User talk:Dmcdysan/sandbox

6/30/22 VERSION WITH SPECIFICS FOR A CURRENT LOOP - REPLACED W/ MAGNETIC MOMENT

The magnetic field at magnetopause at distance $L$ (m) for a magnetic dipole generated by a magnetic moment $\mathbf {m}$current of $I_{c}$ (A) in a loop of radius $R_{c}$ is

$B_{mp}={\frac {\mu _{0}I_{c}R_{c}^{2}}{4L^{3}}}$ (MHD.2)

where ${\textstyle \mathbf {m} =I_{c}\ R_{c}^{2}}$ is the magnetic moment of a current loop. Substituting this into Equation MHD.1 and solving for $L$ yields the following result

$$L={\biggl (}{\frac {\mu _{0}}{2C_{SO}\rho }}I_{c}^{2}\ R_{c}^{4}{\biggr )}^{1/6}$$ (MHD.3)

The Force derived by a magnetic sail for a plasma environment is determined from MHD and kinetic equations is:

$F_{w}=C_{d}\ \rho {\frac {u_{pe}^{2}}{2}}S$ (MHD.4)

where ${\textstyle C_{d}}$ is a coefficient of drag determined by numerical analysis, ${\textstyle \rho u^{2}/2}$ (Pa) is the dynamic wind pressure, and $S=\pi L^{2}$ (m²) is the effective blocking area of the plasma magnet sail with characteristic length ${\textstyle L}$ (m) also known as the magnetopause radius ${\textstyle R_{mp}}$ (m). Note that this equation has exactly the same form as the drag equation in fluid dynamics. For a dipole magnetic field ${\textstyle C_{d}}$ is a function of sail tilt angle (0 degrees perpendicular to flow, 90 degrees in line with flow) determined by numerical calculation. For a large current loop following the Biot–Savart law ${\textstyle C_{d}}$ is a smaller value than that of a dipole and is also a function of tilt angle. Through analysis, numerical calculation, simulation and experimentation several conditions must be met before a magnetic sail can generate significant force. An important one is that in order to achieve significant thrust as determined by MHD equations the standoff distance $L$ must be significantly greater than the Gyroradius, also called the Larmor radius or cyclotron radius as follows:

$$r_{g}={\frac {m_{i}v_{\perp }}{\mid q\mid B_{x}C_{Li}}}$$ (MHD.5)

where ${\textstyle m_{i}}$ (kg) is the ion mass, $v_{\perp }$ (m/s) is the velocity of ions perpendicular to the magnetic field, $|q|$ (C) is the elementary charge of the ion, ${\textstyle B_{x}}$ (T) is the magnetic field strength at the point of reference ${\textstyle x}$ and ${\textstyle C_{Li}}$ is a constant that differs by source with ${\textstyle C_{Li}}$ =1 and Wikipedia gyroradius and ${\textstyle C_{Li}}$ =2. At magnetopause let $v_{\perp }=u_{pe}$ and $B_{x}=B_{mp}$in the above equation, form the inequality ${\textstyle r_{g}/L>a\ ,a>1}$ using ${\textstyle L}$ from Equation MHD.3, and after rearrangement yields the following condition on magnetic moment $\mathbf {m}$ for MHD applicability:

$$\mathbf {m} =I_{c}R_{c}^{2}>a^{3}{\frac {u}{\rho }}C_{a}$$ (MHD.6)

where the constant $C_{a}={\frac {4m_{i}^{3}}{\mid q\mid ^{3}C_{SO}\mu _{0}^{2}}}\approx 2.9\times 10^{-}12$ for C_SO=1.

Note that this MHD applicability test depends upon the ratio of effective plasma wind velocity $u=u_{pe}$ and plasma density $\rho =\rho _{pe}$y for a specific plasma environment and use case. If this condition is not met then the effective sail blocking area $A_{e}$ can be much smaller than $S=\pi L^{2}$. In some cases $A_{e}$ can be determined from a kinetic model specific to the plasma environment. Dmcdysan (talk) 19:10, 1 July 2022 (UTC)

Solar wind example - Need to decide placement

Although the ram pressure of the solar wind is relatively small at 1-6 nPa at 1 AU, the force exerted on a magnetic sail is the product of this pressure, the effective area of the sail, which for a sail of radius 100 km is  m², and a coefficient of dragranging between 3.6 and 5. This corresponds to a force on the order of 2,000-8,000 Newtons, which is on the order of the acceleration of a small car. However, since no propellant is exhausted the specific impulse is effectively infinite and even with very small acceleration a very high velocity is potentially achievable. Dmcdysan (talk) 19:16, 1 July 2022 (UTC)

Equation line formatting and numbering in Wikipedia

Template MathFormula$$r_{Li}={\frac {m_{i}v_{\perp }}{|q|B_{x}C_{Li}}}$$r_{Li}=\frac {m_i v_\perp}{|q| B_x C_{Li}

Template NumBlk with Template "{{EquationRef|A1|1}}"

$$r_{Li}={\frac {m_{i}v_{\perp }}{{|}q{|}B_{x}C_{Li}}}$$ (1)

Template NumBlk with Template "{{Anchor|1}}"

$$r_{Li}={\frac {m_{i}v_{\perp }}{\mid q\mid B_{x}C_{Li}}}$$ (1)

Issues: Equations with "|" or "}}" create ??

