Lunar eclipse

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lunar_eclipse 

Totality during the lunar eclipse of 21 January 2019. Direct sunlight is being blocked by the Earth, and the only light reaching it is sunlight refracted by Earth's atmosphere, producing a reddish color.

 

Latter phases of the partial lunar eclipse on 17 July 2019 taken from Gloucestershire, United Kingdom

A lunar eclipse occurs when the Moon moves into the Earth's shadow. This can occur only when the Sun, Earth, and Moon are exactly or very closely aligned (in syzygy) with Earth between the other two, and only on the night of a full moon. The type and length of a lunar eclipse depend on the Moon's proximity to either node of its orbit.

The reddish color of totally eclipsed Moon is caused by Earth completely blocking direct sunlight from reaching the Moon, with the only light reflected from the lunar surface has been refracted by Earth's atmosphere. This light appears reddish for the same reason that a sunset or sunrise does: the Rayleigh scattering of blue light.

Unlike a solar eclipse, which can only be viewed from a relatively small area of the world, a lunar eclipse may be viewed from anywhere on the night side of Earth. A total lunar eclipse can last up to nearly 2 hours, while a total solar eclipse lasts only up to a few minutes at any given place, because the Moon's shadow is smaller. Also unlike solar eclipses, lunar eclipses are safe to view without any eye protection or special precautions, as they are dimmer than the full Moon.

For the date of the next eclipse, see § Recent and forthcoming lunar eclipses.

Types of lunar eclipse

A schematic diagram of the shadow cast by Earth. Within the umbra, the central region, the planet totally shields direct sunlight. In contrast, within the penumbra, the outer portion, the sunlight is only partially blocked. (Neither the Sun, Moon, and Earth sizes nor the distances between the bodies are to scale.)

 

A total penumbral lunar eclipse dims the Moon in direct proportion to the area of the Sun's disk covered by Earth. This comparison of the Moon (within the southern part of Earth's shadow) during the penumbral lunar eclipse of January 1999 (left) and the Moon outside the shadow (right) shows this slight darkening.

Earth's shadow can be divided into two distinctive parts: the umbra and penumbra. Earth totally occludes direct solar radiation within the umbra, the central region of the shadow. However, since the Sun's diameter appears about one-quarter of Earth's in the lunar sky, the planet only partially blocks direct sunlight within the penumbra, the outer portion of the shadow.

Penumbral lunar eclipse

This occurs when the Moon passes through Earth's penumbra. The penumbra causes a subtle dimming of the lunar surface, which is only visible to the naked eye when about 70% of the Moon's diameter has immersed into Earth's penumbra. A special type of penumbral eclipse is a total penumbral lunar eclipse, during which the Moon lies exclusively within Earth's penumbra. Total penumbral eclipses are rare, and when these occur, the portion of the Moon closest to the umbra may appear slightly darker than the rest of the lunar disk.

Partial lunar eclipse

This occurs when only a portion of the Moon enters Earth's umbra, while a total lunar eclipse occurs when the entire Moon enters the planet's umbra. The Moon's average orbital speed is about 1.03 km/s (2,300 mph), or a little more than its diameter per hour, so totality may last up to nearly 107 minutes. Nevertheless, the total time between the first and the last contacts of the Moon's limb with Earth's shadow is much longer and could last up to 236 minutes.

Total lunar eclipse

This occurs when the moon falls entirely within the earth's umbra. Just prior to complete entry, the brightness of the lunar limb-- the curved edge of the moon still being hit by direct sunlight-- will cause the rest of the moon to appear comparatively dim. The moment the moon enters a complete eclipse, the entire surface will become more or less uniformly bright. Later, as the moon's opposite limb is struck by sunlight, the overall disk will again become obscured. This is because as viewed from the Earth, the brightness of a lunar limb is generally greater than that of the rest of the surface due to reflections from the many surface irregularities within the limb: sunlight striking these irregularities is always reflected back in greater quantities than that striking more central parts, and is why the edges of full moons generally appear brighter than the rest of the lunar surface. This is similar to the effect of velvet fabric over a convex curved surface which to an observer will appear darkest at the center of the curve. It will be true of any planetary body with little or no atmosphere and an irregular cratered surface (e.g., Mercury) when viewed opposite the Sun.

Central lunar eclipse

This is a total lunar eclipse during which the Moon passes through the centre of Earth's shadow, contacting the antisolar point. This type of lunar eclipse is relatively rare.

The relative distance of the Moon from Earth at the time of an eclipse can affect the eclipse's duration. In particular, when the Moon is near apogee, the farthest point from Earth in its orbit, its orbital speed is the slowest. The diameter of Earth's umbra does not decrease appreciably within the changes in the Moon's orbital distance. Thus, the concurrence of a totally eclipsed Moon near apogee will lengthen the duration of totality.

