Biological half-life

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Biological_half-life

Biological half-life (elimination half-life, pharmacological half-life) is the time taken for concentration of a biological substance (such as a medication) to decrease from its maximum concentration (C_max) to half of C_max in the blood plasma. It is denoted by the abbreviation ${\displaystyle t_{\frac{1}{2}}}$.

This is used to measure the removal of things such as metabolites, drugs, and signalling molecules from the body. Typically, the biological half-life refers to the body's natural detoxification (cleansing) through liver metabolism and through the excretion of the measured substance through the kidneys and intestines. This concept is used when the rate of removal is roughly exponential.

In a medical context, half-life explicitly describes the time it takes for the blood plasma concentration of a substance to halve (plasma half-life) its steady-state when circulating in the full blood of an organism. This measurement is useful in medicine, pharmacology and pharmacokinetics because it helps determine how much of a drug needs to be taken and how frequently it needs to be taken if a certain average amount is needed constantly. By contrast, the stability of a substance in plasma is described as plasma stability. This is essential to ensure accurate analysis of drugs in plasma and for drug discovery.

The relationship between the biological and plasma half-lives of a substance can be complex depending on the substance in question, due to factors including accumulation in tissues, protein binding, active metabolites, and receptor interactions.

Examples

Water

The biological half-life of water in a human is about 7 to 14 days. It can be altered by behavior. Drinking large amounts of alcohol will reduce the biological half-life of water in the body. This has been used to decontaminate patients who are internally contaminated with tritiated water. The basis of this decontamination method is to increase the rate at which the water in the body is replaced with new water.

Alcohol

The removal of ethanol (drinking alcohol) through oxidation by alcohol dehydrogenase in the liver from the human body is limited. Hence the removal of a large concentration of alcohol from blood may follow zero-order kinetics. Also the rate-limiting steps for one substance may be in common with other substances. For instance, the blood alcohol concentration can be used to modify the biochemistry of methanol and ethylene glycol. In this way the oxidation of methanol to the toxic formaldehyde and formic acid in the human body can be prevented by giving an appropriate amount of ethanol to a person who has ingested methanol. Methanol is very toxic and causes blindness and death. A person who has ingested ethylene glycol can be treated in the same way. Half life is also relative to the subjective metabolic rate of the individual in question.

Common prescription medications

Substance	Biological half-life
Adenosine	Less than 10 seconds (estimate)
Norepinephrine	2 minutes
Oxaliplatin	14 minutes
Zaleplon	1 hour
Morphine	1.5–4.5 hours
Flurazepam	2.3 hours Active metabolite (N-desalkylflurazepam): 47–100 hours
Methotrexate	3–10 hours (lower doses), 8–15 hours (higher doses)
Methadone	15–72 hours in rare cases up to 8 days
Diazepam	20–50 hours Active metabolite (nordazepam): 30–200 hours
Phenytoin	20–60 hours
Buprenorphine	28–35 hours
Clonazepam	30–40 hours
Donepezil	3 days (70 hours)
Fluoxetine	4–6 days (under continuous administration) Active lipophilic metabolite (norfluoxetine): 4–16 days
Amiodarone	14–107 days
Vandetanib	19 days
Dutasteride	21–35 days (under continuous administration)
Bedaquiline	165 days

Metals

The biological half-life of caesium in humans is between one and four months. This can be shortened by feeding the person prussian blue. The prussian blue in the digestive system acts as a solid ion exchanger which absorbs the caesium while releasing potassium ions.

For some substances, it is important to think of the human or animal body as being made up of several parts, each with their own affinity for the substance, and each part with a different biological half-life (physiologically-based pharmacokinetic modelling). Attempts to remove a substance from the whole organism may have the effect of increasing the burden present in one part of the organism. For instance, if a person who is contaminated with lead is given EDTA in a chelation therapy, then while the rate at which lead is lost from the body will be increased, the lead within the body tends to relocate into the brain where it can do the most harm.

• Polonium in the body has a biological half-life of about 30 to 50 days.
• Caesium in the body has a biological half-life of about one to four months.
• Mercury (as methylmercury) in the body has a half-life of about 65 days.
• Lead in the blood has a half life of 28–36 days.
• Lead in bone has a biological half-life of about ten years.
• Cadmium in bone has a biological half-life of about 30 years.
• Plutonium in bone has a biological half-life of about 100 years.
• Plutonium in the liver has a biological half-life of about 40 years.

