opinion: Justice White
  announced the judgment of the Court and delivered the opinion of the Court with respect to Parts I, II, and V, and an opinion with respect to Parts III and IV, in which The Chief Justice, Justice O’Connor, and Justice Kennedy joined.
  At issue in this case is the validity of the death sentence imposed by an Arizona trial court after a jury found petitioner Jeffrey Walton guilty of committing first-degree murder.
  The Arizona statutes provide that a person commits first-degree murder if “[ijntending or knowing that his conduct will cause death, such person causes the death of another with premeditation” or if in the course of committing certain specified offenses and without any mental state other than what is required for the commission of such offenses, he causes the death of any person. Ariz. Rev. Stat. Ann. § 13-1105 (1989). After a person has been found guilty of first-degree murder, the sentence for such crime is determined in accordance with the provisions of § 13-703(B). It is there directed that a “separate sentencing hearing . . . shall be conducted before the court alone” to determine whether the sentence shall be death or life imprisonment. In the course of such hearing, the judge is instructed to determine the existence or nonexistence of any of the aggravating or mitigating circumstances defined in subsections (F) and (G) of §13-703. Subsection (F) defines 10 aggravating circumstances that may be considered. One of them is whether the offense was committed with the expectation of receiving anything of pecuniary value. § 13-703(F)(5). Another is whether the defendant committed the offense in an especially heinous, cruel, or depraved manner. § 13-703(F)(6). Subsection (G) defines mitigating circumstances as any factors “which are relevant in determining whether to impose a sentence less than death, including any aspect of the defendant’s character, propensities or record and any of the circumstances of the offense, including but not limited to” five specified factors. The burden of establishing the existence of any of the aggravating circumstances is on the prosecution, while the burden of establishing mitigating circumstances is on the defendant. § 13-703(C). The court is directed to return a special verdict setting forth its findings as to aggravating and mitigating circumstances and then “shall impose a sentence of death if the court finds one or more of the aggravating circumstances enumerated in subsection (F) of this section and that there are no mitigating circumstances sufficiently substantial to call for leniency.” § 13-703(E).
  I
  Petitioner Walton and his two codefendants, Robert Hoover and Sharold Ramsey, went to a bar in Tucson, Arizona, on the night of March 2, 1986, intending to find and rob someone at random, steal his car, tie him up, and leave him in the desert while they fled the State in the car. In the bar’s parking lot, the trio encountered Thomas Powell, a young, off-duty Marine. The three robbed Powell at gunpoint and forced him into his car which they then drove out into the desert. While driving out of Tucson, the three asked Powell questions about where he lived and whether he had any more money. When the car stopped, Ramsey told a frightened Powell that he would not be hurt. Walton and Hoover then forced Powell out of the car and had him lie face down on the ground near the car while they debated what to do with him. Eventually, Walton instructed Hoover and Ramsey to sit in the car and turn the radio up loud. Walton then took a .22 caliber derringer and marched Powell off into the desert. After walking a short distance, Walton forced Powell to lie down on the ground, placed his foot on Powell’s neck, and shot Powell once in the head. Walton later told Hoover and Ramsey that he had shot Powell and that he had “never seen a man pee in his pants before.” Powell’s body was found approximately a week later, after Walton was arrested and led police to the murder site. A medical examiner determined that Powell had been blinded and rendered unconscious by the shot but was not immediately killed. Instead, Powell regained consciousness, apparently floundered about in the desert, and ultimately died from dehydration, starvation, and pneumonia approximately a day before his body was found.