Selenelion

A view of the October 2014 lunar eclipse from Minneapolis, with the setting and partially eclipsed Moon appearing squashed just above the horizon just after sunrise (seen as sunlight shining on the tree in the right image)

A selenelion or selenehelion, also called a horizontal eclipse, occurs where and when both the Sun and an eclipsed Moon can be observed at the same time. The event can only be observed just before sunset or just after sunrise, when both bodies will appear just above opposite horizons at nearly opposite points in the sky. A selenelion occurs during every total lunar eclipse-- it is an experience of the observer, not a planetary event separate from the lunar eclipse itself. Typically, observers on Earth located on high mountain ridges undergoing false sunrise or false sunset at the same moment of a total lunar eclipse will be able to experience it. Although during selenelion the Moon is completely within the Earth's umbra, both it and the Sun can be observed in the sky because atmospheric refraction causes each body to appear higher (i.e., more central) in the sky than its true geometric planetary position.

Timing

As viewed from Earth, the Earth's shadow can be imagined as two concentric circles. As the diagram illustrates, the type of lunar eclipse is defined by the path taken by the Moon as it passes through Earth's shadow. If the Moon passes through the outer circle but does not reach the inner circle, it is a penumbral eclipse; if only a portion of the moon passes through the inner circle, it is a partial eclipse; and if entire Moon passes through the inner circle at some point, it is a total eclipse.

 

Contact points relative to the Earth's umbral and penumbral shadows, here with the Moon near is descending node

The timing of total lunar eclipses is determined by what are known as its "contacts" (moments of contact with Earth's shadow):

P1 (First contact): Beginning of the penumbral eclipse. Earth's penumbra touches the Moon's outer limb.

U1 (Second contact): Beginning of the partial eclipse. Earth's umbra touches the Moon's outer limb.

U2 (Third contact): Beginning of the total eclipse. The Moon's surface is entirely within Earth's umbra.

Greatest eclipse: The peak stage of the total eclipse. The Moon is at its closest to the center of Earth's umbra.

U3 (Fourth contact): End of the total eclipse. The Moon's outer limb exits Earth's umbra.

U4 (Fifth contact): End of the partial eclipse. Earth's umbra leaves the Moon's surface.

P4 (Sixth contact): End of the penumbral eclipse. Earth's penumbra no longer makes contact with the Moon.

Danjon scale

The following scale (the Danjon scale) was devised by André Danjon for rating the overall darkness of lunar eclipses:

L = 0: Very dark eclipse. Moon almost invisible, especially at mid-totality.

L = 1: Dark eclipse, gray or brownish in coloration. Details distinguishable only with difficulty.

L = 2: Deep red or rust-colored eclipse. Very dark central shadow, while outer edge of umbra is relatively bright.

L = 3: Brick-red eclipse. Umbral shadow usually has a bright or yellow rim.

L = 4: Very bright copper-red or orange eclipse. Umbral shadow is bluish and has a very bright rim.

Lunar versus solar eclipse

A solar eclipse occurs in the daytime at new moon, when the Moon is between Earth and the Sun, while a lunar eclipse occurs at night at full moon, when Earth passes between the Sun and the Moon.

 

The Moon does not completely darken as it passes through the umbra because Earth's atmosphere refracts sunlight into the shadow cone.

There is often confusion between a solar eclipse and a lunar eclipse. While both involve interactions between the Sun, Earth, and the Moon, they are very different in their interactions.

Lunar eclipse appearance

In a lunar eclipse, the Moon often passes through two regions of Earth's shadow: an outer penumbra, where direct sunlight is dimmed, and an inner umbra, where indirect and much dimmer sunlight refracted by Earth's atmosphere shines on the Moon, leaving a reddish color. This can be seen in different exposures of a partial lunar eclipse, for example here with exposures of 1/80, 2/5, and 2 seconds.

The Moon does not completely darken as it passes through the umbra because of the refraction of sunlight by Earth's atmosphere into the shadow cone; if Earth had no atmosphere, the Moon would be completely dark during the eclipse. The reddish coloration arises because sunlight reaching the Moon must pass through a long and dense layer of Earth's atmosphere, where it is scattered. Shorter wavelengths are more likely to be scattered by the air molecules and small particles; thus, the longer wavelengths predominate by the time the light rays have penetrated the atmosphere. Human vision perceives this resulting light as red. This is the same effect that causes sunsets and sunrises to turn the sky a reddish color. An alternative way of conceiving this scenario is to realize that, as viewed from the Moon, the Sun would appear to be setting (or rising) behind Earth.

From the Moon, a lunar eclipse would show a ring of reddish-orange light surrounding a silhouetted Earth in the lunar sky.

The amount of refracted light depends on the amount of dust or clouds in the atmosphere; this also controls how much light is scattered. In general, the dustier the atmosphere, the more that other wavelengths of light will be removed (compared to red light), leaving the resulting light a deeper red color. This causes the resulting coppery-red hue of the Moon to vary from one eclipse to the next. Volcanoes are notable for expelling large quantities of dust into the atmosphere, and a large eruption shortly before an eclipse can have a large effect on the resulting color.

Christopher Columbus predicting a lunar eclipse.

