The Hubble law calculator above turns a recession velocity into a cosmological distance, or a distance back into a velocity, using v = H₀d. It does two things a bare formula does not. First, it makes the Hubble constant a live input rather than a hidden assumption, because the measured value of H₀ is genuinely disputed and the distance you get swings by around eight per cent depending on which published figure you adopt. Second, it reports the Hubble time and the Hubble radius next to your answer, so you can see immediately whether your galaxy sits in the range where the simple linear law is trustworthy or well outside it.
Arb Digital publishes free calculators that show the assumptions instead of burying them, and cosmology is the clearest case for that habit. A distance quoted to four significant figures from a constant that two respected measurement programmes disagree about is a false precision worth naming. This page states the constant it used, the unit conversions it applied, and the redshift range over which the answer means what you think it means.
What This Hubble Law Calculator Does
You supply two of the three quantities in Hubble's law and it returns the third. In distance mode you enter a recession velocity in kilometres per second and a Hubble constant in kilometres per second per megaparsec, and the tool divides one by the other to get a distance in megaparsecs, then converts that into light-years. In velocity mode you enter a distance in megaparsecs, light-years or kiloparsecs and it multiplies by H₀.
The supporting grid carries four numbers that are not restatements of the headline. The distance in light-years matters because popular accounts quote light-years while professional catalogues quote parsecs, and the factor between them is 3.2616 rather than anything memorable. The approximate redshift is the raw observable: astronomers measure a wavelength shift, not a velocity. The Hubble time, 1/H₀ in billions of years, is the age the universe would have if it had always expanded at today's rate. The share of the Hubble radius is the best warning flag for when the linear law stops applying.
The tool refuses to return a number for a zero or negative Hubble constant, and it flags any case where the implied velocity reaches or exceeds the speed of light. That second case is not an arithmetic error but a real feature of an expanding universe, and the note says so rather than printing a figure it cannot defend.
How to Use It
- Choose the direction. Distance mode takes a velocity; velocity mode takes a distance. The input panel changes to match, so you never enter a quantity the tool is about to compute.
- Enter the observed quantity. For distance mode this is the recession velocity, usually derived from a measured redshift. If you have z rather than a velocity, multiply z by 299,792.458 for a first estimate, which is accurate to about one per cent below z = 0.02.
- Pick a Hubble constant deliberately. The two preset buttons load the value from the cosmic microwave background analysis and the value from the local distance ladder. Run both. The gap between the two answers is the honest uncertainty in your result.
- Set the distance unit before reading the answer. Megaparsecs and millions of light-years differ by more than a factor of three, and mixing them is the most common way a Hubble law calculation goes wrong by a factor nobody notices.
- Check the share of the Hubble radius. Below a few per cent, the linear law is a sound approximation. Above about a fifth, the answer becomes a rough scale rather than a distance, and you need a cosmological model rather than a straight line.
The Formula: How Hubble's Law Is Calculated
Hubble's law states that v = H₀ × d, where v is the recession velocity, d is the distance and H₀ is the Hubble constant at the present epoch. OpenStax Astronomy 2e, section 26.5 on the expanding universe, gives the law in exactly this form and explains why the same linear relation is seen from every point in an expanding space, which is the reason it is not evidence that we sit at a centre.
Work the default values through. With v = 10,000 km/s and H₀ = 70 km/s per Mpc, the distance is 10,000 ÷ 70 = 142.86 Mpc. Converting with 1 Mpc = 3.2616 million light-years gives 465.9 million light-years. The approximate redshift is 10,000 ÷ 299,792.458 = 0.03336. Swap in H₀ = 73.0 and the same velocity gives 137.0 Mpc, about four per cent nearer; swap in 67.4 and it gives 148.4 Mpc, about four per cent further. That eight per cent spread from a single unresolved measurement disagreement is larger than most other error terms in the calculation.