  A jury convicted Walton of first-degree murder after being given instructions on both premeditated and felony murder. See Ariz. Rev. Stat. Ann. § 13-1105 (1989). The trial judge then conducted the separate sentencing hearing required by §13-703(B). The State argued that two aggravating circumstances were present: (1) The murder was committed “in an especially heinous, cruel or depraved manner,” § 13-703(F)(6), and (2) the murder was committed for pecuniary gain. § 13-703(F)(5). In mitigation Walton presented testimony from a psychiatrist who opined that Walton had a long history of substance abuse which impaired his judgment, see § 13-703(G)(1), and that Walton may have been abused sexually as a child. Walton’s counsel also argued Walton’s age, 20 at the time of sentencing, as a mitigating circumstance. See § 13-703(G)(5). At the conclusion of the hearing, the trial court found “beyond any doubt” that Walton was the one who shot Powell. The court also found that the two aggravating circumstances pressed by the State were present. The court stated that it had considered Walton’s age and his capacity to appreciate the wrongfulness of his conduct, as well as all of the mitigating factors urged by defendant’s counsel. The court then concluded that that there were “no mitigating circumstances sufficiently substantial to call for leniency.” App. 61. See §13-703. The court sentenced Walton to death.
  The Arizona Supreme Court affirmed Walton’s conviction and sentence. 159 Ariz. 571, 769 P. 2d 1017 (1989). Relying on its prior decisions, the court rejected various specific challenges to the constitutionality of the Arizona death penalty statute, some of which are pressed here, and then proceeded to conduct its independent review of Walton’s sentence in order to “ensure that aggravating factors were proven beyond a reasonable doubt and all appropriate mitigation was considered.” Id,., at 586, 769 P. 2d, at 1032. The court began by examining the “especially heinous, cruel or depraved” aggravating circumstance found by the. trial judge. The court pointed out that it previously had determined that a murder is committed in an especially cruel manner when “the perpetrator inflicts mental anguish or physical abuse before the victim’s death,” id., at 586, 769 P. 2d, at 1032, (citations omitted), and that “[mjental anguish includes a victim’s uncertainty as to his ultimate fate.” Ibid. In this case, the court concluded that there was ample evidence that Powell suffered mental anguish prior to his death. The Arizona Supreme Court also found the evidence sufficient to conclude that the crime was committed in an especially depraved manner, pointing out that it had defined a depraved murder as one where “the perpetrator relishes the murder, evidencing debasement or perversion.” Id., at 587, 769 P. 2d, at 1033. Additionally, the court found that the pecuniary gain circumstance was present. Id., at 588, 769 P. 2d, at 1034. After examining Walton’s mitigating evidence regarding his substance abuse and his youth, the court concluded that there were “no mitigating circumstances sufficient to call for lenience.” Id., at 589, 769 P. 2d, at 1035. Finally, the court conducted its proportionality review and determined that Walton’s death sentence was “proportional to sentences imposed in similar cases.” Id., at 590, 769 P. 2d, at 1036.
  Because the United States Court of Appeals for the Ninth Circuit has held the Arizona death penalty statute to be unconstitutional for the reasons submitted by Walton in this case, see Adamson v. Ricketts, 865 F. 2d 1011 (1988) (en banc), we granted certiorari, 493 U. S. 808 (1989), to resolve the conflict and to settle issues that are of importance generally in the administration of the death penalty. We now affirm the judgment of the Arizona Supreme Court.
  h-4 1 — 1
  Walton’s first argument is that every finding of fact underlying the sentencing decision must be made by a jury, not by a judge, and that the Arizona scheme would be constitutional only if a jury decides what aggravating and mitigating circumstances are present in a given case and the trial judge then imposes sentence based on those findings. Contrary to Walton’s assertion, however: “Any argument that the Constitution requires that a jury impose the sentence of death or make the findings prerequisite to imposition of such a sentence has been soundly rejected by prior decisions of this Court.” Clemons v. Mississippi, 494 U. S. 738, 745 (1990).
  We repeatedly have rejected constitutional challenges to Florida’s death sentencing scheme, which provides for sentencing by the judge, not the jury. Hildwin v. Florida, 490 U. S. 638 (1989) (per curiam); Spaziano v. Florida, 468 U. S. 447 (1984); Proffitt v. Florida, 428 U. S. 242 (1976). In Hildwin, for example, we stated that “[t]his case presents us once again with the question whether the Sixth Amendment requires a jury to specify the aggravating factors that permit the imposition of capital punishment in Florida,” 490 U. S., at 638, and we ultimately concluded that “the Sixth Amendment does not require that the specific findings authorizing the imposition of the sentence of death be made by the jury.” Id., at 640-641.