Lunar eclipse in culture

Several cultures have myths related to lunar eclipses or allude to the lunar eclipse as being a good or bad omen. The Egyptians saw the eclipse as a sow swallowing the Moon for a short time; other cultures view the eclipse as the Moon being swallowed by other animals, such as a jaguar in Mayan tradition, or a mythical three-legged toad known as Chan Chu in China. Some societies thought it was a demon swallowing the Moon, and that they could chase it away by throwing stones and curses at it. The Ancient Greeks correctly believed the Earth was round and used the shadow from the lunar eclipse as evidence. Some Hindus believe in the importance of bathing in the Ganges River following an eclipse because it will help to achieve salvation.

Inca

Similarly to the Mayans, the Incans believed that lunar eclipses occurred when a jaguar ate the Moon, which is why a blood moon looks red. The Incans also believed that once the jaguar finished eating the Moon, it could come down and devour all the animals on Earth, so they would take spears and shout at the Moon to keep it away.

Mesopotamians

The ancient Mesopotamians believed that a lunar eclipse was when the Moon was being attacked by seven demons. This attack was more than just one on the Moon, however, for the Mesopotamians linked what happened in the sky with what happened on the land, and because the king of Mesopotamia represented the land, the seven demons were thought to be also attacking the king. In order to prevent this attack on the king, the Mesopotamians made someone pretend to be the king so they would be attacked instead of the true king. After the lunar eclipse was over, the substitute king was made to disappear (possibly by poisoning).

Chinese

In some Chinese cultures, people would ring bells to prevent a dragon or other wild animals from biting the Moon. In the 19th century, during a lunar eclipse, the Chinese navy fired its artillery because of this belief. During the Zhou Dynasty (c. 1046–256 BC) in the Book of Songs, the sight of a Red Moon engulfed in darkness was believed to foreshadow famine or disease.

Blood moon

See also: Blood moon prophecy

 

Change to reddish cast

Certain lunar eclipses have been referred to as "blood moons" in popular articles but this is not a scientifically-recognized term. This term has been given two separate, but overlapping, meanings.

The first, and simpler, meaning relates to the reddish color a totally eclipsed Moon takes on to observers on Earth. As sunlight penetrates the atmosphere of Earth, the gaseous layer filters and refracts the rays in such a way that the green to violet wavelengths on the visible spectrum scatter more strongly than the red, thus giving the Moon a reddish cast.

The second meaning of "blood moon" has been derived from this apparent coloration by two fundamentalist Christian pastors, Mark Blitz and John Hagee. They claimed that the 2014–15 "lunar tetrad" of four lunar eclipses coinciding with the feasts of Passover and Tabernacles matched the "moon turning to blood" described in the Book of Joel of the Hebrew Bible. This tetrad was claimed to herald the Second Coming of Christ and the Rapture as described in the Book of Revelation on the date of the first of the eclipses in this sequence on April 15, 2014.

Occurrence

This multi-exposure sequence shows the August 2017 lunar eclipse visible from the ESO headquarters.

 

This collage shows the transitional stages of a lunar eclipse.

 

See also: Saros (astronomy) and Eclipse cycle

At least two lunar eclipses and as many as five occur every year, although total lunar eclipses are significantly less common. If the date and time of an eclipse is known, the occurrences of upcoming eclipses are predictable using an eclipse cycle, like the saros.

Recent and forthcoming lunar eclipses

Main article: List of 21st-century lunar eclipses

Further information: Lists of lunar eclipses

Eclipses occur only during an eclipse season, when the Sun appears to pass near either node of the Moon's orbit.

Quantum circuit

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_circuit 

 

Circuit that performs teleportation of a qubit. This circuit consists of both quantum gates and measurements. Measurement is a quantum phenomena that does not occur in classical circuits.

In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly other actions. The minimum set of actions that a circuit needs to be able to perform on the qubits to enable quantum computation is known as DiVincenzo's criteria.

Circuits are written such that the horizontal axis is time, starting at the left hand side and ending at the right. Horizontal lines are qubits, doubled lines represent classical bits. The items that are connected by these lines are operations performed on the qubits, such as measurements or gates. These lines define the sequence of events, and are usually not physical cables.

The graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation. Richard Feynman used an early version of the quantum circuit notation in 1986.

Reversible classical logic gates

Most elementary logic gates of a classical computer are not reversible. Thus, for instance, for an AND gate one cannot always recover the two input bits from the output bit; for example, if the output bit is 0, we cannot tell from this whether the input bits are 01 or 10 or 00.

However, reversible gates in classical computers are easily constructed for bit strings of any length; moreover, these are actually of practical interest, since irreversible gates must always increase physical entropy. A reversible gate is a reversible function on n-bit data that returns n-bit data, where an n-bit data is a string of bits x₁,x₂, ...,x_n of length n. The set of n-bit data is the space {0,1}ⁿ, which consists of 2ⁿ strings of 0's and 1's.

More precisely: an n-bit reversible gate is a bijective mapping f from the set {0,1}ⁿ of n-bit data onto itself. An example of such a reversible gate f is a mapping that applies a fixed permutation to its inputs. For reasons of practical engineering, one typically studies gates only for small values of n, e.g. n=1, n=2 or n=3. These gates can be easily described by tables.