Use "{{|}}" for "|", "} }" for "}}"

True first parameter as a unique name for the page searchable in Source editing mode. Use second for display number, use a unique prefix to minimize renumbering.

See 1

Equation template

$y=ax^{2}+bx+c$ (1)

$y=ax^{2}+bx+c$ (2)

See Equation 1

See Equation 2

Insert 1 row x 3 col Table, col 1 is indent, col 2 is Math, col 3 is anchor

Col1

Col2

Col3

Table formatting is rather complex. Dmcdysan (talk) 20:14, 4 July 2022 (UTC)

Ashida 2014 Thesis Issues

Ashida 2014 thesis has issues stemming from Equation (2).Dmcdysan (talk) 18:34, 11 July 2022 (UTC)

In 2011[Citation] and 2014[Citation] Ashida and others documented Particle In Cell (PIC) simulation results for a kinematic model for cases where $r_{g}>>L$ where MHD is not applicable.

Their model for magnetosphere radius $L$ in Equation (2) for characteristic length L had a form unlike that of related papers that make it difficult to easily compare with other results.

Equation (12) of their study included the additional force of the injected plasma jet $F_{jet}$ comprised of momentum and static pressure of ions and electrons and defined thrust gain as ${\textstyle F_{MPS}/(F_{mag}+F_{jet})}$, which differs from the definition of a term by the same name in other studies. [Cite Funaki 2012, 2013]. It represents the gain of MPS over that of simply adding the magnetic sail force and the plasma injection jet force.

For the values cited in the conclusion, $F_{MPS}/F_{mag}$ is 7.5 in the radial orientation. Note that the attack angle defined by Ashida is the angle of the magnetic moment and not the orientation of the coil as defined by Nishida and is therefore differs by 90 degrees.

DIFFERENT $C_{d}$ From Ashida 2014 with erroneous Equation (2) see Equation (8) from the force simulation results for a magnetosphere radius $L$ of 4300 m, a coil radius $R_{c}$ of 75 m and $I_{c}$ selected to yield a magnetic moment ${\textstyle \mathbf {m} =I_{c}\pi R_{c}^{2}}$ corresponding to the specified value of $L$ in accordance with Equation MHD.6. The coefficient of drag $C_{d}$ determined the relative thrust with an attack angle ${\textstyle \alpha _{t}=90}$ degrees $C_{d}=1.8$ and with ${\textstyle \alpha _{t}=0}$ degrees $C_{d}=1.$Dmcdysan (talk) 19:36, 11 July 2022 (UTC)

MPS Edits

Edits from MPS not used or cuts. Dmcdysan (talk) 19:33, 17 July 2022 (UTC) MPS edits ??? More details on a proposed demonstrator spacecraft - Paper available on request. Some info in Funaki 2015 press.

[Ueno?] Preliminary results reported in 2019 indicated a 50% increase above the theoretical magnetosphere size. [Citation]

[Ashida 2014 with Error in Eqn 2] Note that the attack angle defined by Ashida is the angle of the magnetic moment and not the orientation of the coil as defined by Nishida and is therefore differs by 90 degrees. MODIFIED IN ANOTHER 2014 PAPER TO ALIGN.

Dmcdysan (talk) 19:34, 17 July 2022 (UTC)

Plasma magnet edits

Plasma magnet text equations not used. Dmcdysan (talk) 19:35, 17 July 2022 (UTC) NOT USED

Equation (15) of [Slough06] defines the magnetic field B_RMF (T) near an antenna coil of radius R_A (m) in terms of the electron skin depth $\delta$ (m) as follows

where R_A is the antenna coil radius (m). Since the RMF falls off as 1/r².  Note that for the solar wind the skin depth  d is only a few meters depending upon w_RMF (rad/s). — Preceding unsigned comment added by Dmcdysan (talk • contribs) 19:40, 17 July 2022 (UTC)

MHD.6

$F_{w}(f_{o})={\frac {C_{d}}{2}}\,\pi \,p_{w}\,{\biggl (}{\frac {\mathbf {m} }{\pi }}{\sqrt {\frac {\mu _{0}}{2C_{SO}p_{w}}}}{\biggr )}^{2/f_{o}}$