  The distinctions Walton attempts to draw between the Florida and Arizona statutory schemes are not persuasive. It is true that in Florida the jury recommends a sentence, but it does not make specific factual findings with regard to the existence of mitigating or aggravating circumstances and its recommendation is not binding on the trial judge. A Florida trial court no more has the assistance of a jury’s findings of fact with respect to sentencing issues than does a trial judge in Arizona.
  Walton also suggests that in Florida aggravating factors are only sentencing “considerations” while in Arizona they are “elements of the offense.” But as we observed in Poland v. Arizona, 476 U. S. 147 (1986), an Arizona capital punishment case: “Aggravating circumstances are not separate penalties or offenses, but are ‘standards to guide the making of [the] choice’ between the alternative verdicts of death and life imprisonment. Thus, under Arizona’s capital sentencing scheme, the judge’s finding of any particular aggravating circumstance does not of itself ‘convict’ a defendant (i. e., require the death penalty), and the failure to find any particular aggravating circumstance does not ‘acquit’ a defendant (i. e., preclude the death penalty).” Id., at 156 (citation omitted).
  Our holding in Cabana v. Bidlock, 474 U. S. 376 (1986), provides further support for our conclusion. Cabana held that an appellate court could constitutionally make the Enmund v. Florida, 458 U. S. 782 (1982), finding — that the defendant killed, attempted to kill, or intended to kill — in the first instance. We noted that “Enmund, ‘does not affect the state’s definition of any substantive offense, even a capital offense,”’ 474 U. S., at 385 (citations omitted), and that “while the Eighth Amendment prohibits the execution of such defendants, it does not supply a new element of the crime of capital murder that must be found by the jury.” Id., at 385, n. 3. Enmund only places “a substantive limitation on sentencing, and like other such limits it need not be enforced by the jury.” 474 U. S., at 386. If the Constitution does not require that the Enmund finding be proved as an element of the offense of capital murder, and does not require a jury to make that finding, we cannot conclude that a State is required to denominate aggravating circumstances “elements” of the offense or permit only a jury to determine the existence of such circumstances.
  We thus conclude that the Arizona capital sentencing scheme does not violate the Sixth Amendment.
  Ill
  Also unpersuasive is Walton’s contention that the Arizona statute violates the Eighth and Fourteenth Amendments because it imposes on defendants the burden of establishing, by a preponderance of the evidence, the existence of mitigating circumstances sufficiently substantial to call for leniency. See Ariz. Rev. Stat. Ann. §§ 13-703(0 and 13-703(E) (1989). It is true that the Court has refused to countenance state-imposed restrictions on what mitigating circumstances may be considered in deciding whether to impose the death penalty. See, e. g., Lockett v. Ohio, 438 U. S. 586, 604 (1978) (plurality opinion). But Walton is not complaining that the Arizona statute or practice excludes from consideration any particular type of mitigating evidence; and it does not follow from Lockett and its progeny that a State is precluded from specifying how mitigating circumstances are to be proved. Indeed, in Lockett itself, we expressly reserved opinion on whether “it violates the Constitution to require defendants to bear the risk of nonpersuasion as to the existence of mitigating circumstances in capital cases.” Id., at 609, and n. 16 (plurality opinion).
  In Martin v. Ohio, 480 U. S. 228 (1987), we upheld the Ohio practice of imposing on a capital defendant the burden of proving by a preponderance of the evidence that, she was acting in self-defense when she allegedly committed the murder. In Leland v. Oregon, 343 U. S. 790 (1952), the Court upheld, in a capital case, a requirement that the defense of insanity be proved beyond a reasonable doubt by the defendant, see also Rivera v. Delaware, 429 U. S. 877 (1976), and in Patterson v. New York, 432 U. S. 197 (1977), we rejected the argument that a State violated due process by imposing a preponderance of the evidence standard on a defendant to prove the affirmative defense of extreme emotional disturbance.