The Hubble time is 1/H₀, and the unit conversion is the whole of the work. H₀ = 70 km/s per Mpc is 70 ÷ (3.0857 × 1019) per second, which is 2.2685 × 10−18 s−1. Its reciprocal is 4.408 × 1017 seconds, and dividing by the 3.1557 × 1016 seconds in a billion years gives 13.97 billion years. The Hubble radius is c/H₀ = 299,792.458 ÷ 70 = 4,283 Mpc, or about 13.97 billion light-years. Those two numbers are numerically the same because a light-year is defined as the distance light covers in a year.
Why the Hubble Constant Has Two Different Values
Most physical constants have one accepted number and a small uncertainty. H₀ does not, and the disagreement is the reason this tool makes it an editable field rather than a fixed coefficient. Two independent methods measure it, both with quoted precision of a couple of per cent, and their results do not overlap.
The early-universe route fits the pattern of temperature ripples in the cosmic microwave background to a cosmological model and extrapolates forward to today, producing a value near 67 to 68. The late-universe route builds a distance ladder from variable stars and supernovae in relatively nearby galaxies and measures the expansion directly, producing a value near 73. Georgia State University's HyperPhysics page on the Hubble law sets out the history of these measurements and the different techniques behind them.
The practical consequence for anyone using this calculator is simple: your distance is uncertain by at least the spread between those methods, whatever precision the display shows. Quoting 142.86 Mpc implies you know the distance to a tenth of a megaparsec. You do not. Round to 140 or 150 and state which constant you used. That is the same discipline the significant figures calculator enforces for any measurement, and cosmology needs it more than most fields.
Velocity, Redshift, and What Is Actually Measured
No telescope measures a galaxy's speed. It measures a spectrum, and the emission lines in that spectrum sit at longer wavelengths than the same lines measured in a laboratory. The fractional shift is the redshift, z = Δλ/λ₀, and everything else is inference. Converting z into a velocity with v = cz is a low-redshift approximation that this tool inverts to give you the redshift figure in the grid. Our redshift calculator handles the wavelength side of the same problem, including the relativistic formula that takes over when z is no longer small.
The approximation degrades in a predictable way. At z = 0.01 it is accurate to about half a per cent. At z = 0.1 it overstates the velocity by roughly five per cent. Beyond z = 0.3 it is misleading, and beyond z = 1 it produces velocities above the speed of light, which is why the tool flags that case. The resolution is that cosmological redshift is not a Doppler shift at all. Nothing is moving through space at those speeds. Space between the galaxies is expanding, and the wavelength stretches with it, so the relativistic velocity limit does not apply. Our Doppler effect calculator covers the genuine motion-through-a-medium case, which is a different physical mechanism with a different formula.
Where the Linear Law Stops Working
Hubble's law is the first-order term of a more complicated relationship. It holds cleanly for galaxies close enough that the expansion rate has not changed appreciably during the light travel time, which in practice means out to a few hundred megaparsecs. Beyond that, two effects break it. The expansion rate was different in the past, so the light you receive was emitted when H was larger, and the notion of distance itself splits into several distinct definitions that agree locally and diverge badly at high redshift.
There is also a floor. Within about 20 Mpc, galaxies have local motions of a few hundred kilometres per second caused by the gravity of neighbouring clusters, and these peculiar velocities swamp the expansion signal. The Andromeda galaxy is approaching us, not receding, because at that separation gravity wins outright. Feeding a negative velocity into Hubble's law returns a negative distance, which is why the tool declines to accept one. The usable window for a linear Hubble law is roughly 20 to a few hundred megaparsecs, and both edges matter.
Reading the Hubble Time Correctly
The Hubble time is the most misread number this page produces. It is not the age of the universe. It is the age the universe would have had if the expansion rate had been constant forever, which it has not been. Gravity slowed the expansion for the first several billion years and dark energy has been accelerating it since. Those two effects work in opposite directions, and with the currently favoured parameters they very nearly cancel, so the true age lands close to the Hubble time.