Quantum logic gates

The quantum logic gates are reversible unitary transformations on at least one qubit. Multiple qubits taken together are referred to as quantum registers. To define quantum gates, we first need to specify the quantum replacement of an n-bit datum. The quantized version of classical n-bit space {0,1}ⁿ is the Hilbert space

${\displaystyle H_{\operatorname {QB} (n)}=\ell ^{2}(\{0,1\}^{n}).}$

This is by definition the space of complex-valued functions on {0,1}ⁿ and is naturally an inner product space. ${\displaystyle \ell ^{2}}$ refers to the 2-norm. This space can also be regarded as consisting of linear combinations, or superpositions, of classical bit strings. Note that H_QB(n) is a vector space over the complex numbers of dimension 2ⁿ. The elements of this vector space are the possible state-vectors of n-qubit quantum registers.

Using Dirac ket notation, if x₁,x₂, ...,x_n is a classical bit string, then

${\displaystyle |x_{1},x_{2},\cdots ,x_{n}\rangle \quad }$

is a special n-qubit register corresponding to the function which maps this classical bit string to 1 and maps all other bit strings to 0; these 2ⁿ special n-qubit registers are called computational basis states. All n-qubit registers are complex linear combinations of these computational basis states.

Quantum logic gates, in contrast to classical logic gates, are always reversible. One requires a special kind of reversible function, namely a unitary mapping, that is, a linear transformation of a complex inner product space that preserves the Hermitian inner product. An n-qubit (reversible) quantum gate is a unitary mapping U from the space H_QB(n) of n-qubit registers onto itself.

Typically, we are only interested in gates for small values of n.

A reversible n-bit classical logic gate gives rise to a reversible n-bit quantum gate as follows: to each reversible n-bit logic gate f corresponds a quantum gate W_f defined as follows:

${\displaystyle W_{f}(|x_{1},x_{2},\cdots ,x_{n}\rangle )=|f(x_{1},x_{2},\cdots ,x_{n})\rangle .}$

Note that W_f permutes the computational basis states.

Of particular importance is the controlled NOT gate (also called CNOT gate) W_CNOT defined on a quantized 2 qubit. Other examples of quantum logic gates derived from classical ones are the Toffoli gate and the Fredkin gate.

However, the Hilbert-space structure of the qubits permits many quantum gates that are not induced by classical ones. For example, a relative phase shift is a 1 qubit gate given by multiplication by the phase shift operator:

${\displaystyle P(\varphi )={\begin{bmatrix}1&0\\0&e^{i\varphi }\end{bmatrix}},}$

so

${\displaystyle P(\varphi )|0\rangle =|0\rangle \quad P(\varphi )|1\rangle =e^{i\varphi }|1\rangle .}$

Reversible logic circuits

Main article: reversible computing

Again, we consider first reversible classical computation. Conceptually, there is no difference between a reversible n-bit circuit and a reversible n-bit logic gate: either one is just an invertible function on the space of n bit data. However, as mentioned in the previous section, for engineering reasons we would like to have a small number of simple reversible gates, that can be put together to assemble any reversible circuit.

To explain this assembly process, suppose we have a reversible n-bit gate f and a reversible m-bit gate g. Putting them together means producing a new circuit by connecting some set of k outputs of f to some set of k inputs of g as in the figure below. In that figure, n=5, k=3 and m=7. The resulting circuit is also reversible and operates on n+m−k bits.

We will refer to this scheme as a classical assemblage (This concept corresponds to a technical definition in Kitaev's pioneering paper cited below). In composing these reversible machines, it is important to ensure that the intermediate machines are also reversible. This condition assures that intermediate "garbage" is not created (the net physical effect would be to increase entropy, which is one of the motivations for going through this exercise).

Note that each horizontal line on the above picture represents either 0 or 1, not these probabilities. Since quantum computations are reversible, at each 'step' the number of lines must be the same number of input lines. Also, each input combination must be mapped to a single combination at each 'step'. This means that each intermediate combination in a quantum circuit is a bijective function of the input.

Now it is possible to show that the Toffoli gate is a universal gate. This means that given any reversible classical n-bit circuit h, we can construct a classical assemblage of Toffoli gates in the above manner to produce an (n+m)-bit circuit f such that

${\displaystyle f(x_{1},\ldots ,x_{n},\underbrace {0,\dots ,0} )=(y_{1},\ldots ,y_{n},\underbrace {0,\ldots ,0} )}$

where there are m underbraced zeroed inputs and

${\displaystyle (y_{1},\ldots ,y_{n})=h(x_{1},\ldots ,x_{n})}$.

Notice that the end result always has a string of m zeros as the ancilla bits. No "rubbish" is ever produced, and so this computation is indeed one that, in a physical sense, generates no entropy. This issue is carefully discussed in Kitaev's article.

More generally, any function f (bijective or not) can be simulated by a circuit of Toffoli gates. Obviously, if the mapping fails to be injective, at some point in the simulation (for example as the last step) some "garbage" has to be produced.

For quantum circuits a similar composition of qubit gates can be defined. That is, associated to any classical assemblage as above, we can produce a reversible quantum circuit when in place of f we have an n-qubit gate U and in place of g we have an m-qubit gate W. See illustration below:

The fact that connecting gates this way gives rise to a unitary mapping on n+m−k qubit space is easy to check. In a real quantum computer the physical connection between the gates is a major engineering challenge, since it is one of the places where decoherence may occur.