  The basic principle of these cases controls the result in this case. So long as a State’s method of allocating the burdens of proof does not lessen the State’s burden to prove every element of the offense charged, or in this case to prove the existence of aggravating circumstances, a defendant’s constitutional rights are not violated by placing on him the burden of proving mitigating circumstances sufficiently substantial to call for leniency. Mullaney v. Wilbur, 421 U. S. 684 (1975), is not to the contrary. Mullaney struck down on due process grounds a state statute that required a convicted murder defendant to negate an element of the offense of murder in order to be entitled to a sentence for voluntary manslaughter. No such burden is placed on defendants by Arizona’s capital sentencing scheme. We therefore decline to adopt as a constitutional imperative a rule that would require the court to consider the mitigating circumstances claimed by a defendant unless the State negated them by a preponderance of the evidence.
  Neither does Mills v. Maryland, 486 U. S. 367 (1988), lend support to Walton’s position. There this Court reversed a death sentence because it concluded that the jury instructions given at the sentencing phase likely led the jury to believe that any particular mitigating circumstance could not be considered unless the jurors unanimously agreed that such circumstance was present. The Court’s focus was on whether reasonable jurors would have read the instructions to require unanimity and, if so, the possible consequences of such an understanding. Here, of course, the judge alone is the sentencer, and Mills is therefore beside the point.
  Furthermore, Mills did not suggest that it would be forbidden to require each individual juror, before weighing a claimed mitigating circumstance in the balance, to be convinced in his or her own mind that the mitigating circumstance has been proved by a preponderance of the evidence. To the contrary, the jury in that case was instructed that it had to find that any mitigating circumstances had been proved by a preponderance of the evidence. Id., at 387. Neither the petitioner in Mills nor the Court in its opinion hinted that there was any constitutional objection to that aspect of the instructions.
  We therefore reject Walton’s argument that Arizona’s allocation of the burdens of proof in a capital sentencing proceeding violates the Constitution.
  IV
  Walton insists that because § 13 — 703(E) provides that the court “shall” impose the death penalty if one or more aggravating circumstances are found and mitigating circumstances are held insufficient to call for leniency, the statute creates an unconstitutional presumption that death is the proper sentence. Our recent decisions in Blystone v. Pennsylvania, 494 U. S. 299 (1990), and Boyde v. California, 494 U. S. 370 (1990), foreclose this submission. Blystone rejected a challenge to a jury instruction based on a Pennsylvania statute requiring the imposition of the death penalty if aggravating circumstances were found to exist but no mitigating circumstances were present. We pointed out that “[t]he requirement of individualized sentencing in capital cases is satisfied by allowing the jury to consider all relevant mitigating evidence,” 494 U. S., at 307 (footnote omitted), and concluded that because the Pennsylvania statute did not preclude the sentencer from considering any type of mitigating evidence, id., at 308, it was consonant with that principle. In addition, the Court concluded that the statute was not “impermissibly ‘mandatory’ as that term was understood” in Woodson v. North Carolina, 428 U. S. 280 (1976), and Roberts v. Louisiana, 428 U. S. 325 (1976), because it did not automatically impose death upon conviction for certain types of murder. 494 U. S., at 305. The same is true of the Arizona statute.
  Similarly, Boyde v. California, supra, upheld a pattern jury instruction which stated that “[i]f you conclude that the aggravating circumstances outweigh the mitigating circumstances, you shall impose a sentence of death.” See 494 U. S., at 374 (emphasis omitted). The Court specifically noted that “there is no . . . constitutional requirement of unfettered sentencing discretion in the jury, and States are free to structure and shape consideration of mitigating evidence ‘in an effort to achieve a more rational and equitable administration of the death penalty.’” Id., at 377 (quoting Franklin v. Lynaugh, 487 U. S. 164, 181 (1988) (plurality opinion)). Walton’s arguments in this case are no more persuasive than those made in Blystone and Boyde.
  V
  Walton’s final contention is that the especially heinous, cruel, or depraved aggravating circumstance as interpreted by the Arizona courts fails to channel the sentencer’s discretion as required by the Eighth and Fourteenth Amendments. Walton contends that the Arizona factor fails to pass constitutional muster for the same reasons this Court found Oklahoma’s “especially heinous, atrocious, or cruel” aggravating circumstance to be invalid in Maynard v. Cartwright, 486 U. S. 356 (1988), and Georgia’s “outrageously or wantonly vile, horrible or inhuman” circumstance to be invalid in Godfrey v. Georgia, 446 U. S. 420 (1980).
