. Website by
<a href="http://www.jenbayne.com" target="_blank">Jen Bayne</a>.
</p>
</footer>
</div>
</body>
</html>