MHD.1

$p_{w}=\rho u^{2}=p_{B}={\frac {B_{mp}^{2}}{C_{SO}\mu _{0}}}$

$F_{w}(f_{o})={\frac {C_{d}}{2}}\,\pi \,p_{w}\,{\biggl (}{\frac {\mathbf {m} }{\pi }}{\sqrt {\frac {\mu _{0}}{2C_{SO}p_{w}}}}{\biggr )}^{2/f_{o}}$

Note that Equation (B-1) defines the standoff magnetic field  with r (kg/m³) the plasma density and u=w-v_s (m/s) the apparent wind speed and leaves selection of R_mp(m) as a design point.

If R_mp/R₀ is held constant , then w_RMF would be constant as well. If R_mp is held constant, then increasing R₀ would mean that w_RMF should be decreased.

=== Cuts, Equations not used ===

$P_{w}=u\,F_{w}=C_{d}\ \rho {\frac {u^{3}}{2}}\pi R_{mp}^{2}={\frac {C_{d}u\pi }{2\mu _{0}C_{SO}}}R_{mp}^{2}B_{mp}^{2}$

$={\frac {Ze}{m_{i}}}\,{\frac {C_{BR}(f_{o})}{(R_{c})^{f_{o}}}}$

$\omega _{RMF}>\omega _{ci}={\frac {ZeB(0)}{m_{i}}}={\frac {Ze}{m_{i}}}\,{\frac {C_{BR}(f_{o})}{(R_{0})^{f_{o}}}}$

${\frac {C_{BR}}{(R_{c})^{f_{o}}}}$

$L\approx R_{mp}\approx {\biggl (}{\frac {B(R_{0})}{B_{mp}}}{\biggr )}^{1/f_{o}}={\biggl (}{\frac {{\sqrt {\mu _{0}}}\,\,\mathbf {m} }{4\,\pi \,u\,{\sqrt {C_{SO}\rho }}}}{\biggr )}^{1/f_{o}}={\biggl (}{\frac {{\sqrt {\mu _{0}}}\,\,\mathbf {m} }{4\,\pi \,{\sqrt {C_{SO}p_{w}}}}}{\biggr )}^{1/f_{o}}$ — Preceding unsigned comment added by Dmcdysan (talk • contribs) 19:39, 17 July 2022 (UTC)

Magnetic Field Model

Edits to MFM. Dmcdysan (talk) 18:48, 24 July 2022 (UTC)

When the magnetic field source strength $B(R_{0})$ is specified, substituting $B_{mp}$ from the pressure balance analysis from Equation MHD.2 into the above and solving for ${\textstyle R_{mp}}$ yields the following:

$L\approx R_{mp}\approx R_{0}{\biggl (}{\frac {B(R_{0})}{B_{mp}}}{\biggr )}^{1/f_{o}}\approx {\biggl (}{\frac {C_{0}{\sqrt {\mu _{0}}}\,\,\mathbf {m} }{4\,\pi \,u\,{\sqrt {C_{SO}\rho }}}}{\biggr )}^{1/f_{o}}$ (MFM.4)

$L\approx R_{mp}\approx R_{0}{\biggl (}{\frac {B(R_{0})}{B_{mp}}}{\biggr )}^{1/f_{o}}\approx {\biggl (}{\frac {C_{0}{\sqrt {\mu _{0}}}\,\,\mathbf {m} }{4\,\pi \,u\,{\sqrt {C_{SO}\rho }}}}{\biggr )}^{1/f_{o}}$

This is the expression for $L$ when $f_{o}=3$ with $C_{SO}=1/2$ for Equation (4), with $C_{SO}=2$ for Equation (4), and the magnetopause distance of the Earth. This equation shows directly how the falloff rate $f_{o}$ dramatically increases the effective sail area $S=\pi R_{mp}^{2}$ for decreasing values of ${\textstyle f_{o}}$ for a given field source magnetic moment $\mathbf {m}$ and $B_{mp}$determined from the pressure balance equation MHD.1. Substituting this into Equation MHD.3 yields the plasma wind force as a function of falloff rate $f_{o}$ as follows

$F_{w}(f_{o})={\frac {C_{d}}{2}}\,\pi \,p_{w}\,{\biggl (}{\frac {C_{0}\mathbf {m} }{\pi }}{\sqrt {\frac {\mu _{0}}{2C_{SO}p_{w}}}}{\biggr )}^{2/f_{o}}$ (MFM.5)

where $p_{w}=\rho u^{2}$ is the wind pressure from Equation MHD.1. This is the same expression as Equation (10b) when $f_{o}=3$ and $C_{SO}=1/2$. Note that force increases as falloff rate decreases. Dmcdysan (talk) 18:49, 24 July 2022 (UTC)