  Maynard v. Cartwright and Godfrey v. Georgia, however, are distinguishable in two constitutionally significant respects. First, in both Maynard and Godfrey the defendant was sentenced by a jury and the jury either was instructed only in the bare terms of the relevant statute or in terms nearly as vague. See 486 U. S., at 358-359, 363-364; 446 U. S., at 426 (plurality opinion). Neither jury was given a constitutional limiting definition of the challenged aggravating factor. Second, in neither case did the state appellate court, in reviewing the propriety of the death sentence, purport to affirm the death sentence by applying a limiting definition of the aggravating circumstance to the facts presented. 486 U. S., at 364; 446 U. S., at 429 (plurality opinion). These points were crucial to the conclusion we reached in Maynard. See 486 U. S., at 363-364. They are equally crucial to our decision in this case.
  When a jury is the final sentencer, it is essential that the jurors be properly instructed regarding all facets of the sentencing process. It is not enough to instruct the jury in the bare terms of an aggravating circumstance that is unconstitutionally vague on its face. That is the import of our holdings in Maynard and Godfrey. But the logic of those cases has no place in the context of sentencing by a trial judge. Trial judges are presumed to know the law and to apply it in making their decisions. If the Arizona Supreme Court has narrowed the definition of the “especially heinous, cruel or depraved” aggravating circumstance, we presume that Arizona trial judges are applying the narrower definition. It is irrelevant that the statute itself may not narrow the construction of the factor. Moreover, even if a trial judge fails to apply the narrowing construction or applies an improper construction, the Constitution does not necessarily require that a state appellate court vacate a death sentence based on that factor. Rather, as we held in Clemons v. Mississippi, 494 U. S. 738 (1990), a state appellate court may itself determine whether the evidence supports the existence of the aggravating circumstance as properly defined or the court may eliminate consideration of the factor altogether and determine whether any remaining aggravating circumstances are sufficient to warrant the death penalty.
  When a federal court is asked to review a state court’s application of an individual statutory aggravating or mitigating circumstance in a particular case, it must first determine whether the statutory language defining the circumstance is itself too vague to provide any guidance to the sentencer. If so, then the federal court must attempt to determine whether the state courts have further defined the vague terms and, if they have done so, whether those definitions are constitutionally sufficient, /. e., whether they provide some guidance to the sentencer. In this case there is no serious argument that Arizona’s “especially heinous, cruel or depraved” aggravating factor is not facially vague. But the Arizona Supreme Court has sought to give substance to the operative terms, and we find that its construction meets constitutional requirements.
  The Arizona Supreme Court stated that “a crime is committed in an especially cruel manner when the perpetrator inflicts mental anguish or physical abuse before the victim’s death,” and that “[m]ental anguish includes a victim’s uncertainty as to his ultimate fate.” 159 Ariz., at 586, 769 P. 2d, at 1032. The court rejected the State’s argument that the six days Powell suffered after being shot constituted cruelty within the meaning of the statute. The court pointed out that it had limited the cruelty circumstance in prior cases to situations where the suffering of the victim was intended by or foreseeable to the killer. Id., at 587, 769 P. 2d, at 1033.
  In Maynard v. Cartwright, we expressed approval of a definition that would limit Oklahoma’s “especially heinous, atrocious, or cruel” aggravating circumstance to murders involving “some kind of torture or physical abuse,” 486 U. S., at 364-365, but we also noted that such a construction was not the only one “that would be constitutionally acceptable.” Id., at 365. The construction given by the Arizona Supreme Court to the cruelty aspect of the Arizona aggravating circumstance is virtually identical to the- construction we approved in Maynard.