Peripheral half-life

Some substances may have different half-lives in different parts of the body. For example, oxytocin has a half-life of typically about three minutes in the blood when given intravenously. Peripherally administered (e.g. intravenous) peptides like oxytocin cross the blood-brain-barrier very poorly, although very small amounts (< 1%) do appear to enter the central nervous system in humans when given via this route. In contrast to peripheral administration, when administered intranasally via a nasal spray, oxytocin reliably crosses the blood–brain barrier and exhibits psychoactive effects in humans. In addition, also unlike the case of peripheral administration, intranasal oxytocin has a central duration of at least 2.25 hours and as long as 4 hours. In likely relation to this fact, endogenous oxytocin concentrations in the brain have been found to be as much as 1000-fold higher than peripheral levels.

Rate equations

Main article: Rate equation

See also: Pharmacokinetics § Metrics

First-order elimination

Timeline of an exponential decay process

Time (t)	Percent of initial value	Percent completion
t½	50%	50%
t½ × 2	25%	75%
t½ × 3	12.5%	87.5%
t½ × 3.322	10.00%	90.00%
t½ × 4	6.25%	93.75%
t½ × 4.322	5.00%	95.00%
t½ × 5	3.125%	96.875%
t½ × 6	1.5625%	98.4375%
t½ × 7	0.78125%	99.21875%
t½ × 10	~0.09766%	~99.90234%

Half-times apply to processes where the elimination rate is exponential. If ${\displaystyle C(t)}$ is the concentration of a substance at time ${\displaystyle t}$, its time dependence is given by

${\displaystyle C(t)=C(0)e^{-kt}}$

where k is the reaction rate constant. Such a decay rate arises from a first-order reaction where the rate of elimination is proportional to the amount of the substance:

${\displaystyle \frac{dC}{dt}=-kC.}$

The half-life for this process is

${\displaystyle t_{\frac{1}{2}}=\frac{\ln 2}{k}.}$

Alternatively, half-life is given by

${\displaystyle t_{\frac{1}{2}}=\frac{\ln 2}{{\lambda }_z}}$

where λ_z is the slope of the terminal phase of the time–concentration curve for the substance on a semilogarithmic scale.

Half-life is determined by clearance (CL) and volume of distribution (V_D) and the relationship is described by the following equation:

${\displaystyle t_{\frac{1}{2}}=\frac{\ln 2\cdot V_D}{CL}}$

In clinical practice, this means that it takes 4 to 5 times the half-life for a drug's serum concentration to reach steady state after regular dosing is started, stopped, or the dose changed. So, for example, digoxin has a half-life (or t_½) of 24–36 h; this means that a change in the dose will take the best part of a week to take full effect. For this reason, drugs with a long half-life (e.g., amiodarone, elimination t_½ of about 58 days) are usually started with a loading dose to achieve their desired clinical effect more quickly.

Biphasic half-life

Many drugs follow a biphasic elimination curve — first a steep slope then a shallow slope:

STEEP (initial) part of curve —> initial distribution of the drug in the body.

SHALLOW part of curve —> ultimate excretion of drug, which is dependent on the release of the drug from tissue compartments into the blood.

The longer half-life is called the terminal half-life and the half-life of the largest component is called the dominant half-life. For a more detailed description see Pharmacokinetics § Multi-compartmental models.

Very-long-baseline interferometry

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Very-long-baseline_interferometry

Some of the Atacama Large Millimeter Array radio telescopes.

The eight radio telescopes of the Smithsonian Submillimeter Array, located at the Mauna Kea Observatory in Hawai'i.

VLBI was used to create the first image of a black hole, imaged by the Event Horizon Telescope and published in April 2019.

Very-long-baseline interferometry (VLBI) is a type of astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescopes on Earth or in space. The distance between the radio telescopes is then calculated using the time difference between the arrivals of the radio signal at different telescopes. This allows observations of an object that are made simultaneously by many radio telescopes to be combined, emulating a telescope with a size equal to the maximum separation between the telescopes.

Data received at each antenna in the array include arrival times from a local atomic clock, such as a hydrogen maser. At a later time, the data are correlated with data from other antennas that recorded the same radio signal, to produce the resulting image. The resolution achievable using interferometry is proportional to the observing frequency. The VLBI technique enables the distance between telescopes to be much greater than that possible with conventional interferometry, which requires antennas to be physically connected by coaxial cable, waveguide, optical fiber, or other type of transmission line. The greater telescope separations are possible in VLBI due to the development of the closure phase imaging technique by Roger Jennison in the 1950s, allowing VLBI to produce images with superior resolution.