There are also universality theorems for certain sets of well-known gates; such a universality theorem exists, for instance, for the pair consisting of the single qubit phase gate U_θ mentioned above (for a suitable value of θ), together with the 2-qubit CNOT gate W_CNOT. However, the universality theorem for the quantum case is somewhat weaker than the one for the classical case; it asserts only that any reversible n-qubit circuit can be approximated arbitrarily well by circuits assembled from these two elementary gates. Note that there are uncountably many possible single qubit phase gates, one for every possible angle θ, so they cannot all be represented by a finite circuit constructed from {U_θ, W_CNOT}.

Quantum computations

So far we have not shown how quantum circuits are used to perform computations. Since many important numerical problems reduce to computing a unitary transformation U on a finite-dimensional space (the celebrated discrete Fourier transform being a prime example), one might expect that some quantum circuit could be designed to carry out the transformation U. In principle, one needs only to prepare an n qubit state ψ as an appropriate superposition of computational basis states for the input and measure the output Uψ. Unfortunately, there are two problems with this:

• One cannot measure the phase of ψ at any computational basis state so there is no way of reading out the complete answer. This is in the nature of measurement in quantum mechanics.
• There is no way to efficiently prepare the input state ψ.

This does not prevent quantum circuits for the discrete Fourier transform from being used as intermediate steps in other quantum circuits, but the use is more subtle. In fact quantum computations are probabilistic.

We now provide a mathematical model for how quantum circuits can simulate probabilistic but classical computations. Consider an r-qubit circuit U with register space H_QB(r). U is thus a unitary map

${\displaystyle H_{\operatorname {QB} (r)}\rightarrow H_{\operatorname {QB} (r)}.}$

In order to associate this circuit to a classical mapping on bitstrings, we specify

• An input register X = {0,1}^m of m (classical) bits.
• An output register Y = {0,1}ⁿ of n (classical) bits.

The contents x = x₁, ..., x_m of the classical input register are used to initialize the qubit register in some way. Ideally, this would be done with the computational basis state

${\displaystyle |{\vec {x}},0\rangle =|x_{1},x_{2},\cdots ,x_{m},\underbrace {0,\dots ,0} \rangle ,}$

where there are r-m underbraced zeroed inputs. Nevertheless, this perfect initialization is completely unrealistic. Let us assume therefore that the initialization is a mixed state given by some density operator S which is near the idealized input in some appropriate metric, e.g.

${\displaystyle \operatorname {Tr} \left({\big |}|{\vec {x}},0\rangle \langle {\vec {x}},0|-S{\big |}\right)\leq \delta .}$

Similarly, the output register space is related to the qubit register, by a Y valued observable A. Note that observables in quantum mechanics are usually defined in terms of projection valued measures on R; if the variable happens to be discrete, the projection valued measure reduces to a family {E_λ} indexed on some parameter λ ranging over a countable set. Similarly, a Y valued observable, can be associated with a family of pairwise orthogonal projections {E_y} indexed by elements of Y. such that

${\displaystyle I=\sum _{y\in Y}\operatorname {E} _{y}.}$

Given a mixed state S, there corresponds a probability measure on Y given by

${\displaystyle \operatorname {Pr} \{y\}=\operatorname {Tr} (S\operatorname {E} _{y}).}$

The function F:X → Y is computed by a circuit U:H_QB(r) → H_QB(r) to within ε if and only if for all bitstrings x of length m

${\displaystyle \left\langle {\vec {x}},0{\big |}U^{*}\operatorname {E} _{F(x)}U{\big |}{\vec {x}},0\right\rangle =\left\langle \operatorname {E} _{F(x)}U(|{\vec {x}},0\rangle ){\big |}U(|{\vec {x}},0\rangle )\right\rangle \geq 1-\epsilon .}$

Now

${\displaystyle \left|\operatorname {Tr} (SU^{*}\operatorname {E} _{F(x)}U)-\left\langle {\vec {x}},0{\big |}U^{*}\operatorname {E} _{F(x)}U{\big |}{\vec {x}},0\right\rangle \right|\leq \operatorname {Tr} ({\big |}|{\vec {x}},0\rangle \langle {\vec {x}},0|-S{\big |})\|U^{*}\operatorname {E} _{F(x)}U\|\leq \delta }$

so that

${\displaystyle \operatorname {Tr} (SU^{*}\operatorname {E} _{F(x)}U)\geq 1-\epsilon -\delta .}$

Theorem. If ε + δ < 1/2, then the probability distribution

${\displaystyle \operatorname {Pr} \{y\}=\operatorname {Tr} (SU^{*}\operatorname {E} _{y}U)}$

on Y can be used to determine F(x) with an arbitrarily small probability of error by majority sampling, for a sufficiently large sample size. Specifically, take k independent samples from the probability distribution Pr on Y and choose a value on which more than half of the samples agree. The probability that the value F(x) is sampled more than k/2 times is at least

${\displaystyle 1-e^{-2\gamma ^{2}k},}$

where γ = 1/2 - ε - δ.