Cut overview text from magnetic sail

Physical principles involved include: plasma characteristics for the Solar wind, a planetary ionosphere and the interstellar medium; interaction of magnetic fields with charged particles in a plasma; analogies with the Earth's magnetopause; and performance measures; such as, force achieved, energy requirements and the mass of the magnetic sail system. A number of proposals using this concept have been developed since 1988 starting with the Magsail proposed by Andrews and Zubrin, which although analyzed by many sources as theoretically possible has the practical disadvantage of a mass on the order of 100 tonnes (100,000 kg). Subsequent designs strove to achieve similar benefits with reduced mass by injecting plasma, use of a rotating magnetic field, and combining the technology of a magnetic sail with that of an electric sail. This article summarizes the literature regarding performance of these designs in theoretical models, numerical analyses, simulations and laboratory experiments. Some criticisms regarding these results have been rebutted but other issues remain unresolved. Trials for several of these designs have been proposed. This article concludes with a performance comparison of these magnetic sail designs with other each other as well as other technologies for the use cases of: acceleration or deceleration in a stellar plasma wind, deceleration in the interstellar medium, deceleration in a planetary ionosphere, a comparison with electric and solar sails, and a summary of advantages and disadvantages. Dmcdysan (talk) 01:33, 1 September 2022 (UTC)

A simulation of magnetic field lines around a circular current

Deleted by Constant314 from Biot-Savart Law on 9/26/2021.

Assuming a circular current is in xy-plane with center at the origin, thus the magnetic field in yz-plane will have x-component = 0 due to symmetrical cancellation from the circular current. The current runs counterclockwise. At a point, P, in yz-plane (y, z), the contribution from a segment of the circular current at R is,$$d\mathbf {B} =\mathbf {j} dJ+\mathbf {k} dK$$with,$$dJ={\frac {zR\sin \theta \ d\theta }{|R^{2}+P^{2}-2yR\sin \theta |^{\frac {3}{2}}}}$$$$dK={\frac {(-yR\sin \theta +R^{2})d\theta }{|R^{2}+P^{2}-2yR\sin \theta |^{\frac {3}{2}}}}$$The magnetic field B can be obtained by line integration over the circular current with ${\textstyle \theta =[0,\ 2\pi ]}$. Thus,$$\mathbf {B} =J\mathbf {j} +K\mathbf {k}$$$$J=\int _{C}dJ=\int _{0}^{2\pi }{\frac {zR\sin \theta \ d\theta }{|R^{2}+P^{2}-2yR\sin \theta |^{\frac {3}{2}}}}$$$$K=\int _{C}dK=\int _{0}^{2\pi }{\frac {(-yR\sin \theta +R^{2})d\theta }{|R^{2}+P^{2}-2yR\sin \theta |^{\frac {3}{2}}}}$$:Here is a short Python function to execute the numerical calculation:

def cirmag(yb, zb, dt):
	jb, kb, m = 0.0, 0.0, 64
	for i in range(0, m, 1):
	t = i*dt
	dn = (abs(r**2 + yb**2 -
	2*yb*r*math.sin(t)))**(3/2)
	jb = jb + (zb*r*math.sin(t))*dt/dn
	kb = kb + (r**2 - yb*r*math.sin(t))*dt/dn
	bb = math.sqrt(jb**2 + kb**2)
	return jb, kb, bb

To draw complete field lines with direction markers, the following Python function is used:<syntaxhighlight lang="python3">

	def fieldline(yf0, zf0, pt, ds):
	ysign, yf1, zf1, cm = 1, 0.0, 0.0, 0
	ysign = np.sign(yf0)
	for i in range(1, pt, 1):
	jf0, kf0, bf0 = cirmag(yf0, zf0)
	yf1 = yf0 + jf0/bf0*ds
	zf1 = zf0 + kf0/bf0*ds
	if abs(kf0) < 6.0 and zf1 > 0:
	cm += 1
	if cm < 2:
	if ysign > 0:
	ax.plot(yf1, zf1, "g>",
	markersize=4)
	else:
	ax.plot(yf1, zf1, "g<",
	markersize=4)
	if zf1 < 0 and zf1 > -30*ds:
	if ((ysign > 0 and yf1 < r)
	or (ysign < 0 and yf1 > -r)):
	print(c)
	break
	ax.plot([yf0,yf1], [zf0,zf1],
	color="g", linewidth=0.4)
	yf0 = yf1
	zf0 = zf1

Dmcdysan (talk) 03:14, 6 September 2022 (UTC)