VLBI is best known for imaging distant cosmic radio sources, spacecraft tracking, and for applications in astrometry. However, since the VLBI technique measures the time differences between the arrival of radio waves at separate antennas, it can also be used "in reverse" to perform Earth rotation studies, map movements of tectonic plates very precisely (within millimetres), and perform other types of geodesy. Using VLBI in this manner requires large numbers of time difference measurements from distant sources (such as quasars) observed with a global network of antennas over a period of time.

Method

Recording data at each of the telescopes in a VLBI array. Extremely accurate high-frequency clocks are recorded alongside the astronomical data in order to help get the synchronization correct

In VLBI, the digitized antenna data are usually recorded at each of the telescopes (in the past this was done on large magnetic tapes, but nowadays it is usually done on large arrays of computer disk drives). The antenna signal is sampled with an extremely precise and stable atomic clock (usually a hydrogen maser) that is additionally locked onto a GPS time standard. Alongside the astronomical data samples, the output of this clock is recorded. The recorded media are then transported to a central location. More recent experiments have been conducted with "electronic" VLBI (e-VLBI) where the data are sent by fibre-optics (e.g., 10 Gbit/s fiber-optic paths in the European GEANT2 research network) and not recorded at the telescopes, speeding up and simplifying the observing process significantly. Even though the data rates are very high, the data can be sent over normal Internet connections taking advantage of the fact that many of the international high speed networks have significant spare capacity at present.

At the location of the correlator, the data is played back. The timing of the playback is adjusted according to the atomic clock signals, and the estimated times of arrival of the radio signal at each of the telescopes. A range of playback timings over a range of nanoseconds are usually tested until the correct timing is found.

Playing back the data from each of the telescopes in a VLBI array. Great care must be taken to synchronize the play back of the data from different telescopes. Atomic clock signals recorded with the data help in getting the timing correct.

Each antenna will be a different distance from the radio source, and as with the short baseline radio interferometer the delays incurred by the extra distance to one antenna must be added artificially to the signals received at each of the other antennas. The approximate delay required can be calculated from the geometry of the problem. The tape playback is synchronized using the recorded signals from the atomic clocks as time references, as shown in the drawing on the right. If the position of the antennas is not known to sufficient accuracy or atmospheric effects are significant, fine adjustments to the delays must be made until interference fringes are detected. If the signal from antenna A is taken as the reference, inaccuracies in the delay will lead to errors ${\displaystyle {ϵ}_B}$ and ${\displaystyle {ϵ}_C}$ in the phases of the signals from tapes B and C respectively (see drawing on right). As a result of these errors the phase of the complex visibility cannot be measured with a very-long-baseline interferometer.

Temperature variations at VLBI sites can deform the structure of the antennas and affect the baseline measurements. Neglecting atmospheric pressure and hydrological loading corrections at the observation level can also contaminate the VLBI measurements by introducing annual and seasonal signals, like in the Global Navigation Satellite System time series.

The phase of the complex visibility depends on the symmetry of the source brightness distribution. Any brightness distribution can be written as the sum of a symmetric component and an anti-symmetric component. The symmetric component of the brightness distribution only contributes to the real part of the complex visibility, while the anti-symmetric component only contributes to the imaginary part. As the phase of each complex visibility measurement cannot be determined with a very-long-baseline interferometer the symmetry of the corresponding contribution to the source brightness distributions is not known.

Roger Clifton Jennison developed a novel technique for obtaining information about visibility phases when delay errors are present, using an observable called the closure phase. Although his initial laboratory measurements of closure phase had been done at optical wavelengths, he foresaw greater potential for his technique in radio interferometry. In 1958 he demonstrated its effectiveness with a radio interferometer, but it only became widely used for long-baseline radio interferometry in 1974. At least three antennas are required. This method was used for the first VLBI measurements, and a modified form of this approach ("Self-Calibration") is still used today.

Scientific results

Some of the scientific results derived from VLBI include:

• High resolution radio imaging of cosmic radio sources.
• Imaging the surfaces of nearby stars at radio wavelengths (see also interferometry) – similar techniques have also been used to make infrared and optical images of stellar surfaces.
• Definition of the celestial reference frame.
• Measurement of the acceleration of the Solar System toward the center of the Milky Way.
• Motion of the Earth's tectonic plates.
• Regional deformation and local uplift or subsidence.
• Earth's orientation parameters and fluctuations in the length of day.
• Maintenance of the terrestrial reference frame.
• Measurement of gravitational forces of the Sun and Moon on the Earth and the deep structure of the Earth.
• Improvement of atmospheric models.
• Measurement of the fundamental speed of gravity.
• The tracking of the Huygens probe as it passed through Titan's atmosphere, allowing wind velocity measurements.
• First imaging of a supermassive black hole.