  The Arizona Supreme Court’s construction also is similar to the construction of Florida’s “especially heinous, atrocious, or cruel” aggravating circumstance that we approved in Proffttt v. Florida, 428 U. S., at 255-256 (joint opinion of Stewart, Powell, and Stevens, JJ.). Recognizing that the proper degree of definition of an aggravating factor of this nature is not susceptible of mathematical precision, we conclude that the definition given to the “especially cruel” provision by the Arizona Supreme Court is constitutionally sufficient because it gives meaningful guidance to the sentencer. Nor can we fault the state court’s statement that a crime is committed in an especially “depraved” manner when the perpetrator “relishes the murder, evidencing debasement or perversion,” or “shows an indifference to the suffering of the victim and evidences a sense of pleasure” in the killing. See 159 Ariz., at 587, 769 P. 2d, at 1033.
  Walton nevertheless contends that the heinous, cruel, or depraved factor has been applied in an arbitrary manner and, as applied, does not distinguish his case from cases in which the death sentence has not been imposed. In effect Walton challenges the proportionality review of the Arizona Supreme Court as erroneous and asks us to overturn it. This we decline to do, for we have just concluded that the challenged factor has been construed by the Arizona courts in a manner that furnishes sufficient guidance to the sentencer. This being so, proportionality review is not constitutionally required, and we “lawfully may presume that [Walton’s] death sentence was not ‘wantonly and freakishly’ imposed — and thus that the sentence is not disproportionate within any-recognized meaning of the Eighth Amendment.” McCleskey v. Kemp, 481 U. S. 279, 306, 308 (1987); Pulley v. Harris, 465 U. S. 37, 43 (1984). Furthermore, the Arizona Supreme Court plainly undertook its proportionality review in good faith and found that Walton’s sentence was proportional to the sentences imposed in cases similar to his. The Constitution does not require us to look behind that conclusion.
  The judgment of the Arizona Supreme Court is affirmed.
  It is so ordered.
  Those factors are as follows:
  “1. The defendant’s capacity to appreciate the wrongfulness of his conduct or to conform his conduct to the requirements of law was significantly impaired, but not so impaired as to constitute a defense to prosecution.
  “2. The defendant was under unusual and substantial duress, although not such as to constitute a defense to prosecution.
  “3. The defendant was legally accountable for the conduct of another under the provisions of § 13-303, but his participation was relatively minor, although not so minor as to constitute a defense to prosecution.
  “4. The defendant could not reasonably have foreseen that his conduct in the course of the commission of the offense for which the defendant was convicted would cause, or would create a grave risk of causing, death to another person.
  “5. The defendant’s age.” Ariz. Rev. Stat. Ann. § 13 — 703(G) (1989).
  In the course of its opinion, the court also rejected Walton’s challenge, not repeated in this Court, that Hoover and not Walton actually shot Powell. The court pointed out that because the jury was instructed on both felony and premeditated murder but entered only a general verdict, the trial court was required under Arizona law to independently make the determination mandated by Enmund v. Florida, 458 U. S. 782 (1982), and Tison v. Arizona, 481 U. S. 137 (1987), that Walton killed, intended to kill, attempted to kill, or as a participant in a felony was recklessly indifferent to the killing of Powell. 159 Ariz., at 585, 769 P. 2d, at 1031. The court then held that the trial court’s Enmund determination was based on substantial evidence. 159 Ariz., at 586, 769 P. 2d, at 1032.
  The court argued that Powell must have realized as he was being driven out of Tucson into the desert that he might be harmed, and the court pointed out that Powell was obviously frightened enough that Ramsey tried to reassure him that he would not be harmed. Then, the court noted, Walton and Hoover forced Powell to lie on the ground while they argued over his fate, and eventually Walton marched Powell off into the desert with a gun but no rope, surely making Powell realize that he was not going to be tied up and left unharmed. The court further observed that Powell was so frightened that he urinated on himself. Id., at 586-587, 769 P. 2d, at 1032-1033.
  The court concluded that Walton’s reference to having “ ‘never seen a man pee in his pants before’ ” constituted evidence of “callous fascination with the murder” and demonstrated “an indifference to the suffering of the victim and ... a sense of pleasure” taken “in the killing." Id., at 587, 769 P. 2d, at 1033.
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