That near-coincidence is a real result rather than a definition, and treating the Hubble time as the age by construction throws away the physics that makes it interesting. The grid figure of about 13.97 billion years at H₀ = 70 is best read as an order-of-magnitude anchor: it tells you the expansion timescale, and any distance approaching c multiplied by that time is a distance where the simple law has stopped applying.
Parsecs, Light-Years and the Unit Trap
A parsec is the distance at which one astronomical unit subtends one arcsecond, which is 3.2616 light-years or about 3.0857 × 1013 km. A megaparsec is a million of those. The number that makes Hubble's law tidy in professional use is exactly the number that makes it confusing everywhere else, because H₀ is quoted in kilometres per second per megaparsec: a mixed unit combining a velocity in kilometres with a distance in parsecs.
The safest routine is to keep everything in megaparsecs while calculating and convert only at the end, which is what this tool does internally. If your source quotes a distance in millions of light-years, divide by 3.2616 before comparing it to a megaparsec figure, or simply switch the unit dropdown and let the tool do it. For unrelated conversions of ordinary distances and speeds, the length converter and the speed converter handle the everyday units, though neither knows anything about parsecs.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing megaparsecs with millions of light-years — they differ by a factor of 3.2616, and the resulting error is large enough to change the conclusion but small enough to look plausible.
- Quoting more digits than the Hubble constant supports — the published values disagree by around eight per cent, so any answer beyond two significant figures is decoration.
- Applying the law to nearby galaxies — within roughly 20 Mpc, local gravitational motions dominate and the expansion term is buried in the noise.
- Treating v = cz as valid at high redshift — the approximation degrades above z of about 0.1 and produces superluminal nonsense above z of 1.
- Calling the Hubble time the age of the universe — it is the age under a constant expansion rate, and the expansion rate has never been constant.
Related Free Tools From Arb Digital
Start from a measured wavelength shift with the redshift calculator, which converts spectral lines into z before you bring the velocity here. For motion within a gravitational system rather than the expansion of space, the escape velocity calculator and the orbital velocity calculator cover the local dynamics. The percent error calculator quantifies how far apart two published Hubble constants actually put your answer. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
There is no single agreed value. Analyses of the cosmic microwave background give about 67 to 68 kilometres per second per megaparsec, while measurements built from nearby variable stars and supernovae give about 73. Run both preset buttons and treat the gap between the two answers as your real uncertainty.
For small redshifts, multiply z by the speed of light, 299,792.458 kilometres per second. That approximation is good to about half a per cent at z of 0.01 and degrades steadily above z of 0.1, where a relativistic or cosmological treatment is needed instead.
Because the simple v = cz conversion breaks down there, not because such recession is impossible. Cosmological redshift comes from space expanding rather than galaxies moving through space, so recession faster than light is allowed, but it requires a full cosmological model rather than this linear law.
No. The Hubble time is the age the universe would have if it had always expanded at today's rate. Gravitational deceleration early on and dark energy acceleration later happen to nearly cancel, so the true age lands close to it, but that is a result rather than a definition.
Because galaxies have their own motions of a few hundred kilometres per second driven by the gravity of nearby clusters. Within about 20 megaparsecs those peculiar velocities are larger than the expansion signal, which is why Andromeda is approaching us rather than receding.
About 3.2616 million light-years, since one parsec is 3.2616 light-years. This is the single most common source of factor-of-three errors in Hubble law calculations, because professional sources quote parsecs and popular sources quote light-years.
The redshift calculator works on the observation itself, turning a measured wavelength shift into z and into a velocity. This page takes the velocity as given and converts it into a cosmological distance using the Hubble constant. They are the two consecutive steps of the same measurement chain.
This tool is provided for educational and study use. It applies the linear Hubble law with a user-supplied constant and makes no correction for cosmological model, peculiar velocity or the several competing definitions of distance used at high redshift.