VLBI arrays

There are several VLBI arrays located in Europe, Canada, the United States, Chile, Russia, China, South Korea, Japan, Mexico, Australia and Thailand. The most sensitive VLBI array in the world is the European VLBI Network (EVN). This is a part-time array that brings together the largest European radiotelescopes and some others outside of Europe for typically weeklong sessions, with the data being processed at the Joint Institute for VLBI in Europe (JIVE). The Very Long Baseline Array (VLBA), which uses ten dedicated, 25-meter telescopes spanning 5351 miles across the United States, is the largest VLBI array that operates all year round as both an astronomical and geodesy instrument. The combination of the EVN and VLBA is known as Global VLBI. When one or both of these arrays are combined with space-based VLBI antennas such as HALCA or Spektr-R, the resolution obtained is higher than any other astronomical instrument, capable of imaging the sky with a level of detail measured in microarcseconds. VLBI generally benefits from the longer baselines afforded by international collaboration, with a notable early example in 1976, when radio telescopes in the United States, USSR and Australia were linked to observe hydroxyl-maser sources. This technique is currently being used by the Event Horizon Telescope, whose goal is to observe the supermassive black holes at the centers of the Milky Way Galaxy and Messier 87.

e-VLBI

Image of the source IRC +10420. The lower resolution image on the left was taken with the UK's MERLIN array and shows the shell of maser emission produced by an expanding shell of gas with a diameter about 200 times that of the Solar System. The shell of gas was ejected from a supergiant star (10 times the mass of the Sun) at the centre of the emission about 900 years ago. The corresponding EVN e-VLBI image (right) shows the much finer structure of the masers made visible with the higher resolution of the VLBI array.

VLBI has traditionally operated by recording the signal at each telescope on magnetic tapes or disks, and shipping those to the correlation center for replay. In 2004 it became possible to connect VLBI radio telescopes in close to real-time, while still employing the local time references of the VLBI technique, in a technique known as e-VLBI. In Europe, six radio telescopes of the European VLBI Network (EVN) were connected with Gigabit per second links via their National Research Networks and the Pan-European research network GEANT2, and the first astronomical experiments using this new technique were successfully conducted.

The image to the right shows the first science produced by the European VLBI Network using e-VLBI. The data from each of the telescopes were routed through the GÉANT2 network and on through SURFnet to be the processed in real time at the European Data Processing centre at JIVE.

Space VLBI

In the quest for even greater angular resolution, dedicated VLBI satellites have been placed in Earth orbit to provide greatly extended baselines. Experiments incorporating such space-borne array elements are termed Space Very Long Baseline Interferometry (SVLBI). The first SVLBI experiment was carried out on Salyut-6 orbital station with KRT-10, a 10-meter radio telescope, which was launched in July 1978.

The first dedicated SVLBI satellite was HALCA, an 8-meter radio telescope, which was launched in February 1997 and made observations until October 2003. Due to the small size of the dish, only very strong radio sources could be observed with SVLBI arrays incorporating it.

Another SVLBI satellite, a 10-meter radio telescope Spektr-R, was launched in July 2011 and made observations until January 2019. It was placed into a highly elliptical orbit, ranging from a perigee of 10,652 km to an apogee of 338,541 km, making RadioAstron, the SVLBI program incorporating the satellite and ground arrays, the biggest radio interferometer to date. The resolution of the system reached 8 microarcseconds.

International VLBI Service for Geodesy and Astrometry

The International VLBI Service for Geodesy and Astrometry (IVS) is an international collaboration whose purpose is to use the observation of astronomical radio sources using VLBI to precisely determine earth orientation parameters (EOP) and celestial reference frames (CRF) and terrestrial reference frames (TRF). IVS is a service operating under the International Astronomical Union (IAU) and the International Association of Geodesy (IAG).

Specific orbital energy

From Wikipedia, the free encyclopedia

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

In the gravitational two-body problem, the specific orbital energy ${\displaystyle \epsilon }$ (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (${\displaystyle {\epsilon }_p}$) and their total kinetic energy (${\displaystyle {\epsilon }_k}$), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time:

$${\displaystyle \begin{array}{rl}\epsilon & ={\epsilon }_k+{\epsilon }_p\\ & =\frac{v^2}{2}-\frac{\mu }{r}=-\frac{1}{2}\frac{{\mu }^2}{h^2}\left(1-e^2\right)=-\frac{\mu }{2a}\end{array}}$$

where

• ${\displaystyle v}$ is the relative orbital speed;
• ${\displaystyle r}$ is the orbital distance between the bodies;
• ${\displaystyle \mu =G(m_1+m_2)}$ is the sum of the standard gravitational parameters of the bodies;
• ${\displaystyle h}$ is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass;
• ${\displaystyle e}$ is the orbital eccentricity;
• ${\displaystyle a}$ is the semi-major axis.

It is typically expressed in ${\displaystyle \frac{\text{MJ}}{\text{kg}}}$ (megajoule per kilogram) or ${\displaystyle \frac{{\text{km}}^2}{{\text{s}}^2}}$ (squared kilometer per squared second). For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as characteristic energy.

Equation forms for different orbits

For an elliptic orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to:

$${\displaystyle \epsilon =-\frac{\mu }{2a}}$$

where

• ${\displaystyle \mu =G\left(m_1+m_2\right)}$ is the standard gravitational parameter;
• ${\displaystyle a}$ is semi-major axis of the orbit.

Proof

For an elliptic orbit with specific angular momentum h given by

$${\displaystyle h^2=\mu p=\mu a\left(1-e^2\right)}$$

we use the general form of the specific orbital energy equation,

$${\displaystyle \epsilon =\frac{v^2}{2}-\frac{\mu }{r}}$$

with the relation that the relative velocity at periapsis is

$${\displaystyle v_p^2=\frac{h^2}{r_p^2}=\frac{h^2}{a^2(1-e{)}^2}=\frac{\mu a\left(1-e^2\right)}{a^2(1-e{)}^2}=\frac{\mu \left(1-e^2\right)}{a(1-e{)}^2}}$$

Thus our specific orbital energy equation becomes

$${\displaystyle \epsilon =\frac{\mu }{a}\left[\frac{1-e^2}{2(1-e{)}^2}-\frac{1}{1-e}\right]=\frac{\mu }{a}\left[\frac{(1-e)(1+e)}{2(1-e{)}^2}-\frac{1}{1-e}\right]=\frac{\mu }{a}\left[\frac{1+e}{2(1-e)}-\frac{2}{2(1-e)}\right]=\frac{\mu }{a}\left[\frac{e-1}{2(1-e)}\right]}$$

and finally with the last simplification we obtain:

$${\displaystyle \epsilon =-\frac{\mu }{2a}}$$

For a parabolic orbit this equation simplifies to

$${\displaystyle \epsilon =0.}$$

For a hyperbolic trajectory this specific orbital energy is either given by

$${\displaystyle \epsilon =\frac{\mu }{2a}.}$$

or the same as for an ellipse, depending on the convention for the sign of a.

In this case the specific orbital energy is also referred to as characteristic energy (or ${\displaystyle C_3}$) and is equal to the excess specific energy compared to that for a parabolic orbit.

It is related to the hyperbolic excess velocity ${\displaystyle v_{\mathrm{\infty }}}$ (the orbital velocity at infinity) by

$${\displaystyle 2\epsilon =C_3=v_{\mathrm{\infty }}^2.}$$

It is relevant for interplanetary missions.

Thus, if orbital position vector (${\displaystyle \mathbf{r}}$) and orbital velocity vector (${\displaystyle \mathbf{v}}$) are known at one position, and ${\displaystyle \mu }$ is known, then the energy can be computed and from that, for any other position, the orbital speed.

Rate of change

For an elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is

$${\displaystyle \frac{\mu }{2a^2}}$$

where

• ${\displaystyle \mu =G(m_1+m_2)}$ is the standard gravitational parameter;
• ${\displaystyle a}$ is semi-major axis of the orbit.

In the case of circular orbits, this rate is one half of the gravitation at the orbit. This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy.

Additional energy

If the central body has radius R, then the additional specific energy of an elliptic orbit compared to being stationary at the surface is

$${\displaystyle -\frac{\mu }{2a}+\frac{\mu }{R}=\frac{\mu (2a-R)}{2aR}}$$

The quantity ${\displaystyle 2a-R}$ is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth). For the Earth and ${\displaystyle a}$ just little more than ${\displaystyle R}$ the additional specific energy is ${\displaystyle (gR/2)}$; which is the kinetic energy of the horizontal component of the velocity, i.e. $\frac{1}{2}{V}^2=\frac{1}{2}gR$, ${\displaystyle V=\sqrt{gR}}$.

