Some equations and constants

16. Some equations and constants#

16.1. Physical constants#

Table 16.1 Physical constants#
Name	Symbol	Value
Speed of light	\(c\)	\(3.00 \cdot 10^{8}\;\mathrm{m}/\mathrm{s}\)
Elementary charge	\(e\)	\(1.60 \cdot 10^{-19}\;\mathrm{C}\)
Electron mass	\(m_e\)	\(9.11 \cdot 10^{-31}\;\mathrm{kg} = 0.511\;\mathrm{MeV}/c^2\)
Proton mass	\(m_p\)	\(1.67 \cdot 10^{-27}\;\mathrm{kg} = 938\;\mathrm{MeV}/c^2\)
Gravitational constant	\(G\)	\(6.67 \cdot 10^{-11}\;\mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2\)
Gravitational acceleration	\(g\)	\(9.81\;\mathrm{m}/\mathrm{s}^2\)
Boltzmann’s constant	\(\kB\)	\(1.38 \cdot 10^{-23}\;\mathrm{J}/\mathrm{K}\)
Gas constant	\(R\)	\(8.31 \; \mathrm{J}/\mathrm{K} \cdot \mathrm{mol}\)
Stefan-Boltzmann constant	\(\sigma\)	\(5.67 \cdot 10^{-8} \; \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}^4\)
Avogadro’s number	\(N_\mathrm{A}\)	\(6.02 \cdot 10^{23} \; \mathrm{mol}^{-1}\)
Planck’s constant	\(h\)	\(6.63 \cdot 10^{-34}\;\mathrm{J}\cdot\mathrm{s}\)
	\(\hbar = h / 2 \pi\)	\(1.05 \cdot 10^{-34}\;\mathrm{J}\cdot\mathrm{s}\)

16.2. Moments of inertia#

Table 16.2 Moments of inertia, all about axes of symmetry through the center of mass.#
Object	Moment of inertia
Thin stick (length \(L\))	\(\frac{1}{12} M L^2\)
Ring or hollow cylinder (radius \(R\))	\(M R^2\)
Disk or solid cylinder (radius \(R\))	\(\frac12 M R^2\)
Hollow sphere (radius \(R\))	\(\frac23 M R^2\)
Solid sphere (radius \(R\))	\(\frac25 M R^2\)
Rectangle (size \(a \times b\)), perpendicular axis	\(\frac{1}{12} M (a^2 + b^2)\)
Rectangle (size \(a \times b\)), axis parallel to side \(b\)	\(\frac{1}{12} M a^2\)

16.3. Solar system objects#

Table 16.3 Characteristics of the Sun, Earth and Moon.#
	Sun	Earth	Moon
Mass (kg)	\(1.99 \cdot 10^{30}\)	\(5.97 \cdot 10^{24}\)	\(7.35 \cdot 10^{22}\)
Mean radius (m)	\(6.96 \cdot 10^8\)	\(6.37 \cdot 10^6\)	\(1.74 \cdot 10^6\)
Orbital period (s)	\(6 \cdot 10^{15}\)	\(3.16 \cdot 10^7\)	\(2.36 \cdot 10^6\)
	(200 My)	(365.25 days)	(27.3 days)
Mean orbital radius (m)	\(2.6 \cdot 10^{20}\)	\(1.50 \cdot 10^{11}\)	\(3.85 \cdot 10^{8}\)
Mean density (kg/m\(^3\))	\(1.4 \cdot 10^3\)	\(5.5 \cdot 10^3\)	\(3.3 \cdot 10^3\)

Table 16.4 Properties of a number of solar system objects.#
Name	Symbol	Equatorial radius	Mass	Mean orbit radius	Orbital period	Inclination	Orbital eccentricity	Rotation period	Confirmed moons	Axial tilt
Mercury	☿	0.382	0.06	0.39	0.24	3.38	0.206	58.64	0	0.04
Venus	♀	0.949	0.82	0.72	0.62	3.86	0.007	-243.02	0	177.36
Earth	♁	1	1	1	1	7.25	0.017	1	1	23.44
Moon	☾	0.272	0.0123	384399	27.32158	18.29-28.58	0.0549	27.32158	0	6.68
Mars	♂	0.532	0.107	1.52	1.88	5.65	0.093	1.03	2	25.19
Ceres	⚳	0.0742	0.00016	2.766	4.599	10.59	0.08	0.3781	0	4
Jupiter	♃	11.209	317.8	5.2	11.86	6.09	0.048	0.41	69	3.13
Io		0.285	0.015	421600	1.769	0.04	0.0041	1.769	0	0
Europa		0.246	0.008	670900	3.551	0.47	0.009	3.551	0	0
Ganymede		0.413	0.025	1070400	7.155	1.85	0.0013	7.155	0	0
Callisto		0.378	0.018	1882700	16.689	0.2	0.0074	16.689	0	0
Saturn	♄	9.449	95.2	9.54	29.46	5.51	0.054	0.43	62	26.73
Titan		0.404	0.023	1221870	15.945	0.33	0.0288	15.945	0	0
Uranus	⛢	4.007	14.6	19.22	84.01	6.48	0.047	-0.72	27	97.77
Oberon		0.119	0.00051	583519	13.46	0.1	0.0014	13.46	0	0
Neptune	♆	3.883	17.2	30.06	164.8	6.43	0.009	0.67	14	28.32
Triton		0.212	0.00358	354759	5.877	157	0.00002	5.877	0	0
Pluto	♇	0.186	0.0022	39.482	247.9	17.14	0.25	6.39	5	119.59
Charon		0.095	0.00025	17536	6.387	0.001	0.0022	6.387	0	unknown
Haumea		0.13	0.0007	43.335	285.4	28.19	0.19	0.167	2	unknown
Makemake		0.11	unknown	45.792	309.9	28.96	0.16	unknown	1	unknown
Eris		0.18	0.0028	67.668	557	44.19	0.44	unknown	1	unknown