This follows by applying the Chernoff bound.

Fetal abduction

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Fetal_abduction 

Fetal abduction refers to the rare crime of child abduction by kidnapping of an at term pregnant woman and extraction of her fetus through a crude cesarean section. Dr. Michael H. Stone and Dr. Gary Brucato have alternatively referred to this crime as "fetus-snatching" or "fetus abduction." Homicide expert Vernon J. Geberth has used the term "fetal kidnapping." In the small number of reported cases, a few pregnant victims and about half of their fetuses survived the assault and non-medically performed cesarean.

Fetal abduction does not refer to medically induced labor or obstetrical extraction. The definition of the subject does not include compulsory cesarean sections for medical reasons nor child removal from parents for court-approved child protection. However, the "Children of the Disappeared" (desaparecidos) in the Argentine Dirty War are an example of criminal fetal abduction in state institutions as detailed by testimonies on cesarean delivery on desaparecidas and child adoption in a military hospital. Historical atrocities of cesarean extraction for fetal murder (not for child adoption) fall outside the subject definition.

Abductor profile

Fetal abduction is mostly perpetrated by women, usually after organized planning. The abductor may befriend the pregnant victim. In some cases, the abductor impersonates a pregnant and later a puerperal mother, using weight gain and a prosthesis to fake a pregnancy and cutting of the reproductive organs to replicate injuries gained during birth. Some abductors then take the neonate to a hospital. The National Center for Missing and Exploited Children’s spokesperson, Cathy Nahirny, stated in 2007, “Many times the abductor fakes a pregnancy and when it is time to deliver the baby, must abduct someone else's child”. Criminal motives include delusions of fulfilling a partner relationship, child-bearing and childbirth.

Statistics

The National Center for Missing and Exploited Children recorded 18 cases of fetal abductions in the United States between 1983 and 2015, which represented 6% of the recorded 302 cases of infant abduction.

List of reported cases and attempts

Of the current list of 25 reported cases (not including attempts), 4 of the mothers and 13 of their fetuses survived. (This list distinguishes an attempted fetal abduction as without either murder of the mother or extraction of the fetus. An attempt can include severe injury to the mother and fetus.)

Fetal abduction cases

1974

• In Philadelphia in November 1974, a 36-year-old woman named Winifred Ransom hacked and shot to death 26-year-old Margaret Sweeney. Sweeney was 8 months pregnant at the time. After first knocking Sweeney unconscious, Ransom cut the fetus out of Sweeney with a butcher knife. Sweeney regained consciousness during the operation, at which point Ransom struck her with a hatchet at least 20 times and then shot her 3 times. Ransom buried Sweeney beneath the floorboards of her kitchen. Ransom's husband eventually alerted authorities roughly 3 days later. Police found the body November 16. The baby girl survived and was cared for by her grandfather. Ransom was acquitted on the grounds of insanity. She was released from Byberry State Hospital mental institution after 20 months.

1987

• In Albuquerque, New Mexico, Cindy Ray was eight months pregnant when she was kidnapped at Kirtland Air Force Base outside a prenatal clinic. Darci Pierce was nineteen years old when she strangled the pregnant woman to death. She used her car keys to open Ray's womb, snatching the unharmed fetus, Millie. Millie survived, and Pierce was sentenced to 30 years to life for her crime.

1995

• In Addison, Illinois, Deborah Evans was murdered in her apartment. Jacqueline Williams, her boyfriend Fedell Caffey and her cousin Lavern Ward went into Evans' home and shot her in the head. She had three children and was pregnant with a fourth. Two of Evans' children were murdered along with their mother. Evans' murderers then proceeded to cut through her womb with scissors and remove the fetus. One of the children, a baby boy, survived, as did the fetus. The three murderers were caught and sentenced to life in prison.

1996

• In Tuscaloosa, Alabama, seventeen-year-old Carethia Curry was murdered by her friend, 29-year-old Felicia Scott. Curry was abducted by her friend on a night out. She was found three months later, stuffed in a garbage can at the bottom of a fifty-foot ravine with several gunshot wounds to the head, her torso sliced open. The baby girl Curry was carrying survived, and Scott was jailed for life.

1998

• In Fresno, California, Margarita Flores was eight months pregnant when she received a phone call from Josephina Saldana, who offered her gifts of baby furniture and a free one-year supply of diapers. Flores went to a warehouse to collect them and was murdered. Saldana was caught at a hospital the day afterwards carrying a dead fetus that she claimed to have just given birth to. Saldana committed suicide three days after her conviction for kidnapping and murdering Flores and her unborn child and three weeks before her sentencing to life imprisonment.

2000

• In Ravenna, Ohio, Theresa Andrews was twenty-three years old and pregnant when she ran into Michelle Bica. Bica was thirty-nine-years-old and was pretending to be pregnant at the time, and the two exchanged addresses. Then Bica started stalking Andrews. On September 27, 2000, Bica invited the woman to her home, then killed her, extracted the fetus she was carrying, and buried the woman in her garage. The baby survived, and Bica claimed he was her son. When Bica was being investigated by the FBI, she became fearful of punishment for her crime and shot herself.

https://www.jw.org/en/library/magazines/g20010722/Coping-With-an-Unspeakable-Tragedy/

2003

• In Okemah, Oklahoma, Carolyn Simpson was twenty-one years old and six months pregnant when she was shot and killed. She worked at a casino, where her murderer, Effie Goodson, age thirty-seven, was a regular customer. Goodson offered to give Simpson a ride home, and Simpson was later found in a ditch two miles away from her abductor. The baby, removed from the mother's womb three months premature, did not survive. When Goodson brought the fetus to the hospital, the child was pronounced dead, and it was discovered that she was not the mother. Goodson was found unable to stand for a trial, and three years later was sentenced to life in prison.

2004

• In Girardot (Cundinamarca), Colombia in April 2004, a case was reported in which both the mother and child survived. (Aseneth Piedrahita drugged Sol Angela Cartagena Bernal and extracted her fetus. Reportedly the perpetrator had medical knowledge.)
• In Skidmore, Missouri, Bobbie Jo Stinnett died of strangulation at the age of twenty-three at the hands of thirty-seven-year-old Lisa M. Montgomery. The two had been in contact previously; they were both rat terrier breeders in a dog show circuit. Montgomery had even e-mailed the victim, telling her that she wished to purchase one of her dogs. Montgomery faked a pregnancy, and on December 16, she drove from her Kansas home to Skidmore, Missouri. After strangling Stinnett to death, Montgomery cut open her abdomen and took her one-month-premature daughter. An hour later, the victim's mother found her body, and less than twenty-four hours later Victoria Jo Stinnett, the victim's stolen fetus, was found healthy in Melvern, Kansas. Lisa Montgomery was incarcerated, and a jury subsequently sentenced Montgomery, 43, to death. Montgomery was executed by lethal injection on January 13, 2021, at the United States Penitentiary in Terre Haute, Indiana.

2006

• In East St. Louis, Illinois on September 15, 2006, the pregnant Jimella Tunstall was murdered by her childhood friend Tiffany Hall. She was knocked unconscious and her unborn baby was cut from her abdomen with a pair of scissors. Neither survived the attack. Tunstall's body was left in a vacant lot. Hall also drowned Tunstall's three children, ages one, two, and seven, and left their bodies in the washer and dryer machines in the family's apartment. Hall was sentenced to life in prison in June 2008.

2008

• In Kennewick, Washington, Araceli Camacho Gomez, age twenty-seven, was stabbed to death by twenty-three-year-old Phiengchai Sisouvanh Synhavong. Gomez's hands and feet were bound with yarn throughout the attack, and her fetus was cut from her womb with a box cutter. The child survived the vicious attack. Synhavong called the police for help and attempted to pass the fetus off as her own. It quickly became apparent to authorities that she was lying and guilty of the crime. Synhavong was sentenced in 2010 to life in prison without parole.
• In Wilkinsburg, Pennsylvania, pregnant eighteen-year-old Kia Johnson was murdered during a fetal abduction by Andrea Curry-Demus, who had previously spent eight years in prison for stabbing another expectant mother to obtain her unborn baby. Curry-Demus had also seized a child from a hospital. Johnson's body was later found in Curry-Demus' apartment. The baby survived.

2009

• In Worcester, Massachusetts, Julie A. Corey, 35, murdered Darlene Haynes, 23, and extracted her fetus. The baby, Sheila Marie survived. Corey was convicted by jury and sentenced to life imprisonment in February 2014.
• In Portland, Oregon, Korena Elaine Roberts, 29, murdered pregnant Heather Megan Snively, 21, before cutting the fetus out of Snively's uterus. Roberts had been faking a pregnancy to her boyfriend and family, claiming she was expecting twins. She posted advertisements for baby items on Craigslist, and after multiple attempts to meet with other pregnant women fell through, she was able to lure Snively to her Beaverton-area home where she lived with her boyfriend and her two children on June 5, 2009. She then murdered Snively in the bathroom and cut the fetus, a baby boy, out of Snively's uterus. After covering Snively's body in carpet and hiding it in a crawlspace beneath the house, Roberts called her boyfriend, Yan Shubin, claiming she needed help delivering her baby. He came home to find Roberts in the bathtub with the water running, crying uncontrollably and holding the baby boy, who was not breathing. Paramedics took Roberts and the baby to the hospital, where doctors determined that Roberts had not given birth. Hospital staff called police, who arrested Roberts and located Snively's body in Roberts' home that night. The baby boy could not be revived and was pronounced dead at the hospital. An autopsy showed that Snively suffered between 15 and 30 blows, mostly to the back of her head, as well as multiple cuts to her right breast and abdomen, and bite marks on her right arm. The medical examiner was able to determine that while the head injuries likely knocked Snively unconscious, it was the abdominal incisions and blood loss that killed her. Roberts pleaded guilty to one count of aggravated murder and agreed to life in prison without the possibility of parole.

2011

• In Bowling Green, Kentucky, Kathy Coy cut out Jamie Stice's fetus and left Stice to bleed to death on a rural road. Coy initially claimed she'd given birth to the baby five weeks premature, but doctors determined that the baby wasn't hers. Police found that Coy was friends on Facebook with Stice and another pregnant woman. The other woman was unharmed, but police grew suspicious when they couldn't find Stice. After intense questioning, Coy led police to Stice's body. Coy pleaded guilty but mentally ill to avoid the death penalty. In March 2012, she was sentenced to life imprisonment.
• In Milwaukee, Wisconsin, Annette Morales-Rodriguez kidnapped Martiza Ramirez-Cruz, beat her to death and cut her fetus out of the womb. The fetus was just days away from being due. According to a criminal complaint, Morales-Rodriguez called police hours later to report that she'd just given birth in her shower and the baby wasn't breathing. The fetus was pronounced dead, and an autopsy determined the baby did not belong to Morales-Rodriguez.
• In Oakdale, Louisiana, Pamela Causey-Fregia killed pregnant Victoria Marie Perez with blunt force trauma. Causey tried to convince her husband, who was leaving her, that she was pregnant, despite her family believing she had a hysterectomy. She burned the body and buried it on her property. Causey's young children witnessed the murder and alerted police in 2015. In March 2018 Causey-Fregia pled guilty to murder and was sentenced to life imprisonment.

2013

• In Mogale City, South Africa in January 2013, Loretta Cooke extracted the fetus of Valencia Behrens. The mother was found dead, the fetus survived.
• In Johannesburg, South Africa on July 31, 2013, Zandile Makulana extracted Pretty Tsanga's fetus. Neither the mother nor her fetus survived.

2015

• In Longmont, Colorado, 34-year-old Dynel Lane posted a Craigslist ad advertising free maternity clothes. When the seven months pregnant 26-year-old Michelle Wilkins responded to the ad, Lane broke a lava lamp over her head and stabbed her in the neck with glass from the broken lamp, before removing the fetus from her body. According to police reports, Lane's husband got home and she claimed to have had a miscarriage. Her husband found the baby in the bath tub, rolled her over and saw her gasping for air; he then took both Lane and the baby to the hospital. The baby was actually dead or died within minutes. Wilkins survived the attack and while in Lane's basement she was able to lock the door, call 911, and get medical assistance. On April 29, 2016 Lane was sentenced to 100 years imprisonment. According to Colorado law, no homicide charge was brought as the mother survived and the neonate was found to have been not viable. The September 14, 2015 episode of the Dr. Phil interview of Michelle details Michelle's story as of then.
• In Bronx, New York City, Ashleigh Wade, 22, was accused of killing Angelikque Sutton, 22, and taking her baby who survived.

2017

• In Fargo, North Dakota, William Hoehn (32 years old) and Brooke Crews (38) were charged on August 28, 2017 with conspiring to kidnap and murder pregnant Savanna Greywind (22) and to kidnap her baby. Greywind's body was found in the river 8 days after she disappeared on August 19. The newborn, named Haisley Jo, survived. Crews pled guilty and said Greywind was still alive when she performed the cesarean on her.

2019

• A nine-month pregnant 19-year-old Chicago woman, Marlen Ochoa-Lopez (surname alternately given as Ochoa-Uriostegui in some reporting), was lured to a house with the promise of free baby clothes on April 23, and then strangled. Police believe that "the baby was forcibly removed following that murder" and a 46-year-old woman living at the residence subsequently called emergency services and stated that she had just given birth to the infant. The baby boy was stated to be in critical condition then. The deceased mother's body was found on the property on May 15. The boy, named Yovanny Jadiel Lopez, died several weeks later from brain damage. Clarisa Figueroa and her daughter, Desiree Figueroa were charged with first degree murder among other counts, Clarisa's husband Piotr Bobak was charged with associated crimes.

2020

• On August 28, Flavia Godinho Mafra was lured to a fake baby shower in Santa Catarina, Brazil. She was later found dead due to cuts from her abdomen due to her baby being removed, and from being struck with a brick. A 26 year old suspect was arrested in connection with the murder.
• 21 year old Reagan Simmons-Hancock, who was 8 months pregnant, was killed on October 9 in New Boston, Texas. Her baby was cut from her body. Taylor Parker, who claimed to only know Reagan by her first name, had been the photographer at Hancock’s wedding the year before and had spent time with Reagan the week of the murder going out for a “girl’s day” and visited Reagan and her husband’s home the night before telling them she was to be induced the following day. The day the murders occurred. Taylor was later arrested in Oklahoma in connection with the case.

Fetal abduction attempts

2009

• In Washington, DC in December 2009, Teka Adams, 29 years old, homeless and nine months pregnant, was abducted by acquaintance Veronica Deramous, aged 40. Deramous enlisted the help of her seventeen-year-old son to tie up Adams and hold her captive for four days. During those four days, Deramous unsuccessfully attempted to extract the fetus. Adams was able to escape, barely clinging to life and severely injured. A neighbor called 911, and both Adams and her fetus survived. The cesarean was completed at a hospital and the baby was named Miracle. In November 2010 Veronica Deramous was sentenced to 25 years imprisonment on a plea bargain for first-degree assault.