Examples

ISS

The International Space Station has an orbital period of 91.74 minutes (5504 s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738 km.

The specific orbital energy associated with this orbit is −29.6 MJ/kg: the potential energy is −59.2 MJ/kg, and the kinetic energy 29.6 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 3.4 MJ/kg, the total extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net delta-v to reach this orbit is 8.1 km/s (the actual delta-v is typically 1.5–2.0 km/s more for atmospheric drag and gravity drag).

The increase per meter would be 4.4 J/kg; this rate corresponds to one half of the local gravity of 8.8 m/s².

For an altitude of 100 km (radius is 6471 km):

The energy is −30.8 MJ/kg: the potential energy is −61.6 MJ/kg, and the kinetic energy 30.8 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the total extra energy is 31.8 MJ/kg.

The increase per meter would be 4.8 J/kg; this rate corresponds to one half of the local gravity of 9.5 m/s². The speed is 7.8 km/s, the net delta-v to reach this orbit is 8.0 km/s.

Taking into account the rotation of the Earth, the delta-v is up to 0.46 km/s less (starting at the equator and going east) or more (if going west).

Voyager 1

For Voyager 1, with respect to the Sun:

• ${\displaystyle \mu =GM}$ = 132,712,440,018 km³⋅s⁻² is the standard gravitational parameter of the Sun
• r = 17 billion kilometers
• v = 17.1 km/s

Hence:

$${\displaystyle \epsilon ={\epsilon }_k+{\epsilon }_p=\frac{v^2}{2}-\frac{\mu }{r}=146\mathrm{k}{\mathrm{m}}^2{\mathrm{s}}^{-2}-8\mathrm{k}{\mathrm{m}}^2{\mathrm{s}}^{-2}=138\mathrm{k}{\mathrm{m}}^2{\mathrm{s}}^{-2}}$$

Thus the hyperbolic excess velocity (the theoretical orbital velocity at infinity) is given by

$${\displaystyle v_{\mathrm{\infty }}=16.6\mathrm{k}\mathrm{m}/\mathrm{s}}$$

However, Voyager 1 does not have enough velocity to leave the Milky Way. The computed speed applies far away from the Sun, but at such a position that the potential energy with respect to the Milky Way as a whole has changed negligibly, and only if there is no strong interaction with celestial bodies other than the Sun.

Applying thrust

Assume:

• a is the acceleration due to thrust (the time-rate at which delta-v is spent)
• g is the gravitational field strength
• v is the velocity of the rocket

Then the time-rate of change of the specific energy of the rocket is ${\displaystyle \mathbf{v}\cdot \mathbf{a}}$: an amount ${\displaystyle \mathbf{v}\cdot (\mathbf{a}-\mathbf{g})}$ for the kinetic energy and an amount ${\displaystyle \mathbf{v}\cdot \mathbf{g}}$ for the potential energy.

The change of the specific energy of the rocket per unit change of delta-v is

$${\displaystyle \frac{\mathbf{v}\cdot \mathbf{a}}{|\mathbf{a}|}}$$

which is |v| times the cosine of the angle between v and a.

Thus, when applying delta-v to increase specific orbital energy, this is done most efficiently if a is applied in the direction of v, and when |v| is large. If the angle between v and g is obtuse, for example in a launch and in a transfer to a higher orbit, this means applying the delta-v as early as possible and at full capacity. See also gravity drag. When passing by a celestial body it means applying thrust when nearest to the body. When gradually making an elliptic orbit larger, it means applying thrust each time when near the periapsis.

When applying delta-v to decrease specific orbital energy, this is done most efficiently if a is applied in the direction opposite to that of v, and again when |v| is large. If the angle between v and g is acute, for example in a landing (on a celestial body without atmosphere) and in a transfer to a circular orbit around a celestial body when arriving from outside, this means applying the delta-v as late as possible. When passing by a planet it means applying thrust when nearest to the planet. When gradually making an elliptic orbit smaller, it means applying thrust each time when near the periapsis.

If a is in the direction of v:

$${\displaystyle \mathrm{\Delta }\epsilon =\int vd(\mathrm{\Delta }v)=\int vadt}$$