Properties of a number of solar system objects. Equatorial radii and masses are compared to those of Earth (see Table 16.3). Orbital properties are around primary (the sun for (dwarf) planets, the planet for moons). Orbital radii and periods for planets again compared to Earth, for moons in kilograms and days. Rotation period for all objects in days. Inclination and axial tilt in degrees. Data from NASA planetary fact sheets ^[1].

16.4. Material properties#

Table 16.5 Properties of some common solid materials.#
material	\(\rho\) (\(\mathrm{kg}/\mathrm{m}^3\))	\(E\) (\(\mathrm{GPa}\))	\(B\) (\(\mathrm{GPa}\))	\(G\) (\(\mathrm{GPa}\))	\(c\) (\(\mathrm{J}/\mathrm{kg}\cdot\mathrm{K}\))	\(k\) (\(\mathrm{W}/\mathrm{m}\cdot\mathrm{K}\))
aluminium	2700	70	75	25	900	237
bone		16	8	80
concrete	2400	20
copper	8940	110	140	44	386	401
glass		60	50	25	753	0.8
granite		45	45	20
hardwood	700	15		10
iron	7870	210	160	77	447	80.4
marble		60	70	20
steel		200	160	75
ice (\(-10^\circ\mathrm{C}\))	916.7				2050
pine wood	373	7.5		5	1400	0.11

Properties of some common solid materials, measured at room temperature and pressure, unless indicated otherwise: densities \(\rho\), Young’s \(E\), bulk \(B\) and shear \(G\) moduli, specific heats per unit mass \(c\) and thermal conductivities per unit length \(k\). The ‘\(\mathrm{G}\)’ in \(\mathrm{GPa}\) stands for ‘giga’, or \(10^9\).

Table 16.6 Properties of some common liquid materials#
material	\(\rho\) (\(\mathrm{kg}/\mathrm{m}^3\))	\(\eta\) (\(\mathrm{Pa}\cdot\mathrm{s}\))	\(T_\mathrm{m}\) (\(\mathrm{K}\))	\(L_\mathrm{f}\) (\(\mathrm{kJ}/\mathrm{kg}\))	\(T_\mathrm{b}\) (\(\mathrm{K}\))	\(L_\mathrm{v}\) (\(\mathrm{kJ}/\mathrm{kg}\))	\(c\) (\(\mathrm{J}/\mathrm{kg}\cdot\mathrm{K}\))
ethanol	789	\(1.07 \cdot 10^{-3}\)	159	109	351	879	2.43
glycerol	1260	1.2	291	201	563	1007	2437
mercury	13530	\(1.53 \cdot 10^{-3}\)	234	11.3	630	296	140
water	1000	\(8.94 \cdot 10^{-4}\)	273	334	373	2257	4184

Properties of some common liquid materials: densities \(\rho\), viscosities \(\eta\), melting and boiling points \(T_\mathrm{m}\) and \(T_\mathrm{b}\), heats of fusion \(L_\mathrm{f}\) and vaporization \(L_\mathrm{v}\) and specific heats per unit mass \(c\). Densities, viscosities and specific heats measured at room temperature and pressure.

Table 16.7 Melting and boiling points of some common materials, with their respective heats of transformation, measured at atmospheric pressure.#
Material	melting point (\(K\))	\(L_\mathrm{f}\;(\mathrm{kJ}/\mathrm{kg})\)	boiling point (\(K\))	\(L_\mathrm{v}\;(\mathrm{kJ}/\mathrm{kg})\)
alcohol	159	109	351	879
copper	1357	205	2840	4726
lead	601	24.7	2013	858
mercury	234	11.3	630	296
water	273	334	373	2257

16.5. Equations#

16.5.1. Vector derivatives#

Gradient:

(16.1)#\[\begin{split} \bm{\nabla} f(\bm{r}) = \bm{\nabla} f(x, y, z) = \left( \begin{array}{c} \partial_x f \\ \partial_y f \\ \partial_z f \end{array}\right) = \left( \frac{\partial f}{\partial x} \hat{x} + \frac{\partial f}{\partial y} \hat{y} + \frac{\partial f}{\partial z} \hat{z} \right). \end{split}\]

Divergence:

(16.2)#\[\begin{split} \bm{\nabla} \cdot \bm{v} = \left( \partial_x, \partial_y, \partial_z \right) \cdot \left( \begin{array}{c} v_x \\ v_y \\ v_z \end{array} \right) = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}. \end{split}\]

Curl:

(16.3)#\[\begin{split} \bm{\nabla} \times \bm{A} = \left( \partial_x, \partial_y, \partial_z \right) \times \left( \begin{array}{c} A_x \\ A_y \\ A_z \end{array} \right) = \left( \begin{array}{c} \partial_y A_z - \partial_z A_y \\ \partial_z A_x - \partial_x A_z \\ \partial_x A_y - \partial_y A_x \end{array} \right). \end